<reponame>LaisRast/sage<filename>src/sage/schemes/elliptic_curves/height.py
r"""
Canonical heights for elliptic curves over number fields

Also, rigorous lower bounds for the canonical height of non-torsion
points, implementing the algorithms in [CS2006]_ (over `\QQ`) and [Tho2010]_,
which also refer to [CPS2006]_.

AUTHORS:

- <NAME> (2010): initial version

- <NAME> (2014): added many docstrings and doctests
"""
##############################################################################
#       Copyright (C) 2010 <NAME> <<EMAIL>>
#                     2014 <NAME> <<EMAIL>>
#
#  Distributed under the terms of the GNU General Public License (GPL)
#
#    This code is distributed in the hope that it will be useful,
#    but WITHOUT ANY WARRANTY; without even the implied warranty of
#    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
#    General Public License for more details.
#
#  The full text of the GPL is available at:
#
#                  https://www.gnu.org/licenses/
##############################################################################
import numpy
import math
import bisect

from sage.rings.integer_ring import ZZ
from sage.rings.rational_field import QQ
from sage.rings.infinity import infinity
from sage.rings.cif import CIF
from sage.rings.cc import CC
from sage.rings.complex_double import CDF
from sage.rings.real_double import RDF
from sage.rings.real_mpfi import RIF
from sage.rings.real_mpfr import RR

from sage.misc.cachefunc import cached_method
from sage.misc.mrange import cartesian_product_iterator
from sage.arith.all import lcm, factorial
from sage.ext.fast_callable import fast_callable
from sage.functions.log import log, exp
from sage.symbolic.ring import SR


class UnionOfIntervals:
    r"""
    A class representing a finite union of closed intervals in
    `\RR` which can be scaled, shifted, intersected, etc.

    The intervals are represented as an ordered list of their
    endpoints, which may include `-\infty` and `+\infty`.

    EXAMPLES::

        sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals
        sage: R = UnionOfIntervals([1,2,3,infinity]); R
        ([1, 2] U [3, +Infinity])
        sage: R + 5
        ([6, 7] U [8, +Infinity])
        sage: ~R
        ([-Infinity, 1] U [2, 3])
        sage: ~R | (10*R + 100)
        ([-Infinity, 1] U [2, 3] U [110, 120] U [130, +Infinity])

    .. TODO::

        Unify :class:`UnionOfIntervals` with the class ``RealSet``
        introduced by :trac:`13125`; see :trac:`16063`.
    """
    def __init__(self, endpoints):
        r"""
        An union of intervals is initialized by giving an increasing list
        of endpoints, the first of which may be `-\infty` and the last of
        which may be `+\infty`.

        EXAMPLES::

            sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals
            sage: UnionOfIntervals([0,1])
            ([0, 1])
            sage: UnionOfIntervals([-infinity, pi, 17, infinity])
            ([-Infinity, pi] U [17, +Infinity])
            sage: UnionOfIntervals([])
            ()

            sage: UnionOfIntervals([1])
            Traceback (most recent call last):
            ...
            ValueError: an even number of endpoints must be given (got 1)
            sage: UnionOfIntervals([3,2,1,0])
            Traceback (most recent call last):
            ...
            ValueError: endpoints must be given in order
        """
        if len(endpoints) % 2:
            raise ValueError("an even number of endpoints must be given (got %s)" % len(endpoints))
        if endpoints != sorted(endpoints):
            raise ValueError("endpoints must be given in order")
        self._endpoints = endpoints

    def finite_endpoints(self):
        r"""
        Returns the finite endpoints of this union of intervals.

        EXAMPLES::

            sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals
            sage: UnionOfIntervals([0,1]).finite_endpoints()
            [0, 1]
            sage: UnionOfIntervals([-infinity, 0, 1, infinity]).finite_endpoints()
            [0, 1]
        """
        return [e for e in self._endpoints if -infinity < e < infinity]

    def intervals(self):
        r"""
        Returns the intervals in self, as a list of 2-tuples.

        EXAMPLES::

            sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals
            sage: UnionOfIntervals(list(range(10))).intervals()
            [(0, 1), (2, 3), (4, 5), (6, 7), (8, 9)]
            sage: UnionOfIntervals([-infinity, pi, 17, infinity]).intervals()
            [(-Infinity, pi), (17, +Infinity)]
        """
        return list(zip(self._endpoints[::2], self._endpoints[1::2]))

    def is_empty(self):
        r"""
        Returns whether self is empty.

        EXAMPLES::

            sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals
            sage: UnionOfIntervals([3,4]).is_empty()
            False
            sage: all = UnionOfIntervals([-infinity, infinity])
            sage: all.is_empty()
            False
            sage: (~all).is_empty()
            True
            sage: A = UnionOfIntervals([0,1]) & UnionOfIntervals([2,3])
            sage: A.is_empty()
            True
        """
        return not self._endpoints

    def __add__(left, right):
        r"""
        If both left an right are unions of intervals, take their union,
        otherwise treat the non-union of intervals as a scalar and shift.

        EXAMPLES::

            sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals
            sage: A = UnionOfIntervals([0, 1/2, 2, infinity]); A
            ([0, 1/2] U [2, +Infinity])
            sage: A + 1
            ([1, 3/2] U [3, +Infinity])
            sage: pi + A
            ([pi, pi + 1/2] U [pi + 2, +Infinity])
            sage: A + UnionOfIntervals([-infinity, -1])
            ([-Infinity, -1] U [0, 1/2] U [2, +Infinity])
        """
        if not isinstance(left, UnionOfIntervals):
            left, right = right, left
        elif not isinstance(right, UnionOfIntervals):
            return UnionOfIntervals([right + e for e in left._endpoints])
        else:
            return left.union([left, right])

    def __mul__(left, right):
        r"""
        Scale a union of intervals on the left or right.

        EXAMPLES::

            sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals
            sage: A = UnionOfIntervals([0, 1/2, 2, infinity]); A
            ([0, 1/2] U [2, +Infinity])
            sage: 2 * A
            ([0, 1] U [4, +Infinity])
            sage: A * 100
            ([0, 50] U [200, +Infinity])
            sage: 1.5 * A
            ([0.000000000000000, 0.750000000000000] U [3.00000000000000, +Infinity])
        """
        if not isinstance(right, UnionOfIntervals):
            return UnionOfIntervals([e*right for e in left._endpoints])
        elif not isinstance(left, UnionOfIntervals):
            return UnionOfIntervals([left*e for e in right._endpoints])
        else:
            return NotImplemented

    def __rmul__(self, other):
        r"""
        Scale by an operand on the left.

        TESTS::

            sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals
            sage: A = UnionOfIntervals([0, 1/2, 2, infinity]); A
            ([0, 1/2] U [2, +Infinity])
            sage: pi * A
            ([0, 1/2*pi] U [2*pi, +Infinity])
        """
        return self * other

    def __radd__(self, other):
        r"""
        Add a scalar operand on the left.

        TESTS::

            sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals
            sage: A = UnionOfIntervals([0, 1/2, 2, infinity]); A
            ([0, 1/2] U [2, +Infinity])
            sage: 100 + A
            ([100, 201/2] U [102, +Infinity])
        """
        return self + other

    def __invert__(self):
        r"""
        Return the closure of the complement of self.

        .. NOTE::

            We take the closure because open intervals are not supported.

        EXAMPLES::

            sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals
            sage: A = UnionOfIntervals([0, 1/2, 2, infinity]); A
            ([0, 1/2] U [2, +Infinity])
            sage: ~A
            ([-Infinity, 0] U [1/2, 2])
            sage: A | ~A
            ([-Infinity, +Infinity])
            sage: A & ~A
            ([0, 0] U [1/2, 1/2] U [2, 2])
        """
        endpoints = list(self._endpoints)
        if endpoints[0] == -infinity:
            del endpoints[0]
        else:
            endpoints.insert(0, -infinity)
        if endpoints[-1] == infinity:
            endpoints.pop()
        else:
            endpoints.append(infinity)
        return UnionOfIntervals(endpoints)

    @staticmethod
    def join(L, condition):
        r"""
        Utility function to form the union or intersection of a list of UnionOfIntervals.

        INPUT:

        - ``L`` (list) -- a list of UnionOfIntervals instances

        - ``condition`` (function) -- either ``any`` or ``all``, or
          some other boolean function of a list of boolean values.

        OUTPUT:

        A new UnionOfIntervals instance representing the subset of
        '\RR' equal to those reals in any/all/condition of the
        UnionOfIntervals in the list.

        .. NOTE::

            This is a static method for the class.

        EXAMPLES::

            sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals
            sage: A = UnionOfIntervals([1,3,5,7]); A
            ([1, 3] U [5, 7])
            sage: B = A+1; B
            ([2, 4] U [6, 8])
            sage: A.join([A,B],any) # union
            ([1, 4] U [5, 8])
            sage: A.join([A,B],all) # intersection
            ([2, 3] U [6, 7])
            sage: A.join([A,B],sum) # symmetric difference
            ([1, 2] U [3, 4] U [5, 6] U [7, 8])
        """
        all = []
        for ix, region in enumerate(L):
            for i, e in enumerate(region._endpoints):
                all.append((e, -(not (i % 2)), ix))
        all.sort()
        join = []
        in_join = False
        in_L = [False] * len(L)
        for e, start, ix in all:
            in_L[ix] = start
            if condition(in_L) != in_join:
                join.append(e)
                in_join = not in_join
        return UnionOfIntervals(join)

    @classmethod
    def union(cls, L):
        r"""
        Return the union of a list of UnionOfIntervals.

        INPUT:

        - ``L`` (list) -- a list of UnionOfIntervals instances

        OUTPUT:

        A new UnionOfIntervals instance representing the union of the
        UnionOfIntervals in the list.

        .. NOTE::

            This is a class method.

        EXAMPLES::

            sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals
            sage: A = UnionOfIntervals([1,3,5,7]); A
            ([1, 3] U [5, 7])
            sage: B = A+1; B
            ([2, 4] U [6, 8])
            sage: A.union([A,B])
            ([1, 4] U [5, 8])
        """
        return cls.join(L, any)

    @classmethod
    def intersection(cls, L):
        r"""
        Return the intersection of a list of UnionOfIntervals.

        INPUT:

        - ``L`` (list) -- a list of UnionOfIntervals instances

        OUTPUT:

        A new UnionOfIntervals instance representing the intersection
        of the UnionOfIntervals in the list.

        .. NOTE::

            This is a class method.

        EXAMPLES::

            sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals
            sage: A = UnionOfIntervals([1,3,5,7]); A
            ([1, 3] U [5, 7])
            sage: B = A+1; B
            ([2, 4] U [6, 8])
            sage: A.intersection([A,B])
            ([2, 3] U [6, 7])
        """
        for R in L:
            if R.is_empty():
                return R
        return cls.join(L, all)

    def __or__(left, right):
        r"""
        Return the union of a two UnionOfIntervals instances.

        INPUT:

        - ``left``, ``right`` (UnionOfIntervals) -- two UnionOfIntervals instances

        OUTPUT:

        A new UnionOfIntervals instance representing the union of ``left`` and ``right``.

        EXAMPLES::

            sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals
            sage: A = UnionOfIntervals([1,3,5,7]); A
            ([1, 3] U [5, 7])
            sage: B = A+1; B
            ([2, 4] U [6, 8])
            sage: A | B
            ([1, 4] U [5, 8])
        """
        return left.union([left, right])

    def __and__(left, right):
        r"""
        Return the intersection of a two UnionOfIntervals instances.

        INPUT:

        - ``left``, ``right`` (UnionOfIntervals) -- two UnionOfIntervals instances

        OUTPUT:

        A new UnionOfIntervals instance representing the intersection of ``left`` and ``right``.

        EXAMPLES::

            sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals
            sage: A = UnionOfIntervals([1,3,5,7]); A
            ([1, 3] U [5, 7])
            sage: B = A+1; B
            ([2, 4] U [6, 8])
            sage: A & B
            ([2, 3] U [6, 7])
        """
        return left.intersection([left, right])

    def __contains__(self, x):
        r"""
        Return True if ``x`` is in the UnionOfIntervals.

        INPUT:

        - ``x`` (real) -- a real number

        OUTPUT:

        Boolean: True if and only if ``x`` is in the union of intervals.

        EXAMPLES::

            sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals
            sage: A = UnionOfIntervals([1,3,5,7]); A
            ([1, 3] U [5, 7])
            sage: 1 in A
            True
            sage: 4 in A
            False
            sage: -infinity in A
            False
        """
        return x in self._endpoints or bisect.bisect_left(self._endpoints, x) % 2 == 1

    def __str__(self):
        r"""
        Return the string representation of this UnionOfIntervals.

        EXAMPLES::

            sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals
            sage: A = UnionOfIntervals([1,3,5,7])
            sage: str(A)
            '([1, 3] U [5, 7])'
        """
        return repr(self)

    def __repr__(self):
        r"""
        Return the string representation of this UnionOfIntervals.

        EXAMPLES::

            sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals
            sage: A = UnionOfIntervals([1,3,5,7]); A
            ([1, 3] U [5, 7])
        """
        return "(%s)" % " U ".join(str(list(I)) for I in self.intervals())


def nonneg_region(f):
    r"""
    Returns the UnionOfIntervals representing the region where ``f`` is non-negative.

    INPUT:

    - ``f`` (polynomial) -- a univariate polynomial over `\RR`.

    OUTPUT:

    A UnionOfIntervals representing the set `\{x \in\RR mid f(x) \ge 0\}`.

    EXAMPLES::

        sage: from sage.schemes.elliptic_curves.height import nonneg_region
        sage: x = polygen(RR)
        sage: nonneg_region(x^2-1)
        ([-Infinity, -1.00000000000000] U [1.00000000000000, +Infinity])
        sage: nonneg_region(1-x^2)
        ([-1.00000000000000, 1.00000000000000])
        sage: nonneg_region(1-x^3)
        ([-Infinity, 1.00000000000000])
        sage: nonneg_region(x^3-1)
        ([1.00000000000000, +Infinity])
        sage: nonneg_region((x-1)*(x-2))
        ([-Infinity, 1.00000000000000] U [2.00000000000000, +Infinity])
        sage: nonneg_region(-(x-1)*(x-2))
        ([1.00000000000000, 2.00000000000000])
        sage: nonneg_region((x-1)*(x-2)*(x-3))
        ([1.00000000000000, 2.00000000000000] U [3.00000000000000, +Infinity])
        sage: nonneg_region(-(x-1)*(x-2)*(x-3))
        ([-Infinity, 1.00000000000000] U [2.00000000000000, 3.00000000000000])
        sage: nonneg_region(x^4+1)
        ([-Infinity, +Infinity])
        sage: nonneg_region(-x^4-1)
        ()
    """
    roots = sorted(f.roots())
    sign_changes = [r for r,e in roots if e%2 == 1]
    if (f.leading_coefficient() * (-1)**f.degree()) > 0:
        sign_changes = [-infinity] + sign_changes
    if f.leading_coefficient() > 0:
        sign_changes += [infinity]
    return UnionOfIntervals(sign_changes)

def inf_max_abs(f, g, D):
    r"""
    Returns `\inf_D(\max(|f|, |g|))`.

    INPUT:

    - ``f``, ``g`` (polynomials) -- real univariate polynomials

    - ``D`` (UnionOfIntervals) -- a subset of `\RR`

    OUTPUT:

    A real number approximating the value of `\inf_D(\max(|f|, |g|))`.

    ALGORITHM:

    The extreme values must occur at an endpoint of a subinterval of
    `D` or at a point where one of `f`, `f'`, `g`, `g'`, `f\pm g` is
    zero.

    EXAMPLES::

        sage: from sage.schemes.elliptic_curves.height import inf_max_abs, UnionOfIntervals
        sage: x = polygen(RR)
        sage: f = (x-10)^4+1
        sage: g = 2*x^3+100
        sage: inf_max_abs(f,g,UnionOfIntervals([1,2,3,4,5,6]))
        425.638201706391
        sage: r0 = (f-g).roots()[0][0]
        sage: r0
        5.46053402234697
        sage: max(abs(f(r0)),abs(g(r0)))
        425.638201706391
    """
    xs =  f.roots() + f.derivative().roots()
    xs += g.roots() + g.derivative().roots()
    xs += (f-g).roots() + (f+g).roots()
    xs = [r for r,e in xs if r in D]  # ignore multiplicities and points outside D
    xs += D.finite_endpoints()        # include endpoints of intervals
    if xs:
        return min([max(abs(f(r)), abs(g(r))) for r in xs])
    return infinity

def min_on_disk(f, tol, max_iter=10000):
    r"""
    Returns the minimum of a real-valued complex function on a square.

    INPUT:

    - ``f`` -- a function from CIF to RIF

    - ``tol`` (real) -- a positive real number

    - ``max_iter`` (integer, default 10000) -- a positive integer
      bounding the number of iterations to be used

    OUTPUT:

    A 2-tuple `(s,t)`, where `t=f(s)` and `s` is a CIF element
    contained in the disk `|z|\le1`, at which `f` takes its minimum
    value.

    EXAMPLES::

        sage: from sage.schemes.elliptic_curves.height import min_on_disk
        sage: f = lambda x: (x^2+100).abs()
        sage: s, t = min_on_disk(f, 0.0001)
        sage: s, f(s), t
        (0.01? + 1.00?*I, 99.01?, 99.0000000000000)
    """
    # L holds a list of 4-tuples (mfs, ds, s, in_disk) where s is a
    # subregion of the initial square, ds its relative diameter,
    # mfs=-f(s) (actually minus the lower bound on f(s)) and in_disk
    # is a flag indicating whether or not s is a subset of the unit
    # disk.

    # We store the negative of the lower bound on f(s) so that we can
    # use the bisect module to sort these 4-tuples.

    # Initially L contains one element, the whole unit box, which is
    # not contained in the unit square.

    s = CIF(RIF(-1,1), RIF(-1,1))
    fs = f(s)
    L = [(-fs.lower(), fs.relative_diameter(), s, False)]

    # min_max holds the minimum over L of fs.upper().

    min_max = fs.upper()

    # We iterate at most max_iter times.  At each step we look at the
    # best so far and return it if is good enough, meaning that its
    # relative diameter is less than the given tolerance; otherwise we
    # bisect this best region (into 4 pieces) and replace the entry in
    # L with at most 4 smaller entries.

    for k in range(max_iter):
        value, err, region, in_disk = L.pop()
        if err < tol:       # reached desired tolerance, so return
            return region, -value
        for s in region.bisection(): # 4 sub-regions
            if in_disk:
                s_in_disk = True     # if the original region si in the disk so are all its children
            else:
                r = abs(s)          # otherwise we test each one
                if r > 1:
                    continue        # skip this subregion if it is entirely outside the disk
                s_in_disk = r < 1   # meaning it is entirely inside the disk

            fs = f(s)

            if fs.upper() < min_max: # we definitely beat the record
                min_max = fs.upper()
                unneeded = bisect.bisect(L, (-min_max,))
                if unneeded > 100:   # discard the worse entries (if there are many)
                    L = L[unneeded:]

            if fs.lower() < min_max: # we may beat the record, cannot yet tell: insert this region
                                     # into the list at the appropriate place to maintain sorting
                bisect.insort(L, (-fs.lower(), fs.relative_diameter(), s, s_in_disk))

    # If we get here, then even after max_iter iterations the tolerance has not been reached.
    raise ValueError("too many iterations")


two_pi_i_CDF = CDF(0, 2 * RDF.pi())
two_pi_i_CIF = CIF(0, 2 * RIF.pi())

# Ideas: We know tau, so we know the direction of the diagonal.
# We can solve for x in p1, will this allow us to find the maxima exactly?


def rat_term_CIF(z, try_strict=True):
    r"""
    Compute the value of `u/(1-u)^2` in ``CIF``, where `u=\exp(2\pi i z)`.

    INPUT:

    - ``z`` (complex) -- a CIF element

    - ``try_strict`` (bool) -- flag

    EXAMPLES::

        sage: from sage.schemes.elliptic_curves.height import rat_term_CIF
        sage: z = CIF(0.5,0.2)
        sage: rat_term_CIF(z)
        -0.172467461182437? + 0.?e-16*I
        sage: rat_term_CIF(z, False)
        -0.172467461182437? + 0.?e-16*I
    """
    two_pi_i_z = two_pi_i_CIF * z
    r = (two_pi_i_z.real()).exp() # = |u|
    x, y = two_pi_i_z.imag().cos(), two_pi_i_z.imag().sin()

    real_part = imag_part = None

    # If there are no local minima the intervals are strictly
    # determined by their values at the endpoints.

    if try_strict:

        # evaluate the function at the four corners:

        corner_reals = []
        corner_imags = []
        for a, b in cartesian_product_iterator([z.real().endpoints(), z.imag().endpoints()]):
            zz = CDF(a,b)
            u = (two_pi_i_CDF*zz).exp()
            f = u/(1-u)**2
            corner_reals.append(f.real())
            corner_imags.append(f.imag())

        p1 = (((((r+2*x)*r - 6)*r + 2*x) * r) + 1)
            # =  r^4 + 2*r^3*x - 6*r^2 + 2*r*x + 1
        p2 = (r*(x*(r+2*x)-4)+x)
            # = r^2*x + 2*r*x^2 - 4*r + x

        df_dr = (r**2-1) * p2
        dg_dr = p1 * y
        dg_dx = r * df_dr / y

        if not dg_dr.contains_zero() or not dg_dx.contains_zero():
            real_part = RIF(min(corner_reals), max(corner_reals))

        if not dg_dr.contains_zero() or not dg_dx.contains_zero():
            imag_part = RIF(min(corner_imags), max(corner_imags))

    if real_part is None or imag_part is None:
        denom = (1-r*(2*x-r))**2
    if real_part is None:
        real_part = r*(x*(1+r**2)-2*r)/denom
    if imag_part is None:
        imag_part = -(r**2-1)*y*r/denom

    return CIF(real_part, imag_part)


def eps(err, is_real):
    r"""
    Return a Real or Complex interval centered on 0 with radius err.

    INPUT:

    - ``err`` (real) -- a positive real number, the radius of the interval

    - ``is_real`` (boolean) -- if True, returns a real interval in
      RIF, else a complex interval in CIF

    OUTPUT:

    An element of RIF or CIF (as specified), centered on 0, with given radius.

    EXAMPLES::

        sage: from sage.schemes.elliptic_curves.height import eps
        sage: eps(0.01, True)
        0.0?
        sage: eps(0.01, False)
        0.0? + 0.0?*I
    """
    e = RIF(-err, err)
    if is_real:
        return e
    else:
        return CIF(e, e)


class EllipticCurveCanonicalHeight:
    r"""
    Class for computing canonical heights of points on elliptic curves
    defined over number fields, including rigorous lower bounds for
    the canonical height of non-torsion points.

    EXAMPLES::

        sage: from sage.schemes.elliptic_curves.height import EllipticCurveCanonicalHeight
        sage: E = EllipticCurve([0,0,0,0,1])
        sage: EllipticCurveCanonicalHeight(E)
        EllipticCurveCanonicalHeight object associated to Elliptic Curve defined by y^2 = x^3 + 1 over Rational Field

    Normally this object would be created like this::

        sage: E.height_function()
        EllipticCurveCanonicalHeight object associated to Elliptic Curve defined by y^2 = x^3 + 1 over Rational Field
    """

    def __init__(self, E):
        r"""
        Initialize the class with an elliptic curve.

        INPUT:

        - `E` -- an elliptic curve defined over a number field

        EXAMPLES::

            sage: from sage.schemes.elliptic_curves.height import EllipticCurveCanonicalHeight
            sage: E = EllipticCurve([0,0,0,0,1])
            sage: EllipticCurveCanonicalHeight(E)
            EllipticCurveCanonicalHeight object associated to Elliptic Curve defined by y^2 = x^3 + 1 over Rational Field

        An example over a number field::

            sage: K.<i>=QuadraticField(-1)
            sage: E = EllipticCurve([0,i,0,i,i])
            sage: EllipticCurveCanonicalHeight(E)
            EllipticCurveCanonicalHeight object associated to Elliptic Curve defined by y^2 = x^3 + i*x^2 + i*x + i over Number Field in i with defining polynomial x^2 + 1 with i = 1*I

        TESTS:

        The base field must be a number field (or `\QQ`)::

            sage: from sage.schemes.elliptic_curves.height import EllipticCurveCanonicalHeight
            sage: E = EllipticCurve(GF(7),[0,0,0,0,1])
            sage: EllipticCurveCanonicalHeight(E)
            Traceback (most recent call last):
            ...
            ValueError: EllipticCurveCanonicalHeight class can only be created from an elliptic curve defined over a number field
        """
        from sage.schemes.elliptic_curves.ell_generic import is_EllipticCurve
        if is_EllipticCurve(E):
            self.E = E
            from sage.rings.number_field.number_field_base import is_NumberField
            K = E.base_ring()
            if is_NumberField(K):
                self.K = K
            else:
                raise ValueError("EllipticCurveCanonicalHeight class can only be created from an elliptic curve defined over a number field")
        else:
            raise ValueError("EllipticCurveCanonicalHeight class can only be created from an elliptic curve")

    def __repr__(self):
        r"""
        Return the string representation.

        EXAMPLES::

            sage: E = EllipticCurve([0,0,0,0,1])
            sage: E.height_function()
            EllipticCurveCanonicalHeight object associated to Elliptic Curve defined by y^2 = x^3 + 1 over Rational Field
        """
        return "EllipticCurveCanonicalHeight object associated to %s" % self.E

    def curve(self):
        r"""
        Return the elliptic curve.

        EXAMPLES::

            sage: E = EllipticCurve([0,0,0,0,1])
            sage: H = E.height_function()
            sage: H.curve()
            Elliptic Curve defined by y^2 = x^3 + 1 over Rational Field
        """
        return self.E

    def base_field(self):
        r"""
        Return the base field.

        EXAMPLES::

            sage: E = EllipticCurve([0,0,0,0,1])
            sage: H = E.height_function()
            sage: H.base_field()
            Rational Field
        """
        return self.K

    def __call__(self, P):
        r"""
        Return the canonical height of the point ``P``.

        INPUT:

        - ``P`` -- a point on the elliptic curve.

        OUTPUT:

        The canonical height of ``P``.

        EXAMPLES::

            sage: E = EllipticCurve([0,0,1,-1,0])
            sage: P = E(0,0)
            sage: P.height()
            0.0511114082399688
            sage: H = E.height_function()
            sage: H(P)
            0.0511114082399688
            sage: H([0,0])
            0.0511114082399688
            sage: H((0,0))
            0.0511114082399688

        Over a number field other than `\QQ`::

            sage: K.<i> = QuadraticField(-1)
            sage: E = EllipticCurve(K, [0,0,0,1,-27])
            sage: H = E.height_function()
            sage: H.base_field()
            Number Field in i with defining polynomial x^2 + 1 with i = 1*I
            sage: H((1,5*i))
            1.22257115164148
        """
        return self.E(P).height()

    @cached_method
    def alpha(self, v, tol=0.01):
        r"""
        Return the constant `\alpha_v` associated to the embedding ``v``.

        INPUT:

        - ``v`` -- an embedding of the base field into `\RR` or `\CC`

        OUTPUT:

        The constant `\alpha_v`.  In the notation of [CPS2006]_ and
        [Tho2010]_ (section 3.2), `\alpha_v^3=\epsilon_v`.  The result is
        cached since it only depends on the curve.

        EXAMPLES:

        Example 1 from [CPS2006]_::

            sage: K.<i>=QuadraticField(-1)
            sage: E = EllipticCurve([0,0,0,1+5*i,3+i])
            sage: H = E.height_function()
            sage: alpha = H.alpha(K.places()[0])
            sage: alpha
            1.12272013439355

        Compare with `\log(\epsilon_v)=0.344562...` in [CPS2006]_::

            sage: 3*alpha.log()
            0.347263296676126
        """
        from sage.rings.polynomial.polynomial_ring import polygen
        b2, b4, b6, b8 = [v(b) for b in self.E.b_invariants()]
        x = polygen(v.codomain())
        f = 4*x**3 + b2*x**2 + 2*b4*x + b6
        g = x**4 - b4*x**2 - 2*b6*x - b8
        F = f.reverse() << (4-f.degree())
        G = g.reverse() << (4-g.degree())

        if v(self.K.gen()) in RR:
            I = UnionOfIntervals([-1,1])
            min_fg = inf_max_abs(f, g, nonneg_region(f) & I)
            min_FG = inf_max_abs(F, G, nonneg_region(F) & I)
            return min(min_fg, min_FG) ** (-1/QQ(3))

        else:
            # def pair_max(f, g):
            #     f = f.change_ring(CIF)
            #     g = g.change_ring(CIF)
            #     max = type(RIF(0)).max
            #     def max_f_g(z):
            #         return max(abs(f(z)), abs(g(z)))
            #     return max_f_g
            def pair_max(f, g):
                f = f.change_ring(CDF)
                g = g.change_ring(CDF)
                dfn = [fast_callable(f.derivative(n)/factorial(n), CDF) for n in range(f.degree()+1)]
                dgn = [fast_callable(g.derivative(n)/factorial(n), CDF) for n in range(g.degree()+1)]
                def max_f_g(s):
                    (a,b),(c,d) = s.real().endpoints(), s.imag().endpoints()
                    dx = a - b
                    dy = c - d
                    eta = RDF(dx*dx + dy*dy).sqrt()
                    z = CDF(s.center())
                    err_f = sum(eta ** n * abs(df(z)) for n, df in enumerate(dfn) if n)
                    err_g = sum(eta ** n * abs(dg(z)) for n, dg in enumerate(dgn) if n)
                    return RIF(max(abs(f(z)), abs(g(z)))) + eps(max(err_f, err_g), True)
                return max_f_g
            _, min_fg = min_on_disk(pair_max(f, g), tol)
            _, min_FG = min_on_disk(pair_max(F, G), tol)
            return min(min_fg, min_FG) ** QQ((-1, 3))

    @cached_method
    def e_p(self, p):
        r"""
        Return the exponent of the group over the residue field at ``p``.

        INPUT:

        - ``p`` -- a prime ideal of `K` (or a prime number if `K=\QQ`).

        OUTPUT:

        A positive integer `e_p`, the exponent of the group of
        nonsingular points on the reduction of the elliptic curve
        modulo `p`.  The result is cached.

        EXAMPLES::

            sage: K.<i>=QuadraticField(-1)
            sage: E = EllipticCurve([0,0,0,1+5*i,3+i])
            sage: H = E.height_function()
            sage: H.e_p(K.prime_above(2))
            2
            sage: H.e_p(K.prime_above(3))
            10
            sage: H.e_p(K.prime_above(5))
            9
            sage: E.conductor().norm().factor()
            2^10 * 20921
            sage: p1, p2 = K.primes_above(20921)
            sage: E.local_data(p1)
            Local data at Fractional ideal (-40*i + 139):
            Reduction type: bad split multiplicative
            ...
            sage: H.e_p(p1)
            20920
            sage: E.local_data(p2)
            Local data at Fractional ideal (40*i + 139):
            Reduction type: good
            ...
            sage: H.e_p(p2)
            20815
        """
        kp = self.K.residue_field(p)
        if self.E.has_bad_reduction(p):
            if self.E.has_additive_reduction(p):
                ep = kp.characteristic()
            elif self.E.has_split_multiplicative_reduction(p):
                ep = len(kp) - 1
            else:
                ep = len(kp) + 1
        else:
            ep = self.E.reduction(p).abelian_group().exponent()
        return ZZ(ep)

    @cached_method
    def DE(self, n):
        r"""
        Return the value `D_E(n)`.

        INPUT:

        - ``n`` (int) -- a positive integer

        OUTPUT:

        The value `D_E(n)` as defined in [Tho2010]_, section 4.

        EXAMPLES::

            sage: K.<i>=QuadraticField(-1)
            sage: E = EllipticCurve([0,0,0,1+5*i,3+i])
            sage: H = E.height_function()
            sage: [H.DE(n) for n in srange(1,6)]
            [0, 2*log(5) + 2*log(2), 0, 2*log(13) + 2*log(5) + 4*log(2), 0]
        """
        s = 0
        B = (n+1) ** max(2, self.K.degree())
        for p in self.K.primes_of_bounded_norm_iter(B):
            ep = self.e_p(p)
            if ep.divides(n):
                kp = self.K.residue_field(p)
                s += 2*(1+(n/ep).valuation(kp.characteristic())) * log(len(kp))
        return s

    @cached_method
    def ME(self):
        r"""
        Return the norm of the ideal `M_E`.

        OUTPUT:

        The norm of the ideal `M_E` as defined in [Tho2010]_, section 3.1.
        This is `1` if `E` is a global minimal model, and in general
        measures the non-minimality of `E`.

        EXAMPLES::

            sage: K.<i>=QuadraticField(-1)
            sage: E = EllipticCurve([0,0,0,1+5*i,3+i])
            sage: H = E.height_function()
            sage: H.ME()
            1
            sage: E = EllipticCurve([0,0,0,0,1])
            sage: E.height_function().ME()
            1
            sage: E = EllipticCurve([0,0,0,0,64])
            sage: E.height_function().ME()
            4096
            sage: E.discriminant()/E.minimal_model().discriminant()
            4096
        """
        from sage.misc.misc_c import prod
        if self.K is QQ:
            return prod([p ** (e - self.E.local_data(p).discriminant_valuation()) for p, e in self.E.discriminant().factor()], QQ.one())

        ME = prod([p.norm() ** (e - self.E.local_data(p).discriminant_valuation()) for p, e in self.K.ideal(self.E.discriminant()).factor()], QQ.one())
        return ME.norm()

    def B(self, n, mu):
        r"""
        Return the value `B_n(\mu)`.

        INPUT:

        - ``n`` (int) -- a positive integer

        - ``mu`` (real) -- a positive real number

        OUTPUT:

        The real value `B_n(\mu)` as defined in [Tho2010]_, section 5.

        EXAMPLES:

        Example 10.2 from [Tho2010]_::

            sage: K.<i>=QuadraticField(-1)
            sage: E = EllipticCurve([0,1-i,i,-i,0])
            sage: H = E.height_function()

        In [Tho2010]_ the value is given as 0.772::

            sage: RealField(12)( H.B(5, 0.01) )
            0.777
        """
        K = self.K
        B = exp(K.degree() * n**2 * mu - RDF(self.DE(n))) / self.ME() ** 6
        for v in K.places():
            if v(K.gen()) in RR:
                B *= self.alpha(v)
            else:
                B *= self.alpha(v) ** 2
        return B

    ######################################
    # Empty real intersection detection. #
    ######################################

    def psi(self, xi, v):
        r"""
        Return the normalised elliptic log of a point with this x-coordinate.

        INPUT:

        - ``xi`` (real) -- the real x-coordinate of a point on the
          curve in the connected component with respect to a real
          embedding.

        - ``v`` (embedding) -- a real embedding of the number field.

        OUTPUT:

        A real number in the interval [0.5,1] giving the elliptic
        logarithm of a point on `E` with `x`-coordinate ``xi``, on the
        connected component with respect to the embedding `v`, scaled
        by the real period.

        EXAMPLES:

        An example over `\QQ`::

            sage: E = EllipticCurve('389a')
            sage: v = QQ.places()[0]
            sage: L = E.period_lattice(v)
            sage: P = E.lift_x(10/9)
            sage: L(P)
            1.53151606047462
            sage: L(P) / L.real_period()
            0.615014189772115
            sage: H = E.height_function()
            sage: H.psi(10/9,v)
            0.615014189772115

        An example over a number field::

            sage: K.<a> = NumberField(x^3-2)
            sage: E = EllipticCurve([0,0,0,0,a])
            sage: P = E.lift_x(1/3*a^2 + a + 5/3)
            sage: v = K.real_places()[0]
            sage: L = E.period_lattice(v)
            sage: L(P)
            3.51086196882538
            sage: L(P) / L.real_period()
            0.867385122699931
            sage: xP = v(P.xy()[0])
            sage: H = E.height_function()
            sage: H.psi(xP,v)
            0.867385122699931
            sage: H.psi(1.23,v)
            0.785854718241495
        """
        if xi > 1e9:
            return 1
        L = self.E.period_lattice(v)
        w1, w2 = L.basis()
        from sage.schemes.elliptic_curves.constructor import EllipticCurve
        ER = EllipticCurve([v(ai) for ai in self.E.a_invariants()])
        xP, yP = ER.lift_x(xi).xy()
        t = L.e_log_RC(xP,yP) / w1
        if t < 0.5:
            t = 1 - t
        return t

    def S(self, xi1, xi2, v):
        r"""
        Return the union of intervals `S^{(v)}(\xi_1,\xi_2)`.

        INPUT:

        - ``xi1, xi2`` (real) -- real numbers with `\xi_1\le\xi_2`.

        - ``v`` (embedding) -- a real embedding of the field.

        OUTPUT:

        The union of intervals `S^{(v)}(\xi_1,\xi_2)` defined in [Tho2010]_
        section 6.1.

        EXAMPLES:

        An example over `\QQ`::

            sage: E = EllipticCurve('389a')
            sage: v = QQ.places()[0]
            sage: H = E.height_function()
            sage: H.S(2,3,v)
            ([0.224512677391895, 0.274544821597130] U [0.725455178402870, 0.775487322608105])

        An example over a number field::

            sage: K.<a> = NumberField(x^3-2)
            sage: E = EllipticCurve([0,0,0,0,a])
            sage: v = K.real_places()[0]
            sage: H = E.height_function()
            sage: H.S(9,10,v)
            ([0.0781194447253472, 0.0823423732016403] U [0.917657626798360, 0.921880555274653])
        """
        L = self.E.period_lattice(v)
        w1, w2 = L.basis(prec = v.codomain().prec())
        beta = L.elliptic_exponential(w1/2)[0]
        if xi2 < beta:
            return UnionOfIntervals([])
        elif xi1 < beta <= xi2:
            a = self.psi(xi2, v)
            return UnionOfIntervals([1-a, a])
        else:
            a, b = self.psi(xi1, v), self.psi(xi2, v)
            return UnionOfIntervals([1-b, 1-a, a, b])

    def Sn(self, xi1, xi2, n, v):
        r"""
        Return the union of intervals `S_n^{(v)}(\xi_1,\xi_2)`.

        INPUT:

        - ``xi1, xi2`` (real) -- real numbers with `\xi_1\le\xi_2`.

        - ``n`` (integer) -- a positive integer.

        - ``v`` (embedding) -- a real embedding of the field.

        OUTPUT:

        The union of intervals `S_n^{(v)}(\xi_1,\xi_2)` defined in [Tho2010]_
        (Lemma 6.1).

        EXAMPLES:

        An example over `\QQ`::

            sage: E = EllipticCurve('389a')
            sage: v = QQ.places()[0]
            sage: H = E.height_function()
            sage: H.S(2,3,v) , H.Sn(2,3,1,v)
            (([0.224512677391895, 0.274544821597130] U [0.725455178402870, 0.775487322608105]),
            ([0.224512677391895, 0.274544821597130] U [0.725455178402870, 0.775487322608105]))
            sage: H.Sn(2,3,6,v)
            ([0.0374187795653158, 0.0457574702661884] U [0.120909196400478, 0.129247887101351] U [0.204085446231982, 0.212424136932855] U [0.287575863067145, 0.295914553768017] U [0.370752112898649, 0.379090803599522] U [0.454242529733812, 0.462581220434684] U [0.537418779565316, 0.545757470266188] U [0.620909196400478, 0.629247887101351] U [0.704085446231982, 0.712424136932855] U [0.787575863067145, 0.795914553768017] U [0.870752112898649, 0.879090803599522] U [0.954242529733812, 0.962581220434684])

        An example over a number field::

            sage: K.<a> = NumberField(x^3-2)
            sage: E = EllipticCurve([0,0,0,0,a])
            sage: v = K.real_places()[0]
            sage: H = E.height_function()
            sage: H.S(2,3,v) , H.Sn(2,3,1,v)
            (([0.142172065860075, 0.172845716928584] U [0.827154283071416, 0.857827934139925]),
            ([0.142172065860075, 0.172845716928584] U [0.827154283071416, 0.857827934139925]))
            sage: H.Sn(2,3,6,v)
            ([0.0236953443100124, 0.0288076194880974] U [0.137859047178569, 0.142971322356654] U [0.190362010976679, 0.195474286154764] U [0.304525713845236, 0.309637989023321] U [0.357028677643346, 0.362140952821431] U [0.471192380511903, 0.476304655689988] U [0.523695344310012, 0.528807619488097] U [0.637859047178569, 0.642971322356654] U [0.690362010976679, 0.695474286154764] U [0.804525713845236, 0.809637989023321] U [0.857028677643346, 0.862140952821431] U [0.971192380511903, 0.976304655689988])
        """
        SS = 1/ZZ(n) * self.S(xi1, xi2, v)
        return UnionOfIntervals.union([t/ZZ(n) + SS for t in range(n)])

    def real_intersection_is_empty(self, Bk, v):
        r"""
        Returns True iff an intersection of `S_n^{(v)}` sets is empty.

        INPUT:

        - ``Bk`` (list) -- a list of reals.

        - ``v`` (embedding) -- a real embedding of the number field.

        OUTPUT:

        True or False, according as the intersection of the unions of
        intervals `S_n^{(v)}(-b,b)` for `b` in the list ``Bk`` is
        empty or not.  When ``Bk`` is the list of `b=B_n(\mu)` for
        `n=1,2,3,\dots` for some `\mu>0` this means that all
        non-torsion points on `E` with everywhere good reduction have
        canonical height strictly greater than `\mu`, by [Tho2010]_,
        Proposition 6.2.

        EXAMPLES:

        An example over `\QQ`::

            sage: E = EllipticCurve('389a')
            sage: v = QQ.places()[0]
            sage: H = E.height_function()

        The following two lines prove that the heights of non-torsion
        points on `E` with everywhere good reduction have canonical
        height strictly greater than 0.2, but fail to prove the same
        for 0.3::

            sage: H.real_intersection_is_empty([H.B(n,0.2) for n in srange(1,10)],v)
            True
            sage: H.real_intersection_is_empty([H.B(n,0.3) for n in srange(1,10)],v)
            False

        An example over a number field::

            sage: K.<a> = NumberField(x^3-2)
            sage: E = EllipticCurve([0,0,0,0,a])
            sage: v = K.real_places()[0]
            sage: H = E.height_function()

        The following two lines prove that the heights of non-torsion
        points on `E` with everywhere good reduction have canonical
        height strictly greater than 0.07, but fail to prove the same
        for 0.08::

            sage: H.real_intersection_is_empty([H.B(n,0.07) for n in srange(1,5)],v) # long time (3.3s)
            True
            sage: H.real_intersection_is_empty([H.B(n,0.08) for n in srange(1,5)],v)
            False
        """
        return UnionOfIntervals.intersection([self.Sn(-B, B, k+1, v) for k,B in enumerate(Bk)]).is_empty()

    ########################################
    # Empty complex intersection detection.#
    ########################################

    def tau(self, v):
        r"""
        Return the normalised upper half-plane parameter `\tau` for
        the period lattice with respect to the embedding `v`.

        INPUT:

        - ``v`` (embedding) -- a real or complex embedding of the number field.

        OUTPUT:

        (Complex) `\tau = \omega_1/\omega_2` in the fundamental region
        of the upper half-plane.

        EXAMPLES::

            sage: E = EllipticCurve('37a')
            sage: H = E.height_function()
            sage: H.tau(QQ.places()[0])
            1.22112736076463*I
        """
        return self.E.period_lattice(v).tau()

    def wp_c(self, v):
        r"""
        Return a bound for the Weierstrass `\wp`-function.

        INPUT:

        - ``v`` (embedding) -- a real or complex embedding of the number field.

        OUTPUT:

        (Real) `c>0` such that

        .. MATH::

            |\wp(z) - z^-2| \le \frac{c^2|z|^2}{1-c|z|^2}

        whenever `c|z|^2<1`. Given the recurrence relations for the
        Laurent series expansion of `\wp`, it is easy to see that
        there is such a constant `c`.  [Reference?]

        EXAMPLES::

            sage: E = EllipticCurve('37a')
            sage: H = E.height_function()
            sage: H.wp_c(QQ.places()[0])
            2.68744508779950

            sage: K.<i>=QuadraticField(-1)
            sage: E = EllipticCurve([0,0,0,1+5*i,3+i])
            sage: H = E.height_function()
            sage: H.wp_c(K.places()[0])
            2.66213425640096
        """
        # Note that we normalise w1, w2 differently from [Tho2010]_!
        w2, w1 = self.E.period_lattice(v).normalised_basis()
        return max(abs(v(self.E.c4()/240)) ** 0.5,
                   abs(v(self.E.c6()/6048)) ** (1.0/3)) * abs(w1)**2

    def fk_intervals(self, v=None, N=20, domain=CIF):
        r"""
        Return a function approximating the Weierstrass function, with error.

        INPUT:

        - ``v`` (embedding) -- an embedding of the number field.  If
          None (default) use the real embedding if the field is `\QQ`
          and raise an error for other fields.

        - ``N`` (int) -- The number of terms to use in the
          `q`-expansion of `\wp`.

        - ``domain`` (complex field) -- the model of `\CC` to use, for
          example ``CDF`` of ``CIF`` (default).

        OUTPUT:

        A pair of functions fk, err which can be evaluated at complex
        numbers `z` (in the correct ``domain``) to give an
        approximation to `\wp(z)` and an upper bound on the error,
        respectively.  The Weierstrass function returned is with
        respect to the normalised lattice `[1,\tau]` associated to the
        given embedding.

        EXAMPLES::

            sage: E = EllipticCurve('37a')
            sage: L = E.period_lattice()
            sage: w1, w2 = L.normalised_basis()
            sage: z = CDF(0.3, 0.4)

        Compare the value give by the standard elliptic exponential
        (scaled since ``fk`` is with respect to the normalised
        lattice)::

            sage: L.elliptic_exponential(z*w2, to_curve=False)[0] * w2 ** 2
            -1.82543539306049 - 2.49336319992847*I

        to the value given by this function, and see the error::

            sage: fk, err = E.height_function().fk_intervals(N=10)
            sage: fk(CIF(z))
            -1.82543539306049? - 2.49336319992847?*I
            sage: err(CIF(z))
            2.71750621458744e-31

        The same, but in the domain ``CDF`` instead of ``CIF``::

            sage: fk, err = E.height_function().fk_intervals(N=10, domain=CDF)
            sage: fk(z)
            -1.8254353930604... - 2.493363199928...*I
        """
        if v is None:
            if self.K is QQ:
                v = QQ.hom(RR)
            else:
                raise ValueError("must specify embedding")
        # pre-compute some constants
        tau = self.tau(v)
        const_term = 1/CC(12)
        qn = q = (2 * CC.gen() * CC.pi() * tau).exp()
        for n in range(1, N):
            const_term -= 2 * qn/(1-qn) ** 2
            qn *= q

        two_pi_i = 2 * domain.gen() * domain.pi()
        neg_four_pi2 = -4 * domain.pi() ** 2
        const_term = domain(const_term)
        tau = domain(tau)

        abs_q = abs(domain(q))
        abs_qN = abs(domain(qn))
        err_factor = abs(neg_four_pi2) / (1-abs_q)
        err_term = 2*abs_qN/(1-abs_qN) ** 2

        # choose u/(1-u)^2 evaluation method
        if domain is CIF:
            rat_term = rat_term_CIF
        else:
            def rat_term(z):
                u = (two_pi_i*z).exp()
                return u/(1-u)**2

        # the actual series
        def fk(z):
            return (const_term +
                    sum([rat_term(z+n*tau) for n in range(1-N,N)])
                    ) * neg_four_pi2

        # the error function
        def err(z):
            alpha = z.imag() / tau.imag()
            qNa = abs_q**(N+alpha)
            qNai = abs_q**(N-alpha)
            return (err_factor * (qNa/(1-qNa) ** 2 + qNai/(1-qNai) ** 2 + err_term)).upper()

        return fk, err

    @cached_method
    def wp_intervals(self, v=None, N=20, abs_only=False):
        r"""
        Return a function approximating the Weierstrass function.

        INPUT:

        - ``v`` (embedding) -- an embedding of the number field.  If
          None (default) use the real embedding if the field is `\QQ`
          and raise an error for other fields.

        - ``N`` (int, default 20) -- The number of terms to use in the
          `q`-expansion of `\wp`.

        - ``abs_only`` (boolean, default False) -- flag to determine
          whether (if True) the error adjustment should use the
          absolute value or (if False) the real and imaginary parts.

        OUTPUT:

        A function wp which can be evaluated at complex numbers `z` to
        give an approximation to `\wp(z)`.  The Weierstrass function
        returned is with respect to the normalised lattice `[1,\tau]`
        associated to the given embedding.  For `z` which are not near
        a lattice point the function ``fk`` is used, otherwise a
        better approximation is used.

        EXAMPLES::

            sage: E = EllipticCurve('37a')
            sage: wp = E.height_function().wp_intervals()
            sage: z = CDF(0.3, 0.4)
            sage: wp(CIF(z))
            -1.82543539306049? - 2.4933631999285?*I

            sage: L = E.period_lattice()
            sage: w1, w2 = L.normalised_basis()
            sage: L.elliptic_exponential(z*w2, to_curve=False)[0] * w2^2
            -1.82543539306049 - 2.49336319992847*I

            sage: z = CDF(0.3, 0.1)
            sage: wp(CIF(z))
            8.5918243572165? - 5.4751982004351?*I
            sage: L.elliptic_exponential(z*w2, to_curve=False)[0] * w2^2
            8.59182435721650 - 5.47519820043503*I
        """
        if v is None:
            if self.K is QQ:
                v = QQ.hom(RR)
            else:
                raise ValueError("must specify embedding")

        tau = self.tau(v)
        fk, fk_err = self.fk_intervals(v, N)
        c = self.wp_c(v)

        def wp(z):

            # center around origin
            offset = (z.imag().lower() / tau.imag()).round()
            if offset:
                z -= CIF(offset * tau)
            offset = z.real().lower().round()
            if offset:
                z -= offset

            # estimate using the series
            approx = fk(z)
            err = fk_err(z)
            if abs_only:
                approx = abs(approx)
            approx += eps(err, abs_only)

            # refine using an estimate that's better near the pole
            z_bound = abs(z).upper()
            cz2 = c * z_bound ** 2
            if cz2 < 1:
                err = (c * cz2) / (1 - cz2)
                if abs_only:
                    pole_approx = abs(z) ** -2
                else:
                    pole_approx = z ** -2
    #            print "pole approx", pole_approx + eps(err, abs_only)
    #            print approx in approx.intersection(pole_approx + eps(err, abs_only))
                approx = approx.intersection(pole_approx + eps(err, abs_only))

            return approx

        return wp

    @cached_method
    def wp_on_grid(self, v, N, half=False):
        r"""
        Return an array of the values of `\wp` on an `N\times N` grid.

        INPUT:

        - ``v`` (embedding) -- an embedding of the number field.

        - ``N`` (int) -- The number of terms to use in the
          `q`-expansion of `\wp`.

        - ``half`` (boolean, default False) -- if True, use an array of
          size `N\times N/2` instead of `N\times N`.

        OUTPUT:

        An array of size either `N\times N/2` or `N\times N` whose
        `(i,j)` entry is the value of the Weierstrass `\wp`-function
        at `(i+.5)/N + (j+.5)*\tau/N`, a grid of points in the
        fundamental region for the lattice `[1,\tau]`.

        EXAMPLES::

            sage: E = EllipticCurve('37a')
            sage: H = E.height_function()
            sage: v = QQ.places()[0]

        The array of values on the grid shows symmetry, since `\wp` is
        even::

            sage: H.wp_on_grid(v,4)
            array([[25.43920182,  5.28760943,  5.28760943, 25.43920182],
                   [ 6.05099485,  1.83757786,  1.83757786,  6.05099485],
                   [ 6.05099485,  1.83757786,  1.83757786,  6.05099485],
                   [25.43920182,  5.28760943,  5.28760943, 25.43920182]])

        The array of values on the half-grid::

            sage: H.wp_on_grid(v,4,True)
            array([[25.43920182,  5.28760943],
                   [ 6.05099485,  1.83757786],
                   [ 6.05099485,  1.83757786],
                   [25.43920182,  5.28760943]])
        """
        tau = self.tau(v)
        fk, err = self.fk_intervals(v, 15, CDF)
        var_z = SR.var('z')
        ff = fast_callable(fk(var_z), CDF, [var_z])
        N_or_half = N // (1+half)         # array is NxN or Nx(N/2)
        vals = numpy.empty((N,N_or_half)) # empty array tp hold values
        for i in range(N):
            for j in range(N_or_half):
                vals[i,j] = abs(ff((i+.5)/N + (j+.5)*tau/N))
        return vals

    def complex_intersection_is_empty(self, Bk, v, verbose=False, use_half=True):
        r"""
        Returns True iff an intersection of `T_n^{(v)}` sets is empty.

        INPUT:

        - ``Bk`` (list) -- a list of reals.

        - ``v`` (embedding) -- a complex embedding of the number field.

        - ``verbose`` (boolean, default False) -- verbosity flag.

        - ``use_half`` (boolean, default False) -- if True, use only half
          the fundamental region.

        OUTPUT:

        True or False, according as the intersection of the unions of
        intervals `T_n^{(v)}(-b,b)` for `b` in the list ``Bk`` (see
        [Tho2010]_, section 7) is empty or not.  When ``Bk`` is the list of
        `b=\sqrt{B_n(\mu)}` for `n=1,2,3,\dots` for some `\mu>0` this
        means that all non-torsion points on `E` with everywhere good
        reduction have canonical height strictly greater than `\mu`,
        by [Tho2010]_, Proposition 7.8.

        EXAMPLES::

            sage: K.<a> = NumberField(x^3-2)
            sage: E = EllipticCurve([0,0,0,0,a])
            sage: v = K.complex_embeddings()[0]
            sage: H = E.height_function()

        The following two lines prove that the heights of non-torsion
        points on `E` with everywhere good reduction have canonical
        height strictly greater than 0.02, but fail to prove the same
        for 0.03.  For the first proof, using only `n=1,2,3` is not
        sufficient::

            sage: H.complex_intersection_is_empty([H.B(n,0.02) for n in [1,2,3]],v) # long time (~6s)
            False
            sage: H.complex_intersection_is_empty([H.B(n,0.02) for n in [1,2,3,4]],v)
            True
            sage: H.complex_intersection_is_empty([H.B(n,0.03) for n in [1,2,3,4]],v) # long time (4s)
            False

        Using `n\le6` enables us to prove the lower bound 0.03.  Note
        that it takes longer when the result is ``False`` than when it
        is ``True``::

            sage: H.complex_intersection_is_empty([H.B(n,0.03) for n in [1..6]],v)
            True
        """
        from sage.schemes.elliptic_curves.period_lattice_region import PeriodicRegion

        b2 = v(self.E.b2())
        # Note that we normalise w1, w2 differently from [Tho2010]_!
        w2, w1 = self.E.period_lattice(v).normalised_basis()
        tau = w2/w1
        bounds = [RDF((B.sqrt() + abs(b2)/12) * abs(w1) ** 2) for B in Bk]
        vals = self.wp_on_grid(v, 30, half=use_half)
        wp = self.wp_intervals(v, abs_only=True)

        k = len(bounds)

        # First try and prove a negative result (cheap).
        if verbose:
            print("trying to prove negative result...")
        intersection = None
        for B, n in sorted(zip(bounds, ZZ.range(1, k+1))):
            T = PeriodicRegion(CDF(1), CDF(tau), vals < B, full=not use_half)
            if intersection is None:
                intersection = PeriodicRegion(CDF(1), CDF(tau), vals < B, full=not use_half)
            else:
                intersection &= T/n
                if intersection.is_empty():
                    break
        else:
            z = CIF(intersection.innermost_point())
            if all(wp((k+1)*z).upper() < B for k, B in enumerate(bounds)):
                return False

        # Now try to prove a positive result.
        if verbose:
            print("trying to prove positive result...")
        intersection = None
        for B, n in sorted(zip(bounds, ZZ.range(1, k+1))):

            T = PeriodicRegion(CDF(1), CDF(tau), vals < B, full=not use_half).expand().refine()
            B = RIF(B)
            leaning_right = tau.real() / tau.imag() >= 0
            def check_line(z):
                wpz = wp(z)
                if wpz > B:
                    return True
                # Try refining once before we declare failure.
                z00, z01, z10, z11 = z.bisection()
                if leaning_right:
                    start, end = z00, z11
                else:
                    start, end = z01, z10
                if wp(start) > B and wp(end) > B:
                    return True
                return False

            # This step here is the bottleneck.
            while not T.verify(check_line):
                if verbose:
                    print("bad")
                T = T.expand()
            if intersection is None:
                intersection = T
            else:
                intersection &= T/n
                if intersection.is_empty():
                    return True

        return False

    def test_mu(self, mu, N, verbose=True):
        r"""
        Return ``True`` if we can prove that `\mu` is a lower bound.

        INPUT:

        - ``mu`` (real) -- a positive real number

        - ``N`` (integer) -- upper bound on the multiples to be used.

        - ``verbose`` (boolean, default True) -- verbosity flag.

        OUTPUT:

        ``True`` or ``False``, according to whether we succeed in
        proving that `\mu` is a lower bound for the canonical heights
        of points of infinite order with everywhere good reduction.

        .. NOTE::

            A ``True`` result is rigorous; ``False`` only means that
            the attempt failed: trying again with larger `N` may yield
            ``True``.

        EXAMPLES::

            sage: K.<a> = NumberField(x^3-2)
            sage: E = EllipticCurve([0,0,0,0,a])
            sage: H = E.height_function()

        This curve does have a point of good reduction whose canonical
        point is approximately 1.68::

            sage: P = E.gens(lim3=5)[0]; P
            (1/3*a^2 + a + 5/3 : -2*a^2 - 4/3*a - 5/3 : 1)
            sage: P.height()
            1.68038085233673
            sage: P.has_good_reduction()
            True

        Using `N=5` we can prove that 0.1 is a lower bound (in fact we
        only need `N=2`), but not that 0.2 is::

            sage: H.test_mu(0.1, 5)
            B_1(0.100000000000000) = 1.51580969677387
            B_2(0.100000000000000) = 0.932072561526720
            True
            sage: H.test_mu(0.2, 5)
            B_1(0.200000000000000) = 2.04612906979932
            B_2(0.200000000000000) = 3.09458988474327
            B_3(0.200000000000000) = 27.6251108409484
            B_4(0.200000000000000) = 1036.24722370223
            B_5(0.200000000000000) = 3.67090854562318e6
            False

        Since 0.1 is a lower bound we can deduce that the point `P` is
        either primitive or divisible by either 2 or 3.  In fact it is
        primitive::

            sage: (P.height()/0.1).sqrt()
            4.09924487233530
            sage: P.division_points(2)
            []
            sage: P.division_points(3)
            []
        """
        # Compute the list of values `B_n(\mu)` for n in 1..N.  If any
        # of these is 1 we can return True right away (see [Tho2010]_,
        # Proposition 5.1).
        Bk = []
        for n in ZZ.range(1, N + 1):
            b = self.B(n, mu)
            if verbose:
                print("B_%s(%s) = %s" % (n, mu, b))
            if b < 1:
                return True
            Bk.append(b)

        # Each real or complex embedding of the number field gives us
        # a chance to prove the lower bound.  We try each in turn,
        # stopping if one gives a True result.

        from sage.rings.number_field.number_field import refine_embedding
        for v in self.K.places():
            ok = False
            while not ok:
                try:
                    if v(self.K.gen()) in RR:
                        if self.real_intersection_is_empty(Bk, v):
                            return True
                    else:
                        if self.complex_intersection_is_empty(Bk, v):
                            return True
                    ok = True
                except ArithmeticError:
                    v = refine_embedding(v)
                    if verbose:
                        print("Refining embedding, codomain now {}".format(v.codomain()))
        return False # Couldn't prove it...

    def min_gr(self, tol, n_max, verbose=False):
        r"""
        Returns a lower bound for points of infinite order with good reduction.

        INPUT:

        - ``tol`` -- tolerance in output (see below).

        - ``n_max`` -- how many multiples to use in iteration.

        - ``verbose`` (boolean, default False) -- verbosity flag.

        OUTPUT:

        A positive real `\mu` for which it has been established
        rigorously that every point of infinite order on the elliptic
        curve (defined over its ground field), which has good
        reduction at all primes, has canonical height greater than
        `\mu`, and such that it is not possible (at least without
        increasing ``n_max``) to prove the same for
        `\mu\cdot\text{tol}`.

        EXAMPLES:

        Example 1 from [CS2006]_ (where a lower bound of 1.9865 was
        given)::

            sage: E = EllipticCurve([1, 0, 1, 421152067, 105484554028056]) # 60490d1
            sage: E.height_function().min_gr(.0001, 5)
            1.98684388146518

        Example 10.1 from [Tho2010]_ (where a lower bound of 0.18 was
        given)::

            sage: K.<i> = QuadraticField(-1)
            sage: E = EllipticCurve([0,0,0,91-26*i,-144-323*i])
            sage: H = E.height_function()
            sage: H.min_gr(0.1,4) # long time (8.1s)
            0.1621049443313762

        Example 10.2 from [Tho2010]_::

            sage: K.<i> = QuadraticField(-1)
            sage: E = EllipticCurve([0,1-i,i,-i,0])
            sage: H = E.height_function()
            sage: H.min_gr(0.01, 5)  # long time
            0.020153685521979152

        In this example the point `P=(0,0)` has height 0.023 so our
        lower bound is quite good::

            sage: P = E((0,0))
            sage: P.has_good_reduction()
            True
            sage: P.height()
            0.0230242154471211

        Example 10.3 from [Tho2010]_ (where the same bound of 0.25 is
        given)::

            sage: K.<a> = NumberField(x^3-2)
            sage: E = EllipticCurve([0,0,0,-3*a-a^2,a^2])
            sage: H = E.height_function()
            sage: H.min_gr(0.1,5) # long time (7.2s)
            0.25

        TESTS:

        This example from the LMFDB gave problems before the fix in :trac:`8829`::

            sage: K.<phi> = NumberField(x^2-x-1)
            sage: E = EllipticCurve([phi + 1, -phi + 1, 1, 20*phi - 39, 196*phi + 237])
            sage: H = E.height_function()
            sage: H.min_gr(.1, 5, verbose=True)  # long time (~22s)
            B_1(1) = 1540.199246369678
            ...
            halving mu to 0.25 and increasing n_max to 6
            ...
            halving mu to 0.001953125 and increasing n_max to 13
            doubling mu to 0.0078125
            doubling mu to 0.015625
            height bound in [0.0078125, 0.015625] using n_max = 13
            ...
            height bound in [0.012048522073499539, 0.01313900648833929] using n_max = 13
            0.012048522073499539
        """
        test = self.test_mu
        if test(1, n_max, verbose):
            mu = 2.0
            while test(mu, n_max, False):
                mu *= 2
            mu *= 0.5
        else:
            mu = 0.5
            while not test(mu, n_max, False):
                mu *= 0.5
                n_max += 1
                if verbose:
                    print("halving mu to %r and increasing n_max to %r" % (mu, n_max))
            # now we have (mu,n_max) which work we can try to increase
            # mu again using this larger n_max:
            mu *= 2
            while test(mu, n_max, False):
                mu *= 2
                if verbose:
                    print("doubling mu to %r" % mu)
            mu *= 0.5

        # The true value lies between mu and eps * mu.
        eps = 2.0
        while eps > tol + 1:
            if verbose:
                print("height bound in [%r, %r] using n_max = %r"
                      % (mu, mu * eps, n_max))
            eps = math.sqrt(eps)
            if test(mu * eps, n_max, False):
                mu = mu * eps
        if verbose:
            print("height bound in [%r, %r] using n_max = %r"
                  % (mu, mu * eps, n_max))
        return RDF(mu)

    def min(self, tol, n_max, verbose=False):
        r"""
        Returns a lower bound for all points of infinite order.

        INPUT:

        - ``tol`` -- tolerance in output (see below).

        - ``n_max`` -- how many multiples to use in iteration.

        - ``verbose`` (boolean, default False) -- verbosity flag.

        OUTPUT:

        A positive real `\mu` for which it has been established
        rigorously that every point of infinite order on the elliptic
        curve (defined over its ground field) has canonical height
        greater than `\mu`, and such that it is not possible (at least
        without increasing ``n_max``) to prove the same for
        `\mu\cdot\text{tol}`.

        EXAMPLES:

        Example 1 from [CS2006]_ (where the same lower bound of 0.1126 was
        given)::

            sage: E = EllipticCurve([1, 0, 1, 421152067, 105484554028056]) # 60490d1
            sage: E.height_function().min(.0001, 5)
            0.0011263287309893311

        Example 10.1 from [Tho2010]_ (where a lower bound of 0.18 was
        given)::

            sage: K.<i> = QuadraticField(-1)
            sage: E = EllipticCurve([0,0,0,91-26*i,-144-323*i])
            sage: H = E.height_function()
            sage: H.min(0.1,4) # long time (8.1s)
            0.1621049443313762

        Example 10.2 from [Tho2010]_::

            sage: K.<i> = QuadraticField(-1)
            sage: E = EllipticCurve([0,1-i,i,-i,0])
            sage: H = E.height_function()
            sage: H.min(0.01,5) # long time (4s)
            0.020153685521979152

        In this example the point `P=(0,0)` has height 0.023 so our
        lower bound is quite good::

            sage: P = E((0,0))
            sage: P.height()
            0.0230242154471211

        Example 10.3 from [Tho2010]_ (where the same bound of 0.0625 is
        given)::

            sage: K.<a> = NumberField(x^3-2)
            sage: E = EllipticCurve([0,0,0,-3*a-a^2,a^2])
            sage: H = E.height_function()
            sage: H.min(0.1,5) # long time (7s)
            0.0625

        More examples over `\QQ`::

            sage: E = EllipticCurve('37a')
            sage: h = E.height_function()
            sage: h.min(.01, 5)
            0.03987318057488725
            sage: E.gen(0).height()
            0.0511114082399688

        After base change the lower bound can decrease::

            sage: K.<a> = QuadraticField(-5)
            sage: E.change_ring(K).height_function().min(0.5, 10) # long time (8s)
            0.04419417382415922

            sage: E = EllipticCurve('389a')
            sage: h = E.height_function()
            sage: h.min(0.1, 5)
            0.05731275270029196
            sage: [P.height() for P in E.gens()]
            [0.686667083305587, 0.327000773651605]
        """
        # The lcm of the exponents of all the component groups at
        # finite places (allowing for everywhere good reduction!)
        tp = lcm([L.tamagawa_exponent() for L in self.E.local_data()] + [ZZ(1)])

        # Include infinite places:
        if tp % 2 == 1:
            if self.K == QQ:
                if self.E.real_components() == 2:
                    tp *= 2
            elif any(v(self.E.discriminant()) > 0
                     for v in self.K.real_places()):
                tp *= 2
        # Now tp is such that tp*P has good reduction at all places
        # for all points P:
        return self.min_gr(tol, n_max, verbose) / tp ** 2

r"""
Canonical heights for elliptic curves over number fields

Also, rigorous lower bounds for the canonical height of non-torsion
points, implementing the algorithms in [CS2006]_ (over `\QQ`) and [Tho2010]_,
which also refer to [CPS2006]_.

AUTHORS:

- <NAME> (2010): initial version

- <NAME> (2014): added many docstrings and doctests
"""
##############################################################################
#       Copyright (C) 2010 <NAME> <<EMAIL>>
#                     2014 <NAME> <<EMAIL>>
#
#  Distributed under the terms of the GNU General Public License (GPL)
#
#    This code is distributed in the hope that it will be useful,
#    but WITHOUT ANY WARRANTY; without even the implied warranty of
#    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
#    General Public License for more details.
#
#  The full text of the GPL is available at:
#
#                  https://www.gnu.org/licenses/
##############################################################################
import numpy
import math
import bisect

from sage.rings.integer_ring import ZZ
from sage.rings.rational_field import QQ
from sage.rings.infinity import infinity
from sage.rings.cif import CIF
from sage.rings.cc import CC
from sage.rings.complex_double import CDF
from sage.rings.real_double import RDF
from sage.rings.real_mpfi import RIF
from sage.rings.real_mpfr import RR

from sage.misc.cachefunc import cached_method
from sage.misc.mrange import cartesian_product_iterator
from sage.arith.all import lcm, factorial
from sage.ext.fast_callable import fast_callable
from sage.functions.log import log, exp
from sage.symbolic.ring import SR


class UnionOfIntervals:
    r"""
    A class representing a finite union of closed intervals in
    `\RR` which can be scaled, shifted, intersected, etc.

    The intervals are represented as an ordered list of their
    endpoints, which may include `-\infty` and `+\infty`.

    EXAMPLES::

        sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals
        sage: R = UnionOfIntervals([1,2,3,infinity]); R
        ([1, 2] U [3, +Infinity])
        sage: R + 5
        ([6, 7] U [8, +Infinity])
        sage: ~R
        ([-Infinity, 1] U [2, 3])
        sage: ~R | (10*R + 100)
        ([-Infinity, 1] U [2, 3] U [110, 120] U [130, +Infinity])

    .. TODO::

        Unify :class:`UnionOfIntervals` with the class ``RealSet``
        introduced by :trac:`13125`; see :trac:`16063`.
    """
    def __init__(self, endpoints):
        r"""
        An union of intervals is initialized by giving an increasing list
        of endpoints, the first of which may be `-\infty` and the last of
        which may be `+\infty`.

        EXAMPLES::

            sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals
            sage: UnionOfIntervals([0,1])
            ([0, 1])
            sage: UnionOfIntervals([-infinity, pi, 17, infinity])
            ([-Infinity, pi] U [17, +Infinity])
            sage: UnionOfIntervals([])
            ()

            sage: UnionOfIntervals([1])
            Traceback (most recent call last):
            ...
            ValueError: an even number of endpoints must be given (got 1)
            sage: UnionOfIntervals([3,2,1,0])
            Traceback (most recent call last):
            ...
            ValueError: endpoints must be given in order
        """
        if len(endpoints) % 2:
            raise ValueError("an even number of endpoints must be given (got %s)" % len(endpoints))
        if endpoints != sorted(endpoints):
            raise ValueError("endpoints must be given in order")
        self._endpoints = endpoints

    def finite_endpoints(self):
        r"""
        Returns the finite endpoints of this union of intervals.

        EXAMPLES::

            sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals
            sage: UnionOfIntervals([0,1]).finite_endpoints()
            [0, 1]
            sage: UnionOfIntervals([-infinity, 0, 1, infinity]).finite_endpoints()
            [0, 1]
        """
        return [e for e in self._endpoints if -infinity < e < infinity]

    def intervals(self):
        r"""
        Returns the intervals in self, as a list of 2-tuples.

        EXAMPLES::

            sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals
            sage: UnionOfIntervals(list(range(10))).intervals()
            [(0, 1), (2, 3), (4, 5), (6, 7), (8, 9)]
            sage: UnionOfIntervals([-infinity, pi, 17, infinity]).intervals()
            [(-Infinity, pi), (17, +Infinity)]
        """
        return list(zip(self._endpoints[::2], self._endpoints[1::2]))

    def is_empty(self):
        r"""
        Returns whether self is empty.

        EXAMPLES::

            sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals
            sage: UnionOfIntervals([3,4]).is_empty()
            False
            sage: all = UnionOfIntervals([-infinity, infinity])
            sage: all.is_empty()
            False
            sage: (~all).is_empty()
            True
            sage: A = UnionOfIntervals([0,1]) & UnionOfIntervals([2,3])
            sage: A.is_empty()
            True
        """
        return not self._endpoints

    def __add__(left, right):
        r"""
        If both left an right are unions of intervals, take their union,
        otherwise treat the non-union of intervals as a scalar and shift.

        EXAMPLES::

            sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals
            sage: A = UnionOfIntervals([0, 1/2, 2, infinity]); A
            ([0, 1/2] U [2, +Infinity])
            sage: A + 1
            ([1, 3/2] U [3, +Infinity])
            sage: pi + A
            ([pi, pi + 1/2] U [pi + 2, +Infinity])
            sage: A + UnionOfIntervals([-infinity, -1])
            ([-Infinity, -1] U [0, 1/2] U [2, +Infinity])
        """
        if not isinstance(left, UnionOfIntervals):
            left, right = right, left
        elif not isinstance(right, UnionOfIntervals):
            return UnionOfIntervals([right + e for e in left._endpoints])
        else:
            return left.union([left, right])

    def __mul__(left, right):
        r"""
        Scale a union of intervals on the left or right.

        EXAMPLES::

            sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals
            sage: A = UnionOfIntervals([0, 1/2, 2, infinity]); A
            ([0, 1/2] U [2, +Infinity])
            sage: 2 * A
            ([0, 1] U [4, +Infinity])
            sage: A * 100
            ([0, 50] U [200, +Infinity])
            sage: 1.5 * A
            ([0.000000000000000, 0.750000000000000] U [3.00000000000000, +Infinity])
        """
        if not isinstance(right, UnionOfIntervals):
            return UnionOfIntervals([e*right for e in left._endpoints])
        elif not isinstance(left, UnionOfIntervals):
            return UnionOfIntervals([left*e for e in right._endpoints])
        else:
            return NotImplemented

    def __rmul__(self, other):
        r"""
        Scale by an operand on the left.

        TESTS::

            sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals
            sage: A = UnionOfIntervals([0, 1/2, 2, infinity]); A
            ([0, 1/2] U [2, +Infinity])
            sage: pi * A
            ([0, 1/2*pi] U [2*pi, +Infinity])
        """
        return self * other

    def __radd__(self, other):
        r"""
        Add a scalar operand on the left.

        TESTS::

            sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals
            sage: A = UnionOfIntervals([0, 1/2, 2, infinity]); A
            ([0, 1/2] U [2, +Infinity])
            sage: 100 + A
            ([100, 201/2] U [102, +Infinity])
        """
        return self + other

    def __invert__(self):
        r"""
        Return the closure of the complement of self.

        .. NOTE::

            We take the closure because open intervals are not supported.

        EXAMPLES::

            sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals
            sage: A = UnionOfIntervals([0, 1/2, 2, infinity]); A
            ([0, 1/2] U [2, +Infinity])
            sage: ~A
            ([-Infinity, 0] U [1/2, 2])
            sage: A | ~A
            ([-Infinity, +Infinity])
            sage: A & ~A
            ([0, 0] U [1/2, 1/2] U [2, 2])
        """
        endpoints = list(self._endpoints)
        if endpoints[0] == -infinity:
            del endpoints[0]
        else:
            endpoints.insert(0, -infinity)
        if endpoints[-1] == infinity:
            endpoints.pop()
        else:
            endpoints.append(infinity)
        return UnionOfIntervals(endpoints)

    @staticmethod
    def join(L, condition):
        r"""
        Utility function to form the union or intersection of a list of UnionOfIntervals.

        INPUT:

        - ``L`` (list) -- a list of UnionOfIntervals instances

        - ``condition`` (function) -- either ``any`` or ``all``, or
          some other boolean function of a list of boolean values.

        OUTPUT:

        A new UnionOfIntervals instance representing the subset of
        '\RR' equal to those reals in any/all/condition of the
        UnionOfIntervals in the list.

        .. NOTE::

            This is a static method for the class.

        EXAMPLES::

            sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals
            sage: A = UnionOfIntervals([1,3,5,7]); A
            ([1, 3] U [5, 7])
            sage: B = A+1; B
            ([2, 4] U [6, 8])
            sage: A.join([A,B],any) # union
            ([1, 4] U [5, 8])
            sage: A.join([A,B],all) # intersection
            ([2, 3] U [6, 7])
            sage: A.join([A,B],sum) # symmetric difference
            ([1, 2] U [3, 4] U [5, 6] U [7, 8])
        """
        all = []
        for ix, region in enumerate(L):
            for i, e in enumerate(region._endpoints):
                all.append((e, -(not (i % 2)), ix))
        all.sort()
        join = []
        in_join = False
        in_L = [False] * len(L)
        for e, start, ix in all:
            in_L[ix] = start
            if condition(in_L) != in_join:
                join.append(e)
                in_join = not in_join
        return UnionOfIntervals(join)

    @classmethod
    def union(cls, L):
        r"""
        Return the union of a list of UnionOfIntervals.

        INPUT:

        - ``L`` (list) -- a list of UnionOfIntervals instances

        OUTPUT:

        A new UnionOfIntervals instance representing the union of the
        UnionOfIntervals in the list.

        .. NOTE::

            This is a class method.

        EXAMPLES::

            sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals
            sage: A = UnionOfIntervals([1,3,5,7]); A
            ([1, 3] U [5, 7])
            sage: B = A+1; B
            ([2, 4] U [6, 8])
            sage: A.union([A,B])
            ([1, 4] U [5, 8])
        """
        return cls.join(L, any)

    @classmethod
    def intersection(cls, L):
        r"""
        Return the intersection of a list of UnionOfIntervals.

        INPUT:

        - ``L`` (list) -- a list of UnionOfIntervals instances

        OUTPUT:

        A new UnionOfIntervals instance representing the intersection
        of the UnionOfIntervals in the list.

        .. NOTE::

            This is a class method.

        EXAMPLES::

            sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals
            sage: A = UnionOfIntervals([1,3,5,7]); A
            ([1, 3] U [5, 7])
            sage: B = A+1; B
            ([2, 4] U [6, 8])
            sage: A.intersection([A,B])
            ([2, 3] U [6, 7])
        """
        for R in L:
            if R.is_empty():
                return R
        return cls.join(L, all)

    def __or__(left, right):
        r"""
        Return the union of a two UnionOfIntervals instances.

        INPUT:

        - ``left``, ``right`` (UnionOfIntervals) -- two UnionOfIntervals instances

        OUTPUT:

        A new UnionOfIntervals instance representing the union of ``left`` and ``right``.

        EXAMPLES::

            sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals
            sage: A = UnionOfIntervals([1,3,5,7]); A
            ([1, 3] U [5, 7])
            sage: B = A+1; B
            ([2, 4] U [6, 8])
            sage: A | B
            ([1, 4] U [5, 8])
        """
        return left.union([left, right])

    def __and__(left, right):
        r"""
        Return the intersection of a two UnionOfIntervals instances.

        INPUT:

        - ``left``, ``right`` (UnionOfIntervals) -- two UnionOfIntervals instances

        OUTPUT:

        A new UnionOfIntervals instance representing the intersection of ``left`` and ``right``.

        EXAMPLES::

            sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals
            sage: A = UnionOfIntervals([1,3,5,7]); A
            ([1, 3] U [5, 7])
            sage: B = A+1; B
            ([2, 4] U [6, 8])
            sage: A & B
            ([2, 3] U [6, 7])
        """
        return left.intersection([left, right])

    def __contains__(self, x):
        r"""
        Return True if ``x`` is in the UnionOfIntervals.

        INPUT:

        - ``x`` (real) -- a real number

        OUTPUT:

        Boolean: True if and only if ``x`` is in the union of intervals.

        EXAMPLES::

            sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals
            sage: A = UnionOfIntervals([1,3,5,7]); A
            ([1, 3] U [5, 7])
            sage: 1 in A
            True
            sage: 4 in A
            False
            sage: -infinity in A
            False
        """
        return x in self._endpoints or bisect.bisect_left(self._endpoints, x) % 2 == 1

    def __str__(self):
        r"""
        Return the string representation of this UnionOfIntervals.

        EXAMPLES::

            sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals
            sage: A = UnionOfIntervals([1,3,5,7])
            sage: str(A)
            '([1, 3] U [5, 7])'
        """
        return repr(self)

    def __repr__(self):
        r"""
        Return the string representation of this UnionOfIntervals.

        EXAMPLES::

            sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals
            sage: A = UnionOfIntervals([1,3,5,7]); A
            ([1, 3] U [5, 7])
        """
        return "(%s)" % " U ".join(str(list(I)) for I in self.intervals())


def nonneg_region(f):
    r"""
    Returns the UnionOfIntervals representing the region where ``f`` is non-negative.

    INPUT:

    - ``f`` (polynomial) -- a univariate polynomial over `\RR`.

    OUTPUT:

    A UnionOfIntervals representing the set `\{x \in\RR mid f(x) \ge 0\}`.

    EXAMPLES::

        sage: from sage.schemes.elliptic_curves.height import nonneg_region
        sage: x = polygen(RR)
        sage: nonneg_region(x^2-1)
        ([-Infinity, -1.00000000000000] U [1.00000000000000, +Infinity])
        sage: nonneg_region(1-x^2)
        ([-1.00000000000000, 1.00000000000000])
        sage: nonneg_region(1-x^3)
        ([-Infinity, 1.00000000000000])
        sage: nonneg_region(x^3-1)
        ([1.00000000000000, +Infinity])
        sage: nonneg_region((x-1)*(x-2))
        ([-Infinity, 1.00000000000000] U [2.00000000000000, +Infinity])
        sage: nonneg_region(-(x-1)*(x-2))
        ([1.00000000000000, 2.00000000000000])
        sage: nonneg_region((x-1)*(x-2)*(x-3))
        ([1.00000000000000, 2.00000000000000] U [3.00000000000000, +Infinity])
        sage: nonneg_region(-(x-1)*(x-2)*(x-3))
        ([-Infinity, 1.00000000000000] U [2.00000000000000, 3.00000000000000])
        sage: nonneg_region(x^4+1)
        ([-Infinity, +Infinity])
        sage: nonneg_region(-x^4-1)
        ()
    """
    roots = sorted(f.roots())
    sign_changes = [r for r,e in roots if e%2 == 1]
    if (f.leading_coefficient() * (-1)**f.degree()) > 0:
        sign_changes = [-infinity] + sign_changes
    if f.leading_coefficient() > 0:
        sign_changes += [infinity]
    return UnionOfIntervals(sign_changes)

def inf_max_abs(f, g, D):
    r"""
    Returns `\inf_D(\max(|f|, |g|))`.

    INPUT:

    - ``f``, ``g`` (polynomials) -- real univariate polynomials

    - ``D`` (UnionOfIntervals) -- a subset of `\RR`

    OUTPUT:

    A real number approximating the value of `\inf_D(\max(|f|, |g|))`.

    ALGORITHM:

    The extreme values must occur at an endpoint of a subinterval of
    `D` or at a point where one of `f`, `f'`, `g`, `g'`, `f\pm g` is
    zero.

    EXAMPLES::

        sage: from sage.schemes.elliptic_curves.height import inf_max_abs, UnionOfIntervals
        sage: x = polygen(RR)
        sage: f = (x-10)^4+1
        sage: g = 2*x^3+100
        sage: inf_max_abs(f,g,UnionOfIntervals([1,2,3,4,5,6]))
        425.638201706391
        sage: r0 = (f-g).roots()[0][0]
        sage: r0
        5.46053402234697
        sage: max(abs(f(r0)),abs(g(r0)))
        425.638201706391
    """
    xs =  f.roots() + f.derivative().roots()
    xs += g.roots() + g.derivative().roots()
    xs += (f-g).roots() + (f+g).roots()
    xs = [r for r,e in xs if r in D]  # ignore multiplicities and points outside D
    xs += D.finite_endpoints()        # include endpoints of intervals
    if xs:
        return min([max(abs(f(r)), abs(g(r))) for r in xs])
    return infinity

def min_on_disk(f, tol, max_iter=10000):
    r"""
    Returns the minimum of a real-valued complex function on a square.

    INPUT:

    - ``f`` -- a function from CIF to RIF

    - ``tol`` (real) -- a positive real number

    - ``max_iter`` (integer, default 10000) -- a positive integer
      bounding the number of iterations to be used

    OUTPUT:

    A 2-tuple `(s,t)`, where `t=f(s)` and `s` is a CIF element
    contained in the disk `|z|\le1`, at which `f` takes its minimum
    value.

    EXAMPLES::

        sage: from sage.schemes.elliptic_curves.height import min_on_disk
        sage: f = lambda x: (x^2+100).abs()
        sage: s, t = min_on_disk(f, 0.0001)
        sage: s, f(s), t
        (0.01? + 1.00?*I, 99.01?, 99.0000000000000)
    """
    # L holds a list of 4-tuples (mfs, ds, s, in_disk) where s is a
    # subregion of the initial square, ds its relative diameter,
    # mfs=-f(s) (actually minus the lower bound on f(s)) and in_disk
    # is a flag indicating whether or not s is a subset of the unit
    # disk.

    # We store the negative of the lower bound on f(s) so that we can
    # use the bisect module to sort these 4-tuples.

    # Initially L contains one element, the whole unit box, which is
    # not contained in the unit square.

    s = CIF(RIF(-1,1), RIF(-1,1))
    fs = f(s)
    L = [(-fs.lower(), fs.relative_diameter(), s, False)]

    # min_max holds the minimum over L of fs.upper().

    min_max = fs.upper()

    # We iterate at most max_iter times.  At each step we look at the
    # best so far and return it if is good enough, meaning that its
    # relative diameter is less than the given tolerance; otherwise we
    # bisect this best region (into 4 pieces) and replace the entry in
    # L with at most 4 smaller entries.

    for k in range(max_iter):
        value, err, region, in_disk = L.pop()
        if err < tol:       # reached desired tolerance, so return
            return region, -value
        for s in region.bisection(): # 4 sub-regions
            if in_disk:
                s_in_disk = True     # if the original region si in the disk so are all its children
            else:
                r = abs(s)          # otherwise we test each one
                if r > 1:
                    continue        # skip this subregion if it is entirely outside the disk
                s_in_disk = r < 1   # meaning it is entirely inside the disk

            fs = f(s)

            if fs.upper() < min_max: # we definitely beat the record
                min_max = fs.upper()
                unneeded = bisect.bisect(L, (-min_max,))
                if unneeded > 100:   # discard the worse entries (if there are many)
                    L = L[unneeded:]

            if fs.lower() < min_max: # we may beat the record, cannot yet tell: insert this region
                                     # into the list at the appropriate place to maintain sorting
                bisect.insort(L, (-fs.lower(), fs.relative_diameter(), s, s_in_disk))

    # If we get here, then even after max_iter iterations the tolerance has not been reached.
    raise ValueError("too many iterations")


two_pi_i_CDF = CDF(0, 2 * RDF.pi())
two_pi_i_CIF = CIF(0, 2 * RIF.pi())

# Ideas: We know tau, so we know the direction of the diagonal.
# We can solve for x in p1, will this allow us to find the maxima exactly?


def rat_term_CIF(z, try_strict=True):
    r"""
    Compute the value of `u/(1-u)^2` in ``CIF``, where `u=\exp(2\pi i z)`.

    INPUT:

    - ``z`` (complex) -- a CIF element

    - ``try_strict`` (bool) -- flag

    EXAMPLES::

        sage: from sage.schemes.elliptic_curves.height import rat_term_CIF
        sage: z = CIF(0.5,0.2)
        sage: rat_term_CIF(z)
        -0.172467461182437? + 0.?e-16*I
        sage: rat_term_CIF(z, False)
        -0.172467461182437? + 0.?e-16*I
    """
    two_pi_i_z = two_pi_i_CIF * z
    r = (two_pi_i_z.real()).exp() # = |u|
    x, y = two_pi_i_z.imag().cos(), two_pi_i_z.imag().sin()

    real_part = imag_part = None

    # If there are no local minima the intervals are strictly
    # determined by their values at the endpoints.

    if try_strict:

        # evaluate the function at the four corners:

        corner_reals = []
        corner_imags = []
        for a, b in cartesian_product_iterator([z.real().endpoints(), z.imag().endpoints()]):
            zz = CDF(a,b)
            u = (two_pi_i_CDF*zz).exp()
            f = u/(1-u)**2
            corner_reals.append(f.real())
            corner_imags.append(f.imag())

        p1 = (((((r+2*x)*r - 6)*r + 2*x) * r) + 1)
            # =  r^4 + 2*r^3*x - 6*r^2 + 2*r*x + 1
        p2 = (r*(x*(r+2*x)-4)+x)
            # = r^2*x + 2*r*x^2 - 4*r + x

        df_dr = (r**2-1) * p2
        dg_dr = p1 * y
        dg_dx = r * df_dr / y

        if not dg_dr.contains_zero() or not dg_dx.contains_zero():
            real_part = RIF(min(corner_reals), max(corner_reals))

        if not dg_dr.contains_zero() or not dg_dx.contains_zero():
            imag_part = RIF(min(corner_imags), max(corner_imags))

    if real_part is None or imag_part is None:
        denom = (1-r*(2*x-r))**2
    if real_part is None:
        real_part = r*(x*(1+r**2)-2*r)/denom
    if imag_part is None:
        imag_part = -(r**2-1)*y*r/denom

    return CIF(real_part, imag_part)


def eps(err, is_real):
    r"""
    Return a Real or Complex interval centered on 0 with radius err.

    INPUT:

    - ``err`` (real) -- a positive real number, the radius of the interval

    - ``is_real`` (boolean) -- if True, returns a real interval in
      RIF, else a complex interval in CIF

    OUTPUT:

    An element of RIF or CIF (as specified), centered on 0, with given radius.

    EXAMPLES::

        sage: from sage.schemes.elliptic_curves.height import eps
        sage: eps(0.01, True)
        0.0?
        sage: eps(0.01, False)
        0.0? + 0.0?*I
    """
    e = RIF(-err, err)
    if is_real:
        return e
    else:
        return CIF(e, e)


class EllipticCurveCanonicalHeight:
    r"""
    Class for computing canonical heights of points on elliptic curves
    defined over number fields, including rigorous lower bounds for
    the canonical height of non-torsion points.

    EXAMPLES::

        sage: from sage.schemes.elliptic_curves.height import EllipticCurveCanonicalHeight
        sage: E = EllipticCurve([0,0,0,0,1])
        sage: EllipticCurveCanonicalHeight(E)
        EllipticCurveCanonicalHeight object associated to Elliptic Curve defined by y^2 = x^3 + 1 over Rational Field

    Normally this object would be created like this::

        sage: E.height_function()
        EllipticCurveCanonicalHeight object associated to Elliptic Curve defined by y^2 = x^3 + 1 over Rational Field
    """

    def __init__(self, E):
        r"""
        Initialize the class with an elliptic curve.

        INPUT:

        - `E` -- an elliptic curve defined over a number field

        EXAMPLES::

            sage: from sage.schemes.elliptic_curves.height import EllipticCurveCanonicalHeight
            sage: E = EllipticCurve([0,0,0,0,1])
            sage: EllipticCurveCanonicalHeight(E)
            EllipticCurveCanonicalHeight object associated to Elliptic Curve defined by y^2 = x^3 + 1 over Rational Field

        An example over a number field::

            sage: K.<i>=QuadraticField(-1)
            sage: E = EllipticCurve([0,i,0,i,i])
            sage: EllipticCurveCanonicalHeight(E)
            EllipticCurveCanonicalHeight object associated to Elliptic Curve defined by y^2 = x^3 + i*x^2 + i*x + i over Number Field in i with defining polynomial x^2 + 1 with i = 1*I

        TESTS:

        The base field must be a number field (or `\QQ`)::

            sage: from sage.schemes.elliptic_curves.height import EllipticCurveCanonicalHeight
            sage: E = EllipticCurve(GF(7),[0,0,0,0,1])
            sage: EllipticCurveCanonicalHeight(E)
            Traceback (most recent call last):
            ...
            ValueError: EllipticCurveCanonicalHeight class can only be created from an elliptic curve defined over a number field
        """
        from sage.schemes.elliptic_curves.ell_generic import is_EllipticCurve
        if is_EllipticCurve(E):
            self.E = E
            from sage.rings.number_field.number_field_base import is_NumberField
            K = E.base_ring()
            if is_NumberField(K):
                self.K = K
            else:
                raise ValueError("EllipticCurveCanonicalHeight class can only be created from an elliptic curve defined over a number field")
        else:
            raise ValueError("EllipticCurveCanonicalHeight class can only be created from an elliptic curve")

    def __repr__(self):
        r"""
        Return the string representation.

        EXAMPLES::

            sage: E = EllipticCurve([0,0,0,0,1])
            sage: E.height_function()
            EllipticCurveCanonicalHeight object associated to Elliptic Curve defined by y^2 = x^3 + 1 over Rational Field
        """
        return "EllipticCurveCanonicalHeight object associated to %s" % self.E

    def curve(self):
        r"""
        Return the elliptic curve.

        EXAMPLES::

            sage: E = EllipticCurve([0,0,0,0,1])
            sage: H = E.height_function()
            sage: H.curve()
            Elliptic Curve defined by y^2 = x^3 + 1 over Rational Field
        """
        return self.E

    def base_field(self):
        r"""
        Return the base field.

        EXAMPLES::

            sage: E = EllipticCurve([0,0,0,0,1])
            sage: H = E.height_function()
            sage: H.base_field()
            Rational Field
        """
        return self.K

    def __call__(self, P):
        r"""
        Return the canonical height of the point ``P``.

        INPUT:

        - ``P`` -- a point on the elliptic curve.

        OUTPUT:

        The canonical height of ``P``.

        EXAMPLES::

            sage: E = EllipticCurve([0,0,1,-1,0])
            sage: P = E(0,0)
            sage: P.height()
            0.0511114082399688
            sage: H = E.height_function()
            sage: H(P)
            0.0511114082399688
            sage: H([0,0])
            0.0511114082399688
            sage: H((0,0))
            0.0511114082399688

        Over a number field other than `\QQ`::

            sage: K.<i> = QuadraticField(-1)
            sage: E = EllipticCurve(K, [0,0,0,1,-27])
            sage: H = E.height_function()
            sage: H.base_field()
            Number Field in i with defining polynomial x^2 + 1 with i = 1*I
            sage: H((1,5*i))
            1.22257115164148
        """
        return self.E(P).height()

    @cached_method
    def alpha(self, v, tol=0.01):
        r"""
        Return the constant `\alpha_v` associated to the embedding ``v``.

        INPUT:

        - ``v`` -- an embedding of the base field into `\RR` or `\CC`

        OUTPUT:

        The constant `\alpha_v`.  In the notation of [CPS2006]_ and
        [Tho2010]_ (section 3.2), `\alpha_v^3=\epsilon_v`.  The result is
        cached since it only depends on the curve.

        EXAMPLES:

        Example 1 from [CPS2006]_::

            sage: K.<i>=QuadraticField(-1)
            sage: E = EllipticCurve([0,0,0,1+5*i,3+i])
            sage: H = E.height_function()
            sage: alpha = H.alpha(K.places()[0])
            sage: alpha
            1.12272013439355

        Compare with `\log(\epsilon_v)=0.344562...` in [CPS2006]_::

            sage: 3*alpha.log()
            0.347263296676126
        """
        from sage.rings.polynomial.polynomial_ring import polygen
        b2, b4, b6, b8 = [v(b) for b in self.E.b_invariants()]
        x = polygen(v.codomain())
        f = 4*x**3 + b2*x**2 + 2*b4*x + b6
        g = x**4 - b4*x**2 - 2*b6*x - b8
        F = f.reverse() << (4-f.degree())
        G = g.reverse() << (4-g.degree())

        if v(self.K.gen()) in RR:
            I = UnionOfIntervals([-1,1])
            min_fg = inf_max_abs(f, g, nonneg_region(f) & I)
            min_FG = inf_max_abs(F, G, nonneg_region(F) & I)
            return min(min_fg, min_FG) ** (-1/QQ(3))

        else:
            # def pair_max(f, g):
            #     f = f.change_ring(CIF)
            #     g = g.change_ring(CIF)
            #     max = type(RIF(0)).max
            #     def max_f_g(z):
            #         return max(abs(f(z)), abs(g(z)))
            #     return max_f_g
            def pair_max(f, g):
                f = f.change_ring(CDF)
                g = g.change_ring(CDF)
                dfn = [fast_callable(f.derivative(n)/factorial(n), CDF) for n in range(f.degree()+1)]
                dgn = [fast_callable(g.derivative(n)/factorial(n), CDF) for n in range(g.degree()+1)]
                def max_f_g(s):
                    (a,b),(c,d) = s.real().endpoints(), s.imag().endpoints()
                    dx = a - b
                    dy = c - d
                    eta = RDF(dx*dx + dy*dy).sqrt()
                    z = CDF(s.center())
                    err_f = sum(eta ** n * abs(df(z)) for n, df in enumerate(dfn) if n)
                    err_g = sum(eta ** n * abs(dg(z)) for n, dg in enumerate(dgn) if n)
                    return RIF(max(abs(f(z)), abs(g(z)))) + eps(max(err_f, err_g), True)
                return max_f_g
            _, min_fg = min_on_disk(pair_max(f, g), tol)
            _, min_FG = min_on_disk(pair_max(F, G), tol)
            return min(min_fg, min_FG) ** QQ((-1, 3))

    @cached_method
    def e_p(self, p):
        r"""
        Return the exponent of the group over the residue field at ``p``.

        INPUT:

        - ``p`` -- a prime ideal of `K` (or a prime number if `K=\QQ`).

        OUTPUT:

        A positive integer `e_p`, the exponent of the group of
        nonsingular points on the reduction of the elliptic curve
        modulo `p`.  The result is cached.

        EXAMPLES::

            sage: K.<i>=QuadraticField(-1)
            sage: E = EllipticCurve([0,0,0,1+5*i,3+i])
            sage: H = E.height_function()
            sage: H.e_p(K.prime_above(2))
            2
            sage: H.e_p(K.prime_above(3))
            10
            sage: H.e_p(K.prime_above(5))
            9
            sage: E.conductor().norm().factor()
            2^10 * 20921
            sage: p1, p2 = K.primes_above(20921)
            sage: E.local_data(p1)
            Local data at Fractional ideal (-40*i + 139):
            Reduction type: bad split multiplicative
            ...
            sage: H.e_p(p1)
            20920
            sage: E.local_data(p2)
            Local data at Fractional ideal (40*i + 139):
            Reduction type: good
            ...
            sage: H.e_p(p2)
            20815
        """
        kp = self.K.residue_field(p)
        if self.E.has_bad_reduction(p):
            if self.E.has_additive_reduction(p):
                ep = kp.characteristic()
            elif self.E.has_split_multiplicative_reduction(p):
                ep = len(kp) - 1
            else:
                ep = len(kp) + 1
        else:
            ep = self.E.reduction(p).abelian_group().exponent()
        return ZZ(ep)

    @cached_method
    def DE(self, n):
        r"""
        Return the value `D_E(n)`.

        INPUT:

        - ``n`` (int) -- a positive integer

        OUTPUT:

        The value `D_E(n)` as defined in [Tho2010]_, section 4.

        EXAMPLES::

            sage: K.<i>=QuadraticField(-1)
            sage: E = EllipticCurve([0,0,0,1+5*i,3+i])
            sage: H = E.height_function()
            sage: [H.DE(n) for n in srange(1,6)]
            [0, 2*log(5) + 2*log(2), 0, 2*log(13) + 2*log(5) + 4*log(2), 0]
        """
        s = 0
        B = (n+1) ** max(2, self.K.degree())
        for p in self.K.primes_of_bounded_norm_iter(B):
            ep = self.e_p(p)
            if ep.divides(n):
                kp = self.K.residue_field(p)
                s += 2*(1+(n/ep).valuation(kp.characteristic())) * log(len(kp))
        return s

    @cached_method
    def ME(self):
        r"""
        Return the norm of the ideal `M_E`.

        OUTPUT:

        The norm of the ideal `M_E` as defined in [Tho2010]_, section 3.1.
        This is `1` if `E` is a global minimal model, and in general
        measures the non-minimality of `E`.

        EXAMPLES::

            sage: K.<i>=QuadraticField(-1)
            sage: E = EllipticCurve([0,0,0,1+5*i,3+i])
            sage: H = E.height_function()
            sage: H.ME()
            1
            sage: E = EllipticCurve([0,0,0,0,1])
            sage: E.height_function().ME()
            1
            sage: E = EllipticCurve([0,0,0,0,64])
            sage: E.height_function().ME()
            4096
            sage: E.discriminant()/E.minimal_model().discriminant()
            4096
        """
        from sage.misc.misc_c import prod
        if self.K is QQ:
            return prod([p ** (e - self.E.local_data(p).discriminant_valuation()) for p, e in self.E.discriminant().factor()], QQ.one())

        ME = prod([p.norm() ** (e - self.E.local_data(p).discriminant_valuation()) for p, e in self.K.ideal(self.E.discriminant()).factor()], QQ.one())
        return ME.norm()

    def B(self, n, mu):
        r"""
        Return the value `B_n(\mu)`.

        INPUT:

        - ``n`` (int) -- a positive integer

        - ``mu`` (real) -- a positive real number

        OUTPUT:

        The real value `B_n(\mu)` as defined in [Tho2010]_, section 5.

        EXAMPLES:

        Example 10.2 from [Tho2010]_::

            sage: K.<i>=QuadraticField(-1)
            sage: E = EllipticCurve([0,1-i,i,-i,0])
            sage: H = E.height_function()

        In [Tho2010]_ the value is given as 0.772::

            sage: RealField(12)( H.B(5, 0.01) )
            0.777
        """
        K = self.K
        B = exp(K.degree() * n**2 * mu - RDF(self.DE(n))) / self.ME() ** 6
        for v in K.places():
            if v(K.gen()) in RR:
                B *= self.alpha(v)
            else:
                B *= self.alpha(v) ** 2
        return B

    ######################################
    # Empty real intersection detection. #
    ######################################

    def psi(self, xi, v):
        r"""
        Return the normalised elliptic log of a point with this x-coordinate.

        INPUT:

        - ``xi`` (real) -- the real x-coordinate of a point on the
          curve in the connected component with respect to a real
          embedding.

        - ``v`` (embedding) -- a real embedding of the number field.

        OUTPUT:

        A real number in the interval [0.5,1] giving the elliptic
        logarithm of a point on `E` with `x`-coordinate ``xi``, on the
        connected component with respect to the embedding `v`, scaled
        by the real period.

        EXAMPLES:

        An example over `\QQ`::

            sage: E = EllipticCurve('389a')
            sage: v = QQ.places()[0]
            sage: L = E.period_lattice(v)
            sage: P = E.lift_x(10/9)
            sage: L(P)
            1.53151606047462
            sage: L(P) / L.real_period()
            0.615014189772115
            sage: H = E.height_function()
            sage: H.psi(10/9,v)
            0.615014189772115

        An example over a number field::

            sage: K.<a> = NumberField(x^3-2)
            sage: E = EllipticCurve([0,0,0,0,a])
            sage: P = E.lift_x(1/3*a^2 + a + 5/3)
            sage: v = K.real_places()[0]
            sage: L = E.period_lattice(v)
            sage: L(P)
            3.51086196882538
            sage: L(P) / L.real_period()
            0.867385122699931
            sage: xP = v(P.xy()[0])
            sage: H = E.height_function()
            sage: H.psi(xP,v)
            0.867385122699931
            sage: H.psi(1.23,v)
            0.785854718241495
        """
        if xi > 1e9:
            return 1
        L = self.E.period_lattice(v)
        w1, w2 = L.basis()
        from sage.schemes.elliptic_curves.constructor import EllipticCurve
        ER = EllipticCurve([v(ai) for ai in self.E.a_invariants()])
        xP, yP = ER.lift_x(xi).xy()
        t = L.e_log_RC(xP,yP) / w1
        if t < 0.5:
            t = 1 - t
        return t

    def S(self, xi1, xi2, v):
        r"""
        Return the union of intervals `S^{(v)}(\xi_1,\xi_2)`.

        INPUT:

        - ``xi1, xi2`` (real) -- real numbers with `\xi_1\le\xi_2`.

        - ``v`` (embedding) -- a real embedding of the field.

        OUTPUT:

        The union of intervals `S^{(v)}(\xi_1,\xi_2)` defined in [Tho2010]_
        section 6.1.

        EXAMPLES:

        An example over `\QQ`::

            sage: E = EllipticCurve('389a')
            sage: v = QQ.places()[0]
            sage: H = E.height_function()
            sage: H.S(2,3,v)
            ([0.224512677391895, 0.274544821597130] U [0.725455178402870, 0.775487322608105])

        An example over a number field::

            sage: K.<a> = NumberField(x^3-2)
            sage: E = EllipticCurve([0,0,0,0,a])
            sage: v = K.real_places()[0]
            sage: H = E.height_function()
            sage: H.S(9,10,v)
            ([0.0781194447253472, 0.0823423732016403] U [0.917657626798360, 0.921880555274653])
        """
        L = self.E.period_lattice(v)
        w1, w2 = L.basis(prec = v.codomain().prec())
        beta = L.elliptic_exponential(w1/2)[0]
        if xi2 < beta:
            return UnionOfIntervals([])
        elif xi1 < beta <= xi2:
            a = self.psi(xi2, v)
            return UnionOfIntervals([1-a, a])
        else:
            a, b = self.psi(xi1, v), self.psi(xi2, v)
            return UnionOfIntervals([1-b, 1-a, a, b])

    def Sn(self, xi1, xi2, n, v):
        r"""
        Return the union of intervals `S_n^{(v)}(\xi_1,\xi_2)`.

        INPUT:

        - ``xi1, xi2`` (real) -- real numbers with `\xi_1\le\xi_2`.

        - ``n`` (integer) -- a positive integer.

        - ``v`` (embedding) -- a real embedding of the field.

        OUTPUT:

        The union of intervals `S_n^{(v)}(\xi_1,\xi_2)` defined in [Tho2010]_
        (Lemma 6.1).

        EXAMPLES:

        An example over `\QQ`::

            sage: E = EllipticCurve('389a')
            sage: v = QQ.places()[0]
            sage: H = E.height_function()
            sage: H.S(2,3,v) , H.Sn(2,3,1,v)
            (([0.224512677391895, 0.274544821597130] U [0.725455178402870, 0.775487322608105]),
            ([0.224512677391895, 0.274544821597130] U [0.725455178402870, 0.775487322608105]))
            sage: H.Sn(2,3,6,v)
            ([0.0374187795653158, 0.0457574702661884] U [0.120909196400478, 0.129247887101351] U [0.204085446231982, 0.212424136932855] U [0.287575863067145, 0.295914553768017] U [0.370752112898649, 0.379090803599522] U [0.454242529733812, 0.462581220434684] U [0.537418779565316, 0.545757470266188] U [0.620909196400478, 0.629247887101351] U [0.704085446231982, 0.712424136932855] U [0.787575863067145, 0.795914553768017] U [0.870752112898649, 0.879090803599522] U [0.954242529733812, 0.962581220434684])

        An example over a number field::

            sage: K.<a> = NumberField(x^3-2)
            sage: E = EllipticCurve([0,0,0,0,a])
            sage: v = K.real_places()[0]
            sage: H = E.height_function()
            sage: H.S(2,3,v) , H.Sn(2,3,1,v)
            (([0.142172065860075, 0.172845716928584] U [0.827154283071416, 0.857827934139925]),
            ([0.142172065860075, 0.172845716928584] U [0.827154283071416, 0.857827934139925]))
            sage: H.Sn(2,3,6,v)
            ([0.0236953443100124, 0.0288076194880974] U [0.137859047178569, 0.142971322356654] U [0.190362010976679, 0.195474286154764] U [0.304525713845236, 0.309637989023321] U [0.357028677643346, 0.362140952821431] U [0.471192380511903, 0.476304655689988] U [0.523695344310012, 0.528807619488097] U [0.637859047178569, 0.642971322356654] U [0.690362010976679, 0.695474286154764] U [0.804525713845236, 0.809637989023321] U [0.857028677643346, 0.862140952821431] U [0.971192380511903, 0.976304655689988])
        """
        SS = 1/ZZ(n) * self.S(xi1, xi2, v)
        return UnionOfIntervals.union([t/ZZ(n) + SS for t in range(n)])

    def real_intersection_is_empty(self, Bk, v):
        r"""
        Returns True iff an intersection of `S_n^{(v)}` sets is empty.

        INPUT:

        - ``Bk`` (list) -- a list of reals.

        - ``v`` (embedding) -- a real embedding of the number field.

        OUTPUT:

        True or False, according as the intersection of the unions of
        intervals `S_n^{(v)}(-b,b)` for `b` in the list ``Bk`` is
        empty or not.  When ``Bk`` is the list of `b=B_n(\mu)` for
        `n=1,2,3,\dots` for some `\mu>0` this means that all
        non-torsion points on `E` with everywhere good reduction have
        canonical height strictly greater than `\mu`, by [Tho2010]_,
        Proposition 6.2.

        EXAMPLES:

        An example over `\QQ`::

            sage: E = EllipticCurve('389a')
            sage: v = QQ.places()[0]
            sage: H = E.height_function()

        The following two lines prove that the heights of non-torsion
        points on `E` with everywhere good reduction have canonical
        height strictly greater than 0.2, but fail to prove the same
        for 0.3::

            sage: H.real_intersection_is_empty([H.B(n,0.2) for n in srange(1,10)],v)
            True
            sage: H.real_intersection_is_empty([H.B(n,0.3) for n in srange(1,10)],v)
            False

        An example over a number field::

            sage: K.<a> = NumberField(x^3-2)
            sage: E = EllipticCurve([0,0,0,0,a])
            sage: v = K.real_places()[0]
            sage: H = E.height_function()

        The following two lines prove that the heights of non-torsion
        points on `E` with everywhere good reduction have canonical
        height strictly greater than 0.07, but fail to prove the same
        for 0.08::

            sage: H.real_intersection_is_empty([H.B(n,0.07) for n in srange(1,5)],v) # long time (3.3s)
            True
            sage: H.real_intersection_is_empty([H.B(n,0.08) for n in srange(1,5)],v)
            False
        """
        return UnionOfIntervals.intersection([self.Sn(-B, B, k+1, v) for k,B in enumerate(Bk)]).is_empty()

    ########################################
    # Empty complex intersection detection.#
    ########################################

    def tau(self, v):
        r"""
        Return the normalised upper half-plane parameter `\tau` for
        the period lattice with respect to the embedding `v`.

        INPUT:

        - ``v`` (embedding) -- a real or complex embedding of the number field.

        OUTPUT:

        (Complex) `\tau = \omega_1/\omega_2` in the fundamental region
        of the upper half-plane.

        EXAMPLES::

            sage: E = EllipticCurve('37a')
            sage: H = E.height_function()
            sage: H.tau(QQ.places()[0])
            1.22112736076463*I
        """
        return self.E.period_lattice(v).tau()

    def wp_c(self, v):
        r"""
        Return a bound for the Weierstrass `\wp`-function.

        INPUT:

        - ``v`` (embedding) -- a real or complex embedding of the number field.

        OUTPUT:

        (Real) `c>0` such that

        .. MATH::

            |\wp(z) - z^-2| \le \frac{c^2|z|^2}{1-c|z|^2}

        whenever `c|z|^2<1`. Given the recurrence relations for the
        Laurent series expansion of `\wp`, it is easy to see that
        there is such a constant `c`.  [Reference?]

        EXAMPLES::

            sage: E = EllipticCurve('37a')
            sage: H = E.height_function()
            sage: H.wp_c(QQ.places()[0])
            2.68744508779950

            sage: K.<i>=QuadraticField(-1)
            sage: E = EllipticCurve([0,0,0,1+5*i,3+i])
            sage: H = E.height_function()
            sage: H.wp_c(K.places()[0])
            2.66213425640096
        """
        # Note that we normalise w1, w2 differently from [Tho2010]_!
        w2, w1 = self.E.period_lattice(v).normalised_basis()
        return max(abs(v(self.E.c4()/240)) ** 0.5,
                   abs(v(self.E.c6()/6048)) ** (1.0/3)) * abs(w1)**2

    def fk_intervals(self, v=None, N=20, domain=CIF):
        r"""
        Return a function approximating the Weierstrass function, with error.

        INPUT:

        - ``v`` (embedding) -- an embedding of the number field.  If
          None (default) use the real embedding if the field is `\QQ`
          and raise an error for other fields.

        - ``N`` (int) -- The number of terms to use in the
          `q`-expansion of `\wp`.

        - ``domain`` (complex field) -- the model of `\CC` to use, for
          example ``CDF`` of ``CIF`` (default).

        OUTPUT:

        A pair of functions fk, err which can be evaluated at complex
        numbers `z` (in the correct ``domain``) to give an
        approximation to `\wp(z)` and an upper bound on the error,
        respectively.  The Weierstrass function returned is with
        respect to the normalised lattice `[1,\tau]` associated to the
        given embedding.

        EXAMPLES::

            sage: E = EllipticCurve('37a')
            sage: L = E.period_lattice()
            sage: w1, w2 = L.normalised_basis()
            sage: z = CDF(0.3, 0.4)

        Compare the value give by the standard elliptic exponential
        (scaled since ``fk`` is with respect to the normalised
        lattice)::

            sage: L.elliptic_exponential(z*w2, to_curve=False)[0] * w2 ** 2
            -1.82543539306049 - 2.49336319992847*I

        to the value given by this function, and see the error::

            sage: fk, err = E.height_function().fk_intervals(N=10)
            sage: fk(CIF(z))
            -1.82543539306049? - 2.49336319992847?*I
            sage: err(CIF(z))
            2.71750621458744e-31

        The same, but in the domain ``CDF`` instead of ``CIF``::

            sage: fk, err = E.height_function().fk_intervals(N=10, domain=CDF)
            sage: fk(z)
            -1.8254353930604... - 2.493363199928...*I
        """
        if v is None:
            if self.K is QQ:
                v = QQ.hom(RR)
            else:
                raise ValueError("must specify embedding")
        # pre-compute some constants
        tau = self.tau(v)
        const_term = 1/CC(12)
        qn = q = (2 * CC.gen() * CC.pi() * tau).exp()
        for n in range(1, N):
            const_term -= 2 * qn/(1-qn) ** 2
            qn *= q

        two_pi_i = 2 * domain.gen() * domain.pi()
        neg_four_pi2 = -4 * domain.pi() ** 2
        const_term = domain(const_term)
        tau = domain(tau)

        abs_q = abs(domain(q))
        abs_qN = abs(domain(qn))
        err_factor = abs(neg_four_pi2) / (1-abs_q)
        err_term = 2*abs_qN/(1-abs_qN) ** 2

        # choose u/(1-u)^2 evaluation method
        if domain is CIF:
            rat_term = rat_term_CIF
        else:
            def rat_term(z):
                u = (two_pi_i*z).exp()
                return u/(1-u)**2

        # the actual series
        def fk(z):
            return (const_term +
                    sum([rat_term(z+n*tau) for n in range(1-N,N)])
                    ) * neg_four_pi2

        # the error function
        def err(z):
            alpha = z.imag() / tau.imag()
            qNa = abs_q**(N+alpha)
            qNai = abs_q**(N-alpha)
            return (err_factor * (qNa/(1-qNa) ** 2 + qNai/(1-qNai) ** 2 + err_term)).upper()

        return fk, err

    @cached_method
    def wp_intervals(self, v=None, N=20, abs_only=False):
        r"""
        Return a function approximating the Weierstrass function.

        INPUT:

        - ``v`` (embedding) -- an embedding of the number field.  If
          None (default) use the real embedding if the field is `\QQ`
          and raise an error for other fields.

        - ``N`` (int, default 20) -- The number of terms to use in the
          `q`-expansion of `\wp`.

        - ``abs_only`` (boolean, default False) -- flag to determine
          whether (if True) the error adjustment should use the
          absolute value or (if False) the real and imaginary parts.

        OUTPUT:

        A function wp which can be evaluated at complex numbers `z` to
        give an approximation to `\wp(z)`.  The Weierstrass function
        returned is with respect to the normalised lattice `[1,\tau]`
        associated to the given embedding.  For `z` which are not near
        a lattice point the function ``fk`` is used, otherwise a
        better approximation is used.

        EXAMPLES::

            sage: E = EllipticCurve('37a')
            sage: wp = E.height_function().wp_intervals()
            sage: z = CDF(0.3, 0.4)
            sage: wp(CIF(z))
            -1.82543539306049? - 2.4933631999285?*I

            sage: L = E.period_lattice()
            sage: w1, w2 = L.normalised_basis()
            sage: L.elliptic_exponential(z*w2, to_curve=False)[0] * w2^2
            -1.82543539306049 - 2.49336319992847*I

            sage: z = CDF(0.3, 0.1)
            sage: wp(CIF(z))
            8.5918243572165? - 5.4751982004351?*I
            sage: L.elliptic_exponential(z*w2, to_curve=False)[0] * w2^2
            8.59182435721650 - 5.47519820043503*I
        """
        if v is None:
            if self.K is QQ:
                v = QQ.hom(RR)
            else:
                raise ValueError("must specify embedding")

        tau = self.tau(v)
        fk, fk_err = self.fk_intervals(v, N)
        c = self.wp_c(v)

        def wp(z):

            # center around origin
            offset = (z.imag().lower() / tau.imag()).round()
            if offset:
                z -= CIF(offset * tau)
            offset = z.real().lower().round()
            if offset:
                z -= offset

            # estimate using the series
            approx = fk(z)
            err = fk_err(z)
            if abs_only:
                approx = abs(approx)
            approx += eps(err, abs_only)

            # refine using an estimate that's better near the pole
            z_bound = abs(z).upper()
            cz2 = c * z_bound ** 2
            if cz2 < 1:
                err = (c * cz2) / (1 - cz2)
                if abs_only:
                    pole_approx = abs(z) ** -2
                else:
                    pole_approx = z ** -2
    #            print "pole approx", pole_approx + eps(err, abs_only)
    #            print approx in approx.intersection(pole_approx + eps(err, abs_only))
                approx = approx.intersection(pole_approx + eps(err, abs_only))

            return approx

        return wp

    @cached_method
    def wp_on_grid(self, v, N, half=False):
        r"""
        Return an array of the values of `\wp` on an `N\times N` grid.

        INPUT:

        - ``v`` (embedding) -- an embedding of the number field.

        - ``N`` (int) -- The number of terms to use in the
          `q`-expansion of `\wp`.

        - ``half`` (boolean, default False) -- if True, use an array of
          size `N\times N/2` instead of `N\times N`.

        OUTPUT:

        An array of size either `N\times N/2` or `N\times N` whose
        `(i,j)` entry is the value of the Weierstrass `\wp`-function
        at `(i+.5)/N + (j+.5)*\tau/N`, a grid of points in the
        fundamental region for the lattice `[1,\tau]`.

        EXAMPLES::

            sage: E = EllipticCurve('37a')
            sage: H = E.height_function()
            sage: v = QQ.places()[0]

        The array of values on the grid shows symmetry, since `\wp` is
        even::

            sage: H.wp_on_grid(v,4)
            array([[25.43920182,  5.28760943,  5.28760943, 25.43920182],
                   [ 6.05099485,  1.83757786,  1.83757786,  6.05099485],
                   [ 6.05099485,  1.83757786,  1.83757786,  6.05099485],
                   [25.43920182,  5.28760943,  5.28760943, 25.43920182]])

        The array of values on the half-grid::

            sage: H.wp_on_grid(v,4,True)
            array([[25.43920182,  5.28760943],
                   [ 6.05099485,  1.83757786],
                   [ 6.05099485,  1.83757786],
                   [25.43920182,  5.28760943]])
        """
        tau = self.tau(v)
        fk, err = self.fk_intervals(v, 15, CDF)
        var_z = SR.var('z')
        ff = fast_callable(fk(var_z), CDF, [var_z])
        N_or_half = N // (1+half)         # array is NxN or Nx(N/2)
        vals = numpy.empty((N,N_or_half)) # empty array tp hold values
        for i in range(N):
            for j in range(N_or_half):
                vals[i,j] = abs(ff((i+.5)/N + (j+.5)*tau/N))
        return vals

    def complex_intersection_is_empty(self, Bk, v, verbose=False, use_half=True):
        r"""
        Returns True iff an intersection of `T_n^{(v)}` sets is empty.

        INPUT:

        - ``Bk`` (list) -- a list of reals.

        - ``v`` (embedding) -- a complex embedding of the number field.

        - ``verbose`` (boolean, default False) -- verbosity flag.

        - ``use_half`` (boolean, default False) -- if True, use only half
          the fundamental region.

        OUTPUT:

        True or False, according as the intersection of the unions of
        intervals `T_n^{(v)}(-b,b)` for `b` in the list ``Bk`` (see
        [Tho2010]_, section 7) is empty or not.  When ``Bk`` is the list of
        `b=\sqrt{B_n(\mu)}` for `n=1,2,3,\dots` for some `\mu>0` this
        means that all non-torsion points on `E` with everywhere good
        reduction have canonical height strictly greater than `\mu`,
        by [Tho2010]_, Proposition 7.8.

        EXAMPLES::

            sage: K.<a> = NumberField(x^3-2)
            sage: E = EllipticCurve([0,0,0,0,a])
            sage: v = K.complex_embeddings()[0]
            sage: H = E.height_function()

        The following two lines prove that the heights of non-torsion
        points on `E` with everywhere good reduction have canonical
        height strictly greater than 0.02, but fail to prove the same
        for 0.03.  For the first proof, using only `n=1,2,3` is not
        sufficient::

            sage: H.complex_intersection_is_empty([H.B(n,0.02) for n in [1,2,3]],v) # long time (~6s)
            False
            sage: H.complex_intersection_is_empty([H.B(n,0.02) for n in [1,2,3,4]],v)
            True
            sage: H.complex_intersection_is_empty([H.B(n,0.03) for n in [1,2,3,4]],v) # long time (4s)
            False

        Using `n\le6` enables us to prove the lower bound 0.03.  Note
        that it takes longer when the result is ``False`` than when it
        is ``True``::

            sage: H.complex_intersection_is_empty([H.B(n,0.03) for n in [1..6]],v)
            True
        """
        from sage.schemes.elliptic_curves.period_lattice_region import PeriodicRegion

        b2 = v(self.E.b2())
        # Note that we normalise w1, w2 differently from [Tho2010]_!
        w2, w1 = self.E.period_lattice(v).normalised_basis()
        tau = w2/w1
        bounds = [RDF((B.sqrt() + abs(b2)/12) * abs(w1) ** 2) for B in Bk]
        vals = self.wp_on_grid(v, 30, half=use_half)
        wp = self.wp_intervals(v, abs_only=True)

        k = len(bounds)

        # First try and prove a negative result (cheap).
        if verbose:
            print("trying to prove negative result...")
        intersection = None
        for B, n in sorted(zip(bounds, ZZ.range(1, k+1))):
            T = PeriodicRegion(CDF(1), CDF(tau), vals < B, full=not use_half)
            if intersection is None:
                intersection = PeriodicRegion(CDF(1), CDF(tau), vals < B, full=not use_half)
            else:
                intersection &= T/n
                if intersection.is_empty():
                    break
        else:
            z = CIF(intersection.innermost_point())
            if all(wp((k+1)*z).upper() < B for k, B in enumerate(bounds)):
                return False

        # Now try to prove a positive result.
        if verbose:
            print("trying to prove positive result...")
        intersection = None
        for B, n in sorted(zip(bounds, ZZ.range(1, k+1))):

            T = PeriodicRegion(CDF(1), CDF(tau), vals < B, full=not use_half).expand().refine()
            B = RIF(B)
            leaning_right = tau.real() / tau.imag() >= 0
            def check_line(z):
                wpz = wp(z)
                if wpz > B:
                    return True
                # Try refining once before we declare failure.
                z00, z01, z10, z11 = z.bisection()
                if leaning_right:
                    start, end = z00, z11
                else:
                    start, end = z01, z10
                if wp(start) > B and wp(end) > B:
                    return True
                return False

            # This step here is the bottleneck.
            while not T.verify(check_line):
                if verbose:
                    print("bad")
                T = T.expand()
            if intersection is None:
                intersection = T
            else:
                intersection &= T/n
                if intersection.is_empty():
                    return True

        return False

    def test_mu(self, mu, N, verbose=True):
        r"""
        Return ``True`` if we can prove that `\mu` is a lower bound.

        INPUT:

        - ``mu`` (real) -- a positive real number

        - ``N`` (integer) -- upper bound on the multiples to be used.

        - ``verbose`` (boolean, default True) -- verbosity flag.

        OUTPUT:

        ``True`` or ``False``, according to whether we succeed in
        proving that `\mu` is a lower bound for the canonical heights
        of points of infinite order with everywhere good reduction.

        .. NOTE::

            A ``True`` result is rigorous; ``False`` only means that
            the attempt failed: trying again with larger `N` may yield
            ``True``.

        EXAMPLES::

            sage: K.<a> = NumberField(x^3-2)
            sage: E = EllipticCurve([0,0,0,0,a])
            sage: H = E.height_function()

        This curve does have a point of good reduction whose canonical
        point is approximately 1.68::

            sage: P = E.gens(lim3=5)[0]; P
            (1/3*a^2 + a + 5/3 : -2*a^2 - 4/3*a - 5/3 : 1)
            sage: P.height()
            1.68038085233673
            sage: P.has_good_reduction()
            True

        Using `N=5` we can prove that 0.1 is a lower bound (in fact we
        only need `N=2`), but not that 0.2 is::

            sage: H.test_mu(0.1, 5)
            B_1(0.100000000000000) = 1.51580969677387
            B_2(0.100000000000000) = 0.932072561526720
            True
            sage: H.test_mu(0.2, 5)
            B_1(0.200000000000000) = 2.04612906979932
            B_2(0.200000000000000) = 3.09458988474327
            B_3(0.200000000000000) = 27.6251108409484
            B_4(0.200000000000000) = 1036.24722370223
            B_5(0.200000000000000) = 3.67090854562318e6
            False

        Since 0.1 is a lower bound we can deduce that the point `P` is
        either primitive or divisible by either 2 or 3.  In fact it is
        primitive::

            sage: (P.height()/0.1).sqrt()
            4.09924487233530
            sage: P.division_points(2)
            []
            sage: P.division_points(3)
            []
        """
        # Compute the list of values `B_n(\mu)` for n in 1..N.  If any
        # of these is 1 we can return True right away (see [Tho2010]_,
        # Proposition 5.1).
        Bk = []
        for n in ZZ.range(1, N + 1):
            b = self.B(n, mu)
            if verbose:
                print("B_%s(%s) = %s" % (n, mu, b))
            if b < 1:
                return True
            Bk.append(b)

        # Each real or complex embedding of the number field gives us
        # a chance to prove the lower bound.  We try each in turn,
        # stopping if one gives a True result.

        from sage.rings.number_field.number_field import refine_embedding
        for v in self.K.places():
            ok = False
            while not ok:
                try:
                    if v(self.K.gen()) in RR:
                        if self.real_intersection_is_empty(Bk, v):
                            return True
                    else:
                        if self.complex_intersection_is_empty(Bk, v):
                            return True
                    ok = True
                except ArithmeticError:
                    v = refine_embedding(v)
                    if verbose:
                        print("Refining embedding, codomain now {}".format(v.codomain()))
        return False # Couldn't prove it...

    def min_gr(self, tol, n_max, verbose=False):
        r"""
        Returns a lower bound for points of infinite order with good reduction.

        INPUT:

        - ``tol`` -- tolerance in output (see below).

        - ``n_max`` -- how many multiples to use in iteration.

        - ``verbose`` (boolean, default False) -- verbosity flag.

        OUTPUT:

        A positive real `\mu` for which it has been established
        rigorously that every point of infinite order on the elliptic
        curve (defined over its ground field), which has good
        reduction at all primes, has canonical height greater than
        `\mu`, and such that it is not possible (at least without
        increasing ``n_max``) to prove the same for
        `\mu\cdot\text{tol}`.

        EXAMPLES:

        Example 1 from [CS2006]_ (where a lower bound of 1.9865 was
        given)::

            sage: E = EllipticCurve([1, 0, 1, 421152067, 105484554028056]) # 60490d1
            sage: E.height_function().min_gr(.0001, 5)
            1.98684388146518

        Example 10.1 from [Tho2010]_ (where a lower bound of 0.18 was
        given)::

            sage: K.<i> = QuadraticField(-1)
            sage: E = EllipticCurve([0,0,0,91-26*i,-144-323*i])
            sage: H = E.height_function()
            sage: H.min_gr(0.1,4) # long time (8.1s)
            0.1621049443313762

        Example 10.2 from [Tho2010]_::

            sage: K.<i> = QuadraticField(-1)
            sage: E = EllipticCurve([0,1-i,i,-i,0])
            sage: H = E.height_function()
            sage: H.min_gr(0.01, 5)  # long time
            0.020153685521979152

        In this example the point `P=(0,0)` has height 0.023 so our
        lower bound is quite good::

            sage: P = E((0,0))
            sage: P.has_good_reduction()
            True
            sage: P.height()
            0.0230242154471211

        Example 10.3 from [Tho2010]_ (where the same bound of 0.25 is
        given)::

            sage: K.<a> = NumberField(x^3-2)
            sage: E = EllipticCurve([0,0,0,-3*a-a^2,a^2])
            sage: H = E.height_function()
            sage: H.min_gr(0.1,5) # long time (7.2s)
            0.25

        TESTS:

        This example from the LMFDB gave problems before the fix in :trac:`8829`::

            sage: K.<phi> = NumberField(x^2-x-1)
            sage: E = EllipticCurve([phi + 1, -phi + 1, 1, 20*phi - 39, 196*phi + 237])
            sage: H = E.height_function()
            sage: H.min_gr(.1, 5, verbose=True)  # long time (~22s)
            B_1(1) = 1540.199246369678
            ...
            halving mu to 0.25 and increasing n_max to 6
            ...
            halving mu to 0.001953125 and increasing n_max to 13
            doubling mu to 0.0078125
            doubling mu to 0.015625
            height bound in [0.0078125, 0.015625] using n_max = 13
            ...
            height bound in [0.012048522073499539, 0.01313900648833929] using n_max = 13
            0.012048522073499539
        """
        test = self.test_mu
        if test(1, n_max, verbose):
            mu = 2.0
            while test(mu, n_max, False):
                mu *= 2
            mu *= 0.5
        else:
            mu = 0.5
            while not test(mu, n_max, False):
                mu *= 0.5
                n_max += 1
                if verbose:
                    print("halving mu to %r and increasing n_max to %r" % (mu, n_max))
            # now we have (mu,n_max) which work we can try to increase
            # mu again using this larger n_max:
            mu *= 2
            while test(mu, n_max, False):
                mu *= 2
                if verbose:
                    print("doubling mu to %r" % mu)
            mu *= 0.5

        # The true value lies between mu and eps * mu.
        eps = 2.0
        while eps > tol + 1:
            if verbose:
                print("height bound in [%r, %r] using n_max = %r"
                      % (mu, mu * eps, n_max))
            eps = math.sqrt(eps)
            if test(mu * eps, n_max, False):
                mu = mu * eps
        if verbose:
            print("height bound in [%r, %r] using n_max = %r"
                  % (mu, mu * eps, n_max))
        return RDF(mu)

    def min(self, tol, n_max, verbose=False):
        r"""
        Returns a lower bound for all points of infinite order.

        INPUT:

        - ``tol`` -- tolerance in output (see below).

        - ``n_max`` -- how many multiples to use in iteration.

        - ``verbose`` (boolean, default False) -- verbosity flag.

        OUTPUT:

        A positive real `\mu` for which it has been established
        rigorously that every point of infinite order on the elliptic
        curve (defined over its ground field) has canonical height
        greater than `\mu`, and such that it is not possible (at least
        without increasing ``n_max``) to prove the same for
        `\mu\cdot\text{tol}`.

        EXAMPLES:

        Example 1 from [CS2006]_ (where the same lower bound of 0.1126 was
        given)::

            sage: E = EllipticCurve([1, 0, 1, 421152067, 105484554028056]) # 60490d1
            sage: E.height_function().min(.0001, 5)
            0.0011263287309893311

        Example 10.1 from [Tho2010]_ (where a lower bound of 0.18 was
        given)::

            sage: K.<i> = QuadraticField(-1)
            sage: E = EllipticCurve([0,0,0,91-26*i,-144-323*i])
            sage: H = E.height_function()
            sage: H.min(0.1,4) # long time (8.1s)
            0.1621049443313762

        Example 10.2 from [Tho2010]_::

            sage: K.<i> = QuadraticField(-1)
            sage: E = EllipticCurve([0,1-i,i,-i,0])
            sage: H = E.height_function()
            sage: H.min(0.01,5) # long time (4s)
            0.020153685521979152

        In this example the point `P=(0,0)` has height 0.023 so our
        lower bound is quite good::

            sage: P = E((0,0))
            sage: P.height()
            0.0230242154471211

        Example 10.3 from [Tho2010]_ (where the same bound of 0.0625 is
        given)::

            sage: K.<a> = NumberField(x^3-2)
            sage: E = EllipticCurve([0,0,0,-3*a-a^2,a^2])
            sage: H = E.height_function()
            sage: H.min(0.1,5) # long time (7s)
            0.0625

        More examples over `\QQ`::

            sage: E = EllipticCurve('37a')
            sage: h = E.height_function()
            sage: h.min(.01, 5)
            0.03987318057488725
            sage: E.gen(0).height()
            0.0511114082399688

        After base change the lower bound can decrease::

            sage: K.<a> = QuadraticField(-5)
            sage: E.change_ring(K).height_function().min(0.5, 10) # long time (8s)
            0.04419417382415922

            sage: E = EllipticCurve('389a')
            sage: h = E.height_function()
            sage: h.min(0.1, 5)
            0.05731275270029196
            sage: [P.height() for P in E.gens()]
            [0.686667083305587, 0.327000773651605]
        """
        # The lcm of the exponents of all the component groups at
        # finite places (allowing for everywhere good reduction!)
        tp = lcm([L.tamagawa_exponent() for L in self.E.local_data()] + [ZZ(1)])

        # Include infinite places:
        if tp % 2 == 1:
            if self.K == QQ:
                if self.E.real_components() == 2:
                    tp *= 2
            elif any(v(self.E.discriminant()) > 0
                     for v in self.K.real_places()):
                tp *= 2
        # Now tp is such that tp*P has good reduction at all places
        # for all points P:
        return self.min_gr(tol, n_max, verbose) / tp ** 2
Canonical heights for elliptic curves over number fields  Also, rigorous lower bounds for the canonical height of non-torsion points, implementing the algorithms in [CS2006]_ (over `\QQ`) and [Tho2010]_, which also refer to [CPS2006]_.  AUTHORS:  - <NAME> (2010): initial version  - <NAME> (2014): added many docstrings and doctests ############################################################################## #       Copyright (C) 2010 <NAME> <<EMAIL>> #                     2014 <NAME> <<EMAIL>> # #  Distributed under the terms of the GNU General Public License (GPL) # #    This code is distributed in the hope that it will be useful, #    but WITHOUT ANY WARRANTY; without even the implied warranty of #    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU #    General Public License for more details. # #  The full text of the GPL is available at: # #                  https://www.gnu.org/licenses/ ############################################################################## A class representing a finite union of closed intervals in     `\RR` which can be scaled, shifted, intersected, etc.      The intervals are represented as an ordered list of their     endpoints, which may include `-\infty` and `+\infty`.      EXAMPLES::          sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals         sage: R = UnionOfIntervals([1,2,3,infinity]); R         ([1, 2] U [3, +Infinity])         sage: R + 5         ([6, 7] U [8, +Infinity])         sage: ~R         ([-Infinity, 1] U [2, 3])         sage: ~R | (10*R + 100)         ([-Infinity, 1] U [2, 3] U [110, 120] U [130, +Infinity])      .. TODO::          Unify :class:`UnionOfIntervals` with the class ``RealSet``         introduced by :trac:`13125`; see :trac:`16063`. An union of intervals is initialized by giving an increasing list         of endpoints, the first of which may be `-\infty` and the last of         which may be `+\infty`.          EXAMPLES::              sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals             sage: UnionOfIntervals([0,1])             ([0, 1])             sage: UnionOfIntervals([-infinity, pi, 17, infinity])             ([-Infinity, pi] U [17, +Infinity])             sage: UnionOfIntervals([])             ()              sage: UnionOfIntervals([1])             Traceback (most recent call last):             ...             ValueError: an even number of endpoints must be given (got 1)             sage: UnionOfIntervals([3,2,1,0])             Traceback (most recent call last):             ...             ValueError: endpoints must be given in order Returns the finite endpoints of this union of intervals.          EXAMPLES::              sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals             sage: UnionOfIntervals([0,1]).finite_endpoints()             [0, 1]             sage: UnionOfIntervals([-infinity, 0, 1, infinity]).finite_endpoints()             [0, 1] Returns the intervals in self, as a list of 2-tuples.          EXAMPLES::              sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals             sage: UnionOfIntervals(list(range(10))).intervals()             [(0, 1), (2, 3), (4, 5), (6, 7), (8, 9)]             sage: UnionOfIntervals([-infinity, pi, 17, infinity]).intervals()             [(-Infinity, pi), (17, +Infinity)] Returns whether self is empty.          EXAMPLES::              sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals             sage: UnionOfIntervals([3,4]).is_empty()             False             sage: all = UnionOfIntervals([-infinity, infinity])             sage: all.is_empty()             False             sage: (~all).is_empty()             True             sage: A = UnionOfIntervals([0,1]) & UnionOfIntervals([2,3])             sage: A.is_empty()             True If both left an right are unions of intervals, take their union,         otherwise treat the non-union of intervals as a scalar and shift.          EXAMPLES::              sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals             sage: A = UnionOfIntervals([0, 1/2, 2, infinity]); A             ([0, 1/2] U [2, +Infinity])             sage: A + 1             ([1, 3/2] U [3, +Infinity])             sage: pi + A             ([pi, pi + 1/2] U [pi + 2, +Infinity])             sage: A + UnionOfIntervals([-infinity, -1])             ([-Infinity, -1] U [0, 1/2] U [2, +Infinity]) Scale a union of intervals on the left or right.          EXAMPLES::              sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals             sage: A = UnionOfIntervals([0, 1/2, 2, infinity]); A             ([0, 1/2] U [2, +Infinity])             sage: 2 * A             ([0, 1] U [4, +Infinity])             sage: A * 100             ([0, 50] U [200, +Infinity])             sage: 1.5 * A             ([0.000000000000000, 0.750000000000000] U [3.00000000000000, +Infinity]) Scale by an operand on the left.          TESTS::              sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals             sage: A = UnionOfIntervals([0, 1/2, 2, infinity]); A             ([0, 1/2] U [2, +Infinity])             sage: pi * A             ([0, 1/2*pi] U [2*pi, +Infinity]) Add a scalar operand on the left.          TESTS::              sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals             sage: A = UnionOfIntervals([0, 1/2, 2, infinity]); A             ([0, 1/2] U [2, +Infinity])             sage: 100 + A             ([100, 201/2] U [102, +Infinity]) Return the closure of the complement of self.          .. NOTE::              We take the closure because open intervals are not supported.          EXAMPLES::              sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals             sage: A = UnionOfIntervals([0, 1/2, 2, infinity]); A             ([0, 1/2] U [2, +Infinity])             sage: ~A             ([-Infinity, 0] U [1/2, 2])             sage: A | ~A             ([-Infinity, +Infinity])             sage: A & ~A             ([0, 0] U [1/2, 1/2] U [2, 2]) Utility function to form the union or intersection of a list of UnionOfIntervals.          INPUT:          - ``L`` (list) -- a list of UnionOfIntervals instances          - ``condition`` (function) -- either ``any`` or ``all``, or           some other boolean function of a list of boolean values.          OUTPUT:          A new UnionOfIntervals instance representing the subset of         '\RR' equal to those reals in any/all/condition of the         UnionOfIntervals in the list.          .. NOTE::              This is a static method for the class.          EXAMPLES::              sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals             sage: A = UnionOfIntervals([1,3,5,7]); A             ([1, 3] U [5, 7])             sage: B = A+1; B             ([2, 4] U [6, 8])             sage: A.join([A,B],any) # union             ([1, 4] U [5, 8])             sage: A.join([A,B],all) # intersection             ([2, 3] U [6, 7])             sage: A.join([A,B],sum) # symmetric difference             ([1, 2] U [3, 4] U [5, 6] U [7, 8]) Return the union of a list of UnionOfIntervals.          INPUT:          - ``L`` (list) -- a list of UnionOfIntervals instances          OUTPUT:          A new UnionOfIntervals instance representing the union of the         UnionOfIntervals in the list.          .. NOTE::              This is a class method.          EXAMPLES::              sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals             sage: A = UnionOfIntervals([1,3,5,7]); A             ([1, 3] U [5, 7])             sage: B = A+1; B             ([2, 4] U [6, 8])             sage: A.union([A,B])             ([1, 4] U [5, 8]) Return the intersection of a list of UnionOfIntervals.          INPUT:          - ``L`` (list) -- a list of UnionOfIntervals instances          OUTPUT:          A new UnionOfIntervals instance representing the intersection         of the UnionOfIntervals in the list.          .. NOTE::              This is a class method.          EXAMPLES::              sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals             sage: A = UnionOfIntervals([1,3,5,7]); A             ([1, 3] U [5, 7])             sage: B = A+1; B             ([2, 4] U [6, 8])             sage: A.intersection([A,B])             ([2, 3] U [6, 7]) Return the union of a two UnionOfIntervals instances.          INPUT:          - ``left``, ``right`` (UnionOfIntervals) -- two UnionOfIntervals instances          OUTPUT:          A new UnionOfIntervals instance representing the union of ``left`` and ``right``.          EXAMPLES::              sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals             sage: A = UnionOfIntervals([1,3,5,7]); A             ([1, 3] U [5, 7])             sage: B = A+1; B             ([2, 4] U [6, 8])             sage: A | B             ([1, 4] U [5, 8]) Return the intersection of a two UnionOfIntervals instances.          INPUT:          - ``left``, ``right`` (UnionOfIntervals) -- two UnionOfIntervals instances          OUTPUT:          A new UnionOfIntervals instance representing the intersection of ``left`` and ``right``.          EXAMPLES::              sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals             sage: A = UnionOfIntervals([1,3,5,7]); A             ([1, 3] U [5, 7])             sage: B = A+1; B             ([2, 4] U [6, 8])             sage: A & B             ([2, 3] U [6, 7]) Return True if ``x`` is in the UnionOfIntervals.          INPUT:          - ``x`` (real) -- a real number          OUTPUT:          Boolean: True if and only if ``x`` is in the union of intervals.          EXAMPLES::              sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals             sage: A = UnionOfIntervals([1,3,5,7]); A             ([1, 3] U [5, 7])             sage: 1 in A             True             sage: 4 in A             False             sage: -infinity in A             False Return the string representation of this UnionOfIntervals.          EXAMPLES::              sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals             sage: A = UnionOfIntervals([1,3,5,7])             sage: str(A)             '([1, 3] U [5, 7])' Return the string representation of this UnionOfIntervals.          EXAMPLES::              sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals             sage: A = UnionOfIntervals([1,3,5,7]); A             ([1, 3] U [5, 7]) Returns the UnionOfIntervals representing the region where ``f`` is non-negative.      INPUT:      - ``f`` (polynomial) -- a univariate polynomial over `\RR`.      OUTPUT:      A UnionOfIntervals representing the set `\{x \in\RR mid f(x) \ge 0\}`.      EXAMPLES::          sage: from sage.schemes.elliptic_curves.height import nonneg_region         sage: x = polygen(RR)         sage: nonneg_region(x^2-1)         ([-Infinity, -1.00000000000000] U [1.00000000000000, +Infinity])         sage: nonneg_region(1-x^2)         ([-1.00000000000000, 1.00000000000000])         sage: nonneg_region(1-x^3)         ([-Infinity, 1.00000000000000])         sage: nonneg_region(x^3-1)         ([1.00000000000000, +Infinity])         sage: nonneg_region((x-1)*(x-2))         ([-Infinity, 1.00000000000000] U [2.00000000000000, +Infinity])         sage: nonneg_region(-(x-1)*(x-2))         ([1.00000000000000, 2.00000000000000])         sage: nonneg_region((x-1)*(x-2)*(x-3))         ([1.00000000000000, 2.00000000000000] U [3.00000000000000, +Infinity])         sage: nonneg_region(-(x-1)*(x-2)*(x-3))         ([-Infinity, 1.00000000000000] U [2.00000000000000, 3.00000000000000])         sage: nonneg_region(x^4+1)         ([-Infinity, +Infinity])         sage: nonneg_region(-x^4-1)         () Returns `\inf_D(\max(|f|, |g|))`.      INPUT:      - ``f``, ``g`` (polynomials) -- real univariate polynomials      - ``D`` (UnionOfIntervals) -- a subset of `\RR`      OUTPUT:      A real number approximating the value of `\inf_D(\max(|f|, |g|))`.      ALGORITHM:      The extreme values must occur at an endpoint of a subinterval of     `D` or at a point where one of `f`, `f'`, `g`, `g'`, `f\pm g` is     zero.      EXAMPLES::          sage: from sage.schemes.elliptic_curves.height import inf_max_abs, UnionOfIntervals         sage: x = polygen(RR)         sage: f = (x-10)^4+1         sage: g = 2*x^3+100         sage: inf_max_abs(f,g,UnionOfIntervals([1,2,3,4,5,6]))         425.638201706391         sage: r0 = (f-g).roots()[0][0]         sage: r0         5.46053402234697         sage: max(abs(f(r0)),abs(g(r0)))         425.638201706391 # ignore multiplicities and points outside D # include endpoints of intervals Returns the minimum of a real-valued complex function on a square.      INPUT:      - ``f`` -- a function from CIF to RIF      - ``tol`` (real) -- a positive real number      - ``max_iter`` (integer, default 10000) -- a positive integer       bounding the number of iterations to be used      OUTPUT:      A 2-tuple `(s,t)`, where `t=f(s)` and `s` is a CIF element     contained in the disk `|z|\le1`, at which `f` takes its minimum     value.      EXAMPLES::          sage: from sage.schemes.elliptic_curves.height import min_on_disk         sage: f = lambda x: (x^2+100).abs()         sage: s, t = min_on_disk(f, 0.0001)         sage: s, f(s), t         (0.01? + 1.00?*I, 99.01?, 99.0000000000000) # L holds a list of 4-tuples (mfs, ds, s, in_disk) where s is a # subregion of the initial square, ds its relative diameter, # mfs=-f(s) (actually minus the lower bound on f(s)) and in_disk # is a flag indicating whether or not s is a subset of the unit # disk. # We store the negative of the lower bound on f(s) so that we can # use the bisect module to sort these 4-tuples. # Initially L contains one element, the whole unit box, which is # not contained in the unit square. # min_max holds the minimum over L of fs.upper(). # We iterate at most max_iter times.  At each step we look at the # best so far and return it if is good enough, meaning that its # relative diameter is less than the given tolerance; otherwise we # bisect this best region (into 4 pieces) and replace the entry in # L with at most 4 smaller entries. # reached desired tolerance, so return # 4 sub-regions # if the original region si in the disk so are all its children # otherwise we test each one # skip this subregion if it is entirely outside the disk # meaning it is entirely inside the disk # we definitely beat the record # discard the worse entries (if there are many) # we may beat the record, cannot yet tell: insert this region # into the list at the appropriate place to maintain sorting # If we get here, then even after max_iter iterations the tolerance has not been reached. # Ideas: We know tau, so we know the direction of the diagonal. # We can solve for x in p1, will this allow us to find the maxima exactly? Compute the value of `u/(1-u)^2` in ``CIF``, where `u=\exp(2\pi i z)`.      INPUT:      - ``z`` (complex) -- a CIF element      - ``try_strict`` (bool) -- flag      EXAMPLES::          sage: from sage.schemes.elliptic_curves.height import rat_term_CIF         sage: z = CIF(0.5,0.2)         sage: rat_term_CIF(z)         -0.172467461182437? + 0.?e-16*I         sage: rat_term_CIF(z, False)         -0.172467461182437? + 0.?e-16*I # = |u| # If there are no local minima the intervals are strictly # determined by their values at the endpoints. # evaluate the function at the four corners: # =  r^4 + 2*r^3*x - 6*r^2 + 2*r*x + 1 # = r^2*x + 2*r*x^2 - 4*r + x Return a Real or Complex interval centered on 0 with radius err.      INPUT:      - ``err`` (real) -- a positive real number, the radius of the interval      - ``is_real`` (boolean) -- if True, returns a real interval in       RIF, else a complex interval in CIF      OUTPUT:      An element of RIF or CIF (as specified), centered on 0, with given radius.      EXAMPLES::          sage: from sage.schemes.elliptic_curves.height import eps         sage: eps(0.01, True)         0.0?         sage: eps(0.01, False)         0.0? + 0.0?*I Class for computing canonical heights of points on elliptic curves     defined over number fields, including rigorous lower bounds for     the canonical height of non-torsion points.      EXAMPLES::          sage: from sage.schemes.elliptic_curves.height import EllipticCurveCanonicalHeight         sage: E = EllipticCurve([0,0,0,0,1])         sage: EllipticCurveCanonicalHeight(E)         EllipticCurveCanonicalHeight object associated to Elliptic Curve defined by y^2 = x^3 + 1 over Rational Field      Normally this object would be created like this::          sage: E.height_function()         EllipticCurveCanonicalHeight object associated to Elliptic Curve defined by y^2 = x^3 + 1 over Rational Field Initialize the class with an elliptic curve.          INPUT:          - `E` -- an elliptic curve defined over a number field          EXAMPLES::              sage: from sage.schemes.elliptic_curves.height import EllipticCurveCanonicalHeight             sage: E = EllipticCurve([0,0,0,0,1])             sage: EllipticCurveCanonicalHeight(E)             EllipticCurveCanonicalHeight object associated to Elliptic Curve defined by y^2 = x^3 + 1 over Rational Field          An example over a number field::              sage: K.<i>=QuadraticField(-1)             sage: E = EllipticCurve([0,i,0,i,i])             sage: EllipticCurveCanonicalHeight(E)             EllipticCurveCanonicalHeight object associated to Elliptic Curve defined by y^2 = x^3 + i*x^2 + i*x + i over Number Field in i with defining polynomial x^2 + 1 with i = 1*I          TESTS:          The base field must be a number field (or `\QQ`)::              sage: from sage.schemes.elliptic_curves.height import EllipticCurveCanonicalHeight             sage: E = EllipticCurve(GF(7),[0,0,0,0,1])             sage: EllipticCurveCanonicalHeight(E)             Traceback (most recent call last):             ...             ValueError: EllipticCurveCanonicalHeight class can only be created from an elliptic curve defined over a number field Return the string representation.          EXAMPLES::              sage: E = EllipticCurve([0,0,0,0,1])             sage: E.height_function()             EllipticCurveCanonicalHeight object associated to Elliptic Curve defined by y^2 = x^3 + 1 over Rational Field Return the elliptic curve.          EXAMPLES::              sage: E = EllipticCurve([0,0,0,0,1])             sage: H = E.height_function()             sage: H.curve()             Elliptic Curve defined by y^2 = x^3 + 1 over Rational Field Return the base field.          EXAMPLES::              sage: E = EllipticCurve([0,0,0,0,1])             sage: H = E.height_function()             sage: H.base_field()             Rational Field Return the canonical height of the point ``P``.          INPUT:          - ``P`` -- a point on the elliptic curve.          OUTPUT:          The canonical height of ``P``.          EXAMPLES::              sage: E = EllipticCurve([0,0,1,-1,0])             sage: P = E(0,0)             sage: P.height()             0.0511114082399688             sage: H = E.height_function()             sage: H(P)             0.0511114082399688             sage: H([0,0])             0.0511114082399688             sage: H((0,0))             0.0511114082399688          Over a number field other than `\QQ`::              sage: K.<i> = QuadraticField(-1)             sage: E = EllipticCurve(K, [0,0,0,1,-27])             sage: H = E.height_function()             sage: H.base_field()             Number Field in i with defining polynomial x^2 + 1 with i = 1*I             sage: H((1,5*i))             1.22257115164148 Return the constant `\alpha_v` associated to the embedding ``v``.          INPUT:          - ``v`` -- an embedding of the base field into `\RR` or `\CC`          OUTPUT:          The constant `\alpha_v`.  In the notation of [CPS2006]_ and         [Tho2010]_ (section 3.2), `\alpha_v^3=\epsilon_v`.  The result is         cached since it only depends on the curve.          EXAMPLES:          Example 1 from [CPS2006]_::              sage: K.<i>=QuadraticField(-1)             sage: E = EllipticCurve([0,0,0,1+5*i,3+i])             sage: H = E.height_function()             sage: alpha = H.alpha(K.places()[0])             sage: alpha             1.12272013439355          Compare with `\log(\epsilon_v)=0.344562...` in [CPS2006]_::              sage: 3*alpha.log()             0.347263296676126 # def pair_max(f, g): #     f = f.change_ring(CIF) #     g = g.change_ring(CIF) #     max = type(RIF(0)).max #     def max_f_g(z): #         return max(abs(f(z)), abs(g(z))) #     return max_f_g Return the exponent of the group over the residue field at ``p``.          INPUT:          - ``p`` -- a prime ideal of `K` (or a prime number if `K=\QQ`).          OUTPUT:          A positive integer `e_p`, the exponent of the group of         nonsingular points on the reduction of the elliptic curve         modulo `p`.  The result is cached.          EXAMPLES::              sage: K.<i>=QuadraticField(-1)             sage: E = EllipticCurve([0,0,0,1+5*i,3+i])             sage: H = E.height_function()             sage: H.e_p(K.prime_above(2))             2             sage: H.e_p(K.prime_above(3))             10             sage: H.e_p(K.prime_above(5))             9             sage: E.conductor().norm().factor()             2^10 * 20921             sage: p1, p2 = K.primes_above(20921)             sage: E.local_data(p1)             Local data at Fractional ideal (-40*i + 139):             Reduction type: bad split multiplicative             ...             sage: H.e_p(p1)             20920             sage: E.local_data(p2)             Local data at Fractional ideal (40*i + 139):             Reduction type: good             ...             sage: H.e_p(p2)             20815 Return the value `D_E(n)`.          INPUT:          - ``n`` (int) -- a positive integer          OUTPUT:          The value `D_E(n)` as defined in [Tho2010]_, section 4.          EXAMPLES::              sage: K.<i>=QuadraticField(-1)             sage: E = EllipticCurve([0,0,0,1+5*i,3+i])             sage: H = E.height_function()             sage: [H.DE(n) for n in srange(1,6)]             [0, 2*log(5) + 2*log(2), 0, 2*log(13) + 2*log(5) + 4*log(2), 0] Return the norm of the ideal `M_E`.          OUTPUT:          The norm of the ideal `M_E` as defined in [Tho2010]_, section 3.1.         This is `1` if `E` is a global minimal model, and in general         measures the non-minimality of `E`.          EXAMPLES::              sage: K.<i>=QuadraticField(-1)             sage: E = EllipticCurve([0,0,0,1+5*i,3+i])             sage: H = E.height_function()             sage: H.ME()             1             sage: E = EllipticCurve([0,0,0,0,1])             sage: E.height_function().ME()             1             sage: E = EllipticCurve([0,0,0,0,64])             sage: E.height_function().ME()             4096             sage: E.discriminant()/E.minimal_model().discriminant()             4096 Return the value `B_n(\mu)`.          INPUT:          - ``n`` (int) -- a positive integer          - ``mu`` (real) -- a positive real number          OUTPUT:          The real value `B_n(\mu)` as defined in [Tho2010]_, section 5.          EXAMPLES:          Example 10.2 from [Tho2010]_::              sage: K.<i>=QuadraticField(-1)             sage: E = EllipticCurve([0,1-i,i,-i,0])             sage: H = E.height_function()          In [Tho2010]_ the value is given as 0.772::              sage: RealField(12)( H.B(5, 0.01) )             0.777 ###################################### # Empty real intersection detection. # ###################################### Return the normalised elliptic log of a point with this x-coordinate.          INPUT:          - ``xi`` (real) -- the real x-coordinate of a point on the           curve in the connected component with respect to a real           embedding.          - ``v`` (embedding) -- a real embedding of the number field.          OUTPUT:          A real number in the interval [0.5,1] giving the elliptic         logarithm of a point on `E` with `x`-coordinate ``xi``, on the         connected component with respect to the embedding `v`, scaled         by the real period.          EXAMPLES:          An example over `\QQ`::              sage: E = EllipticCurve('389a')             sage: v = QQ.places()[0]             sage: L = E.period_lattice(v)             sage: P = E.lift_x(10/9)             sage: L(P)             1.53151606047462             sage: L(P) / L.real_period()             0.615014189772115             sage: H = E.height_function()             sage: H.psi(10/9,v)             0.615014189772115          An example over a number field::              sage: K.<a> = NumberField(x^3-2)             sage: E = EllipticCurve([0,0,0,0,a])             sage: P = E.lift_x(1/3*a^2 + a + 5/3)             sage: v = K.real_places()[0]             sage: L = E.period_lattice(v)             sage: L(P)             3.51086196882538             sage: L(P) / L.real_period()             0.867385122699931             sage: xP = v(P.xy()[0])             sage: H = E.height_function()             sage: H.psi(xP,v)             0.867385122699931             sage: H.psi(1.23,v)             0.785854718241495 Return the union of intervals `S^{(v)}(\xi_1,\xi_2)`.          INPUT:          - ``xi1, xi2`` (real) -- real numbers with `\xi_1\le\xi_2`.          - ``v`` (embedding) -- a real embedding of the field.          OUTPUT:          The union of intervals `S^{(v)}(\xi_1,\xi_2)` defined in [Tho2010]_         section 6.1.          EXAMPLES:          An example over `\QQ`::              sage: E = EllipticCurve('389a')             sage: v = QQ.places()[0]             sage: H = E.height_function()             sage: H.S(2,3,v)             ([0.224512677391895, 0.274544821597130] U [0.725455178402870, 0.775487322608105])          An example over a number field::              sage: K.<a> = NumberField(x^3-2)             sage: E = EllipticCurve([0,0,0,0,a])             sage: v = K.real_places()[0]             sage: H = E.height_function()             sage: H.S(9,10,v)             ([0.0781194447253472, 0.0823423732016403] U [0.917657626798360, 0.921880555274653]) Return the union of intervals `S_n^{(v)}(\xi_1,\xi_2)`.          INPUT:          - ``xi1, xi2`` (real) -- real numbers with `\xi_1\le\xi_2`.          - ``n`` (integer) -- a positive integer.          - ``v`` (embedding) -- a real embedding of the field.          OUTPUT:          The union of intervals `S_n^{(v)}(\xi_1,\xi_2)` defined in [Tho2010]_         (Lemma 6.1).          EXAMPLES:          An example over `\QQ`::              sage: E = EllipticCurve('389a')             sage: v = QQ.places()[0]             sage: H = E.height_function()             sage: H.S(2,3,v) , H.Sn(2,3,1,v)             (([0.224512677391895, 0.274544821597130] U [0.725455178402870, 0.775487322608105]),             ([0.224512677391895, 0.274544821597130] U [0.725455178402870, 0.775487322608105]))             sage: H.Sn(2,3,6,v)             ([0.0374187795653158, 0.0457574702661884] U [0.120909196400478, 0.129247887101351] U [0.204085446231982, 0.212424136932855] U [0.287575863067145, 0.295914553768017] U [0.370752112898649, 0.379090803599522] U [0.454242529733812, 0.462581220434684] U [0.537418779565316, 0.545757470266188] U [0.620909196400478, 0.629247887101351] U [0.704085446231982, 0.712424136932855] U [0.787575863067145, 0.795914553768017] U [0.870752112898649, 0.879090803599522] U [0.954242529733812, 0.962581220434684])          An example over a number field::              sage: K.<a> = NumberField(x^3-2)             sage: E = EllipticCurve([0,0,0,0,a])             sage: v = K.real_places()[0]             sage: H = E.height_function()             sage: H.S(2,3,v) , H.Sn(2,3,1,v)             (([0.142172065860075, 0.172845716928584] U [0.827154283071416, 0.857827934139925]),             ([0.142172065860075, 0.172845716928584] U [0.827154283071416, 0.857827934139925]))             sage: H.Sn(2,3,6,v)             ([0.0236953443100124, 0.0288076194880974] U [0.137859047178569, 0.142971322356654] U [0.190362010976679, 0.195474286154764] U [0.304525713845236, 0.309637989023321] U [0.357028677643346, 0.362140952821431] U [0.471192380511903, 0.476304655689988] U [0.523695344310012, 0.528807619488097] U [0.637859047178569, 0.642971322356654] U [0.690362010976679, 0.695474286154764] U [0.804525713845236, 0.809637989023321] U [0.857028677643346, 0.862140952821431] U [0.971192380511903, 0.976304655689988]) Returns True iff an intersection of `S_n^{(v)}` sets is empty.          INPUT:          - ``Bk`` (list) -- a list of reals.          - ``v`` (embedding) -- a real embedding of the number field.          OUTPUT:          True or False, according as the intersection of the unions of         intervals `S_n^{(v)}(-b,b)` for `b` in the list ``Bk`` is         empty or not.  When ``Bk`` is the list of `b=B_n(\mu)` for         `n=1,2,3,\dots` for some `\mu>0` this means that all         non-torsion points on `E` with everywhere good reduction have         canonical height strictly greater than `\mu`, by [Tho2010]_,         Proposition 6.2.          EXAMPLES:          An example over `\QQ`::              sage: E = EllipticCurve('389a')             sage: v = QQ.places()[0]             sage: H = E.height_function()          The following two lines prove that the heights of non-torsion         points on `E` with everywhere good reduction have canonical         height strictly greater than 0.2, but fail to prove the same         for 0.3::              sage: H.real_intersection_is_empty([H.B(n,0.2) for n in srange(1,10)],v)             True             sage: H.real_intersection_is_empty([H.B(n,0.3) for n in srange(1,10)],v)             False          An example over a number field::              sage: K.<a> = NumberField(x^3-2)             sage: E = EllipticCurve([0,0,0,0,a])             sage: v = K.real_places()[0]             sage: H = E.height_function()          The following two lines prove that the heights of non-torsion         points on `E` with everywhere good reduction have canonical         height strictly greater than 0.07, but fail to prove the same         for 0.08::              sage: H.real_intersection_is_empty([H.B(n,0.07) for n in srange(1,5)],v) # long time (3.3s)             True             sage: H.real_intersection_is_empty([H.B(n,0.08) for n in srange(1,5)],v)             False ######################################## # Empty complex intersection detection.# ######################################## Return the normalised upper half-plane parameter `\tau` for         the period lattice with respect to the embedding `v`.          INPUT:          - ``v`` (embedding) -- a real or complex embedding of the number field.          OUTPUT:          (Complex) `\tau = \omega_1/\omega_2` in the fundamental region         of the upper half-plane.          EXAMPLES::              sage: E = EllipticCurve('37a')             sage: H = E.height_function()             sage: H.tau(QQ.places()[0])             1.22112736076463*I Return a bound for the Weierstrass `\wp`-function.          INPUT:          - ``v`` (embedding) -- a real or complex embedding of the number field.          OUTPUT:          (Real) `c>0` such that          .. MATH::              |\wp(z) - z^-2| \le \frac{c^2|z|^2}{1-c|z|^2}          whenever `c|z|^2<1`. Given the recurrence relations for the         Laurent series expansion of `\wp`, it is easy to see that         there is such a constant `c`.  [Reference?]          EXAMPLES::              sage: E = EllipticCurve('37a')             sage: H = E.height_function()             sage: H.wp_c(QQ.places()[0])             2.68744508779950              sage: K.<i>=QuadraticField(-1)             sage: E = EllipticCurve([0,0,0,1+5*i,3+i])             sage: H = E.height_function()             sage: H.wp_c(K.places()[0])             2.66213425640096 # Note that we normalise w1, w2 differently from [Tho2010]_! Return a function approximating the Weierstrass function, with error.          INPUT:          - ``v`` (embedding) -- an embedding of the number field.  If           None (default) use the real embedding if the field is `\QQ`           and raise an error for other fields.          - ``N`` (int) -- The number of terms to use in the           `q`-expansion of `\wp`.          - ``domain`` (complex field) -- the model of `\CC` to use, for           example ``CDF`` of ``CIF`` (default).          OUTPUT:          A pair of functions fk, err which can be evaluated at complex         numbers `z` (in the correct ``domain``) to give an         approximation to `\wp(z)` and an upper bound on the error,         respectively.  The Weierstrass function returned is with         respect to the normalised lattice `[1,\tau]` associated to the         given embedding.          EXAMPLES::              sage: E = EllipticCurve('37a')             sage: L = E.period_lattice()             sage: w1, w2 = L.normalised_basis()             sage: z = CDF(0.3, 0.4)          Compare the value give by the standard elliptic exponential         (scaled since ``fk`` is with respect to the normalised         lattice)::              sage: L.elliptic_exponential(z*w2, to_curve=False)[0] * w2 ** 2             -1.82543539306049 - 2.49336319992847*I          to the value given by this function, and see the error::              sage: fk, err = E.height_function().fk_intervals(N=10)             sage: fk(CIF(z))             -1.82543539306049? - 2.49336319992847?*I             sage: err(CIF(z))             2.71750621458744e-31          The same, but in the domain ``CDF`` instead of ``CIF``::              sage: fk, err = E.height_function().fk_intervals(N=10, domain=CDF)             sage: fk(z)             -1.8254353930604... - 2.493363199928...*I # pre-compute some constants # choose u/(1-u)^2 evaluation method # the actual series # the error function Return a function approximating the Weierstrass function.          INPUT:          - ``v`` (embedding) -- an embedding of the number field.  If           None (default) use the real embedding if the field is `\QQ`           and raise an error for other fields.          - ``N`` (int, default 20) -- The number of terms to use in the           `q`-expansion of `\wp`.          - ``abs_only`` (boolean, default False) -- flag to determine           whether (if True) the error adjustment should use the           absolute value or (if False) the real and imaginary parts.          OUTPUT:          A function wp which can be evaluated at complex numbers `z` to         give an approximation to `\wp(z)`.  The Weierstrass function         returned is with respect to the normalised lattice `[1,\tau]`         associated to the given embedding.  For `z` which are not near         a lattice point the function ``fk`` is used, otherwise a         better approximation is used.          EXAMPLES::              sage: E = EllipticCurve('37a')             sage: wp = E.height_function().wp_intervals()             sage: z = CDF(0.3, 0.4)             sage: wp(CIF(z))             -1.82543539306049? - 2.4933631999285?*I              sage: L = E.period_lattice()             sage: w1, w2 = L.normalised_basis()             sage: L.elliptic_exponential(z*w2, to_curve=False)[0] * w2^2             -1.82543539306049 - 2.49336319992847*I              sage: z = CDF(0.3, 0.1)             sage: wp(CIF(z))             8.5918243572165? - 5.4751982004351?*I             sage: L.elliptic_exponential(z*w2, to_curve=False)[0] * w2^2             8.59182435721650 - 5.47519820043503*I # center around origin # estimate using the series # refine using an estimate that's better near the pole #            print "pole approx", pole_approx + eps(err, abs_only) #            print approx in approx.intersection(pole_approx + eps(err, abs_only)) Return an array of the values of `\wp` on an `N\times N` grid.          INPUT:          - ``v`` (embedding) -- an embedding of the number field.          - ``N`` (int) -- The number of terms to use in the           `q`-expansion of `\wp`.          - ``half`` (boolean, default False) -- if True, use an array of           size `N\times N/2` instead of `N\times N`.          OUTPUT:          An array of size either `N\times N/2` or `N\times N` whose         `(i,j)` entry is the value of the Weierstrass `\wp`-function         at `(i+.5)/N + (j+.5)*\tau/N`, a grid of points in the         fundamental region for the lattice `[1,\tau]`.          EXAMPLES::              sage: E = EllipticCurve('37a')             sage: H = E.height_function()             sage: v = QQ.places()[0]          The array of values on the grid shows symmetry, since `\wp` is         even::              sage: H.wp_on_grid(v,4)             array([[25.43920182,  5.28760943,  5.28760943, 25.43920182],                    [ 6.05099485,  1.83757786,  1.83757786,  6.05099485],                    [ 6.05099485,  1.83757786,  1.83757786,  6.05099485],                    [25.43920182,  5.28760943,  5.28760943, 25.43920182]])          The array of values on the half-grid::              sage: H.wp_on_grid(v,4,True)             array([[25.43920182,  5.28760943],                    [ 6.05099485,  1.83757786],                    [ 6.05099485,  1.83757786],                    [25.43920182,  5.28760943]]) # array is NxN or Nx(N/2) # empty array tp hold values Returns True iff an intersection of `T_n^{(v)}` sets is empty.          INPUT:          - ``Bk`` (list) -- a list of reals.          - ``v`` (embedding) -- a complex embedding of the number field.          - ``verbose`` (boolean, default False) -- verbosity flag.          - ``use_half`` (boolean, default False) -- if True, use only half           the fundamental region.          OUTPUT:          True or False, according as the intersection of the unions of         intervals `T_n^{(v)}(-b,b)` for `b` in the list ``Bk`` (see         [Tho2010]_, section 7) is empty or not.  When ``Bk`` is the list of         `b=\sqrt{B_n(\mu)}` for `n=1,2,3,\dots` for some `\mu>0` this         means that all non-torsion points on `E` with everywhere good         reduction have canonical height strictly greater than `\mu`,         by [Tho2010]_, Proposition 7.8.          EXAMPLES::              sage: K.<a> = NumberField(x^3-2)             sage: E = EllipticCurve([0,0,0,0,a])             sage: v = K.complex_embeddings()[0]             sage: H = E.height_function()          The following two lines prove that the heights of non-torsion         points on `E` with everywhere good reduction have canonical         height strictly greater than 0.02, but fail to prove the same         for 0.03.  For the first proof, using only `n=1,2,3` is not         sufficient::              sage: H.complex_intersection_is_empty([H.B(n,0.02) for n in [1,2,3]],v) # long time (~6s)             False             sage: H.complex_intersection_is_empty([H.B(n,0.02) for n in [1,2,3,4]],v)             True             sage: H.complex_intersection_is_empty([H.B(n,0.03) for n in [1,2,3,4]],v) # long time (4s)             False          Using `n\le6` enables us to prove the lower bound 0.03.  Note         that it takes longer when the result is ``False`` than when it         is ``True``::              sage: H.complex_intersection_is_empty([H.B(n,0.03) for n in [1..6]],v)             True # Note that we normalise w1, w2 differently from [Tho2010]_! # First try and prove a negative result (cheap). # Now try to prove a positive result. # Try refining once before we declare failure. # This step here is the bottleneck. Return ``True`` if we can prove that `\mu` is a lower bound.          INPUT:          - ``mu`` (real) -- a positive real number          - ``N`` (integer) -- upper bound on the multiples to be used.          - ``verbose`` (boolean, default True) -- verbosity flag.          OUTPUT:          ``True`` or ``False``, according to whether we succeed in         proving that `\mu` is a lower bound for the canonical heights         of points of infinite order with everywhere good reduction.          .. NOTE::              A ``True`` result is rigorous; ``False`` only means that             the attempt failed: trying again with larger `N` may yield             ``True``.          EXAMPLES::              sage: K.<a> = NumberField(x^3-2)             sage: E = EllipticCurve([0,0,0,0,a])             sage: H = E.height_function()          This curve does have a point of good reduction whose canonical         point is approximately 1.68::              sage: P = E.gens(lim3=5)[0]; P             (1/3*a^2 + a + 5/3 : -2*a^2 - 4/3*a - 5/3 : 1)             sage: P.height()             1.68038085233673             sage: P.has_good_reduction()             True          Using `N=5` we can prove that 0.1 is a lower bound (in fact we         only need `N=2`), but not that 0.2 is::              sage: H.test_mu(0.1, 5)             B_1(0.100000000000000) = 1.51580969677387             B_2(0.100000000000000) = 0.932072561526720             True             sage: H.test_mu(0.2, 5)             B_1(0.200000000000000) = 2.04612906979932             B_2(0.200000000000000) = 3.09458988474327             B_3(0.200000000000000) = 27.6251108409484             B_4(0.200000000000000) = 1036.24722370223             B_5(0.200000000000000) = 3.67090854562318e6             False          Since 0.1 is a lower bound we can deduce that the point `P` is         either primitive or divisible by either 2 or 3.  In fact it is         primitive::              sage: (P.height()/0.1).sqrt()             4.09924487233530             sage: P.division_points(2)             []             sage: P.division_points(3)             [] # Compute the list of values `B_n(\mu)` for n in 1..N.  If any # of these is 1 we can return True right away (see [Tho2010]_, # Proposition 5.1). # Each real or complex embedding of the number field gives us # a chance to prove the lower bound.  We try each in turn, # stopping if one gives a True result. # Couldn't prove it... Returns a lower bound for points of infinite order with good reduction.          INPUT:          - ``tol`` -- tolerance in output (see below).          - ``n_max`` -- how many multiples to use in iteration.          - ``verbose`` (boolean, default False) -- verbosity flag.          OUTPUT:          A positive real `\mu` for which it has been established         rigorously that every point of infinite order on the elliptic         curve (defined over its ground field), which has good         reduction at all primes, has canonical height greater than         `\mu`, and such that it is not possible (at least without         increasing ``n_max``) to prove the same for         `\mu\cdot\text{tol}`.          EXAMPLES:          Example 1 from [CS2006]_ (where a lower bound of 1.9865 was         given)::              sage: E = EllipticCurve([1, 0, 1, 421152067, 105484554028056]) # 60490d1             sage: E.height_function().min_gr(.0001, 5)             1.98684388146518          Example 10.1 from [Tho2010]_ (where a lower bound of 0.18 was         given)::              sage: K.<i> = QuadraticField(-1)             sage: E = EllipticCurve([0,0,0,91-26*i,-144-323*i])             sage: H = E.height_function()             sage: H.min_gr(0.1,4) # long time (8.1s)             0.1621049443313762          Example 10.2 from [Tho2010]_::              sage: K.<i> = QuadraticField(-1)             sage: E = EllipticCurve([0,1-i,i,-i,0])             sage: H = E.height_function()             sage: H.min_gr(0.01, 5)  # long time             0.020153685521979152          In this example the point `P=(0,0)` has height 0.023 so our         lower bound is quite good::              sage: P = E((0,0))             sage: P.has_good_reduction()             True             sage: P.height()             0.0230242154471211          Example 10.3 from [Tho2010]_ (where the same bound of 0.25 is         given)::              sage: K.<a> = NumberField(x^3-2)             sage: E = EllipticCurve([0,0,0,-3*a-a^2,a^2])             sage: H = E.height_function()             sage: H.min_gr(0.1,5) # long time (7.2s)             0.25          TESTS:          This example from the LMFDB gave problems before the fix in :trac:`8829`::              sage: K.<phi> = NumberField(x^2-x-1)             sage: E = EllipticCurve([phi + 1, -phi + 1, 1, 20*phi - 39, 196*phi + 237])             sage: H = E.height_function()             sage: H.min_gr(.1, 5, verbose=True)  # long time (~22s)             B_1(1) = 1540.199246369678             ...             halving mu to 0.25 and increasing n_max to 6             ...             halving mu to 0.001953125 and increasing n_max to 13             doubling mu to 0.0078125             doubling mu to 0.015625             height bound in [0.0078125, 0.015625] using n_max = 13             ...             height bound in [0.012048522073499539, 0.01313900648833929] using n_max = 13             0.012048522073499539 # now we have (mu,n_max) which work we can try to increase # mu again using this larger n_max: # The true value lies between mu and eps * mu. Returns a lower bound for all points of infinite order.          INPUT:          - ``tol`` -- tolerance in output (see below).          - ``n_max`` -- how many multiples to use in iteration.          - ``verbose`` (boolean, default False) -- verbosity flag.          OUTPUT:          A positive real `\mu` for which it has been established         rigorously that every point of infinite order on the elliptic         curve (defined over its ground field) has canonical height         greater than `\mu`, and such that it is not possible (at least         without increasing ``n_max``) to prove the same for         `\mu\cdot\text{tol}`.          EXAMPLES:          Example 1 from [CS2006]_ (where the same lower bound of 0.1126 was         given)::              sage: E = EllipticCurve([1, 0, 1, 421152067, 105484554028056]) # 60490d1             sage: E.height_function().min(.0001, 5)             0.0011263287309893311          Example 10.1 from [Tho2010]_ (where a lower bound of 0.18 was         given)::              sage: K.<i> = QuadraticField(-1)             sage: E = EllipticCurve([0,0,0,91-26*i,-144-323*i])             sage: H = E.height_function()             sage: H.min(0.1,4) # long time (8.1s)             0.1621049443313762          Example 10.2 from [Tho2010]_::              sage: K.<i> = QuadraticField(-1)             sage: E = EllipticCurve([0,1-i,i,-i,0])             sage: H = E.height_function()             sage: H.min(0.01,5) # long time (4s)             0.020153685521979152          In this example the point `P=(0,0)` has height 0.023 so our         lower bound is quite good::              sage: P = E((0,0))             sage: P.height()             0.0230242154471211          Example 10.3 from [Tho2010]_ (where the same bound of 0.0625 is         given)::              sage: K.<a> = NumberField(x^3-2)             sage: E = EllipticCurve([0,0,0,-3*a-a^2,a^2])             sage: H = E.height_function()             sage: H.min(0.1,5) # long time (7s)             0.0625          More examples over `\QQ`::              sage: E = EllipticCurve('37a')             sage: h = E.height_function()             sage: h.min(.01, 5)             0.03987318057488725             sage: E.gen(0).height()             0.0511114082399688          After base change the lower bound can decrease::              sage: K.<a> = QuadraticField(-5)             sage: E.change_ring(K).height_function().min(0.5, 10) # long time (8s)             0.04419417382415922              sage: E = EllipticCurve('389a')             sage: h = E.height_function()             sage: h.min(0.1, 5)             0.05731275270029196             sage: [P.height() for P in E.gens()]             [0.686667083305587, 0.327000773651605] # The lcm of the exponents of all the component groups at # finite places (allowing for everywhere good reduction!) # Include infinite places: # Now tp is such that tp*P has good reduction at all places # for all points P: