# Rational points in the moduli space of genus two

L. Beshaj  Department of Mathematics
The University of Texas at Austin, Austin, TX 78712<EMAIL_ADDRESS>,
R. Hidalgo  Departamento de Matemática y Estadística, Universidad de La
Frontera, Temuco, Chile<EMAIL_ADDRESS>,  S. Kruk  Department of
Mathematics and Statistics
Oakland University
Mathematics and Science Center,
Rochester, MI 48309<EMAIL_ADDRESS>,  A. Malmendier  Department of
Mathematics
Utah State University, Logan, UT 84322<EMAIL_ADDRESS>,  S.
Quispe  Departamento de Matemática y Estadística,
Universidad de La Frontera, Temuco, Chile<EMAIL_ADDRESS>and  T.
Shaska  Department of Mathematics and Statistics
Oakland University
Mathematics and Science Center,
Rochester, MI 48309<EMAIL_ADDRESS>
###### Abstract.

We build a database of genus 2 curves defined over $\mathbb{Q}$ which contains
all curves with minimal absolute height $\operatorname{h}\leq 5$, all curves
with moduli height $\operatorname{\mathfrak{h}}\leq 20$, and all curves with
extra automorphisms in standard form $y^{2}=f(x^{2})$ defined over
$\mathbb{Q}$ with height $\operatorname{h}\leq 101$. For each isomorphism
class in the database, an equation over its minimal field of definition is
provided, the automorphism group of the curve, Clebsch and Igusa invariants.
The distribution of rational points in the moduli space ${\mathcal{M}}_{2}$
for which the field of moduli is a field of definition is discussed and some
open problems are presented.

##  1\. Introduction

In [MR3525576] were introduced concepts of minimum absolute height of a binary
form, the moduli height, and discussed relations between the two. Moreover,
some computations were performed for binary sextics of minimum absolute height
one. A natural problem in that paper was to check whether a large database of
binary forms (equivalently genus 2 curves) of relatively small minimum
absolute height could be constructed. Comparing the minimum absolute height
with the moduli height would be the main point of this database. Moreover,
such a database would shed some light to other problems related to the
rational points of the moduli space ${\mathcal{M}}_{2}$. For example, not
every rational point $\mathfrak{p}\in{\mathcal{M}}_{2}$ has a representative
genus 2 curve $\mathcal{X}$ defined over $\mathbb{Q}$. What is the percentage
of points of bounded height for which the field of moduli is not a field of
definition? How does this ratio is affected when the height increases?

From another point of view, we can list genus two curves based on their moduli
height. For example, traditionally there have been plenty of effort to count
the curves with bounded discriminant. That is not an easy problem, but would
it make more sense to have an estimate on the number of curves with bounded
moduli height? After all, the moduli height is the most natural way of sorting
points in ${\mathcal{M}}_{2}$.

And then, there are also the curves with automorphisms. In [Beshaj:2016ac] it
was shown that such curves can always be written in an equation such that the
corresponding binary sextic is reduced; see [MR3525574] for details. Reduced
usually means minimal absolute height for the binary form. Is this really
supported computationally?

Our goal was to construct a database of genus 2 curves which addresses some of
these questions.

In this paper we construct three main databases:

  * i)

integral binary sextics of minimum absolute height $\leq 4$,

  * ii)

integral binary sextics with moduli height $\mathfrak{h}\leq 20$,

  * iii)

integral binary sextics $f(x^{2},z^{2})$ of height $\operatorname{h}\leq 101$.

All combined we have over 1 million isomorphism classes of genus two curves
(equivalently rational points in ${\mathcal{M}}_{2}$) without counting twists.
We compute the minimal absolute height, the moduli height, the discriminant,
automorphism group, field of definition, and an equation over the field of
definition for all such points. We discuss all the technical details of
organizing the data and some open questions on the distribution of the
rational points with non-trivial obstruction in the moduli space
${\mathcal{M}}_{2}$.

Let $k$ be an algebraically closed field of characteristic zero and
${\mathcal{M}}_{g}$ the moduli space of smooth, projective, genus $g\geq 2$
algebraic curves defined over $k$. The moduli space ${\mathcal{M}}_{2}$ of
genus 2 curves is the most understood moduli space among all moduli spaces.
This is mostly due to two main facts; first all genus two curves are
hyperelliptic and therefore studying them it is easier than general curves,
secondly even among hyperelliptic curves the curves of genus two have a
special place since they correspond to binary sextics which, from the
computational point of view, are relatively well understood compared to higher
degree binary forms.

Some of the main questions related to ${\mathcal{M}}_{2}$ have been to recover
a nice equation for any point $\mathfrak{p}\in{\mathcal{M}}_{2}$. Since
${\mathcal{M}}_{2}$ is a coarse moduli space, such equation is not always
defined over the field of moduli of $\mathfrak{p}$. Can we find a universal
equation for genus two curves over their minimal field of definition? Can such
equation provide a minimal model for the curve? Does the height of this
minimal model has any relation to the projective height of the corresponding
moduli point $\mathfrak{p}\in{\mathcal{M}}_{2}$? What is the distribution in
${\mathcal{M}}_{2}$ of points $\mathfrak{p}$ for which the field of moduli is
not a field of definition? The answers to these questions are still unknown.

In [data] we provide a computational package for computing with genus 2 curves
and a database of genus 2 curves which contains all curves with height
$\operatorname{h}\leq 5$, curves with moduli height $\mathfrak{h}\leq 20$, and
curves with automorphism and height $\leq 101$. They are organized in three
Python directories $\mathcal{L}_{i}$, $i=1,2,3$ as explained in Section 2.

The database is build with the idea of better understanding
${\mathcal{M}}_{2}$, the distribution of points in ${\mathcal{M}}_{2}$ based
on the moduli height, the distribution of points for which the field of moduli
is not a field of definition.

The goal of this paper is twofold: to provide the mathematical background for
most of the algorithms in [data] and to discuss some of the open questions and
problems raised there. Most of the material of the first part has already
appeared in the vast literature on genus two curves some of which is previous
work of these authors. For sake of completeness and straightening out some
notational confusion we define all the basic invariants of genus two curves in
this paper.

The current database and all the functions are implemented in Sage. It
improves and expands a previous Maple genus 2 computational algebra package as
in [arith-gen-2]. There is another database of genus 2 curves in
[Booker:2016aa] which collects all genus 2 curves with discriminants $\leq
1000$. Some remarks on how the two databases overlap can be found in the last
section.

There is a lot of confusion in the literature about the invariants of genus
two curves. We go to great lengths to make sure that all the invariants are
defined explicitly and there is no room for misunderstanding. We like to warn
the reader that our invariants are different from the ones used by Magma and
all the papers which use Magma in their computations.

##  2\. A database of integral binary sextics

Our main goal is to create an extensive database of integral binary sextics
with minimal absolute height and all the twists with minimal height. We will
use the definitions of minimal absolute height, moduli height, and the basic
properties of heights of polynomials from [MR3525576] and will provide details
in the next coming sections. The database will be organized in a Sage/Python
dictionary, where the key will be the moduli point

$\mathfrak{p}=\left\\{\begin{aligned} &(-1,i_{1},i_{2},i_{3}),\;\quad
if\;J_{2}\neq 0\\\ &(0,t_{1},t_{2},t_{3}),\;\qquad if\;J_{2}=0,\\\
\end{aligned}\right.$

see Eq. (15) for definitions of such moduli point. The data is organized in
three main dictionaries:

  * i)

integral binary sextics with minimum absolute height $\operatorname{h}\leq
10$,

  * ii)

decomposable integral binary sextics $f(x^{2},z^{2})$ with minimum absolute
$\operatorname{h}\leq 101$ and,

  * iii)

integral binary sextics with moduli height $\mathfrak{h}\leq 20$

Each point in the database has the following invariants

$\displaystyle\mathfrak{p}=\left(r,i_{1},i_{2},i_{3}\right):$
$\displaystyle\operatorname{h}$ $\displaystyle=$ minimal absolute height
$\displaystyle\mathfrak{h}$ $\displaystyle=$ moduli height
$\displaystyle\Delta$ $\displaystyle=$ minimal discriminant
$\displaystyle\operatorname{Aut}(\mathfrak{p})$ $\displaystyle=$ automorphism
group $\displaystyle C$ $\displaystyle=$ Conductor $\displaystyle
M_{\mathfrak{p}}$ $\displaystyle=$ field of definition of the universal curve
$\displaystyle\operatorname{Twist}$ $\displaystyle=$ List of twists

An entry in each dictionary looks as the following:

$\left(r,i_{1},i_{2},i_{3}\right):\left[\operatorname{h},\,\mathfrak{h},\,\Delta,\,\operatorname{Aut}(\mathfrak{p}),\,C,\,M_{\mathfrak{p}},\,\left[[a_{0},\dots,a_{6}],\dots,[b_{0},\dots,b_{6}]\frac{}{}\right]\frac{}{}\right]$

We illustrate with an example.

######  Example 1.

Let $\mathcal{X}$ be the curve with equation

(1) $y^{2}=x^{6}-14x^{4}-82x^{2}+1$

If we load the database in Sage and let

$f=t^{6}-14t^{4}-82t^{2}+1$

then the command $\mathfrak{p}$=ModPoint (f) displays

$\left(-1,-\frac{49281147}{5410276},\frac{706232480445}{12584301976},\frac{3071021069999403}{17429644021121376256}\right)$

If we ask if $\mathfrak{p}\in\mathcal{L}$, where $\mathcal{L}$ is the second
dictionary from above, the answer will be yes and $\mathcal{L}(\mathfrak{p})$
will display

$\begin{split}&[82,2^{14}\cdot 1163^{5},17^{2}\cdot
12301^{2},[4,2],C,\mathbb{Q},[[1,0,-14,0,-82,0,1]]]\end{split}$

which means that the minimal absolute height is $\operatorname{h}=82$,
automorphism group with GapId $[4,2]$ which is the Klein group $V_{4}$,
minimal field of definition $\mathbb{Q}$, and minimal discriminant
$\Delta=17^{2}\cdot 12301^{2}$. The moduli height is $\mathfrak{h}=2^{14}\cdot
1163^{5}$. We chose not to display the conductor and all the twists.

In the Appendix A is given a list of all the functions used for the genus 2
curves package in Sage. In the next few sections we will go over the necessary
definitions and procedures to construct such databases. The details will be
explained in Section 9.

##  3\. Heights of genus two curves

In this section we define heights on algebraic curves when such curves are
given by some affine equation. Throughout this paper $K$ denotes an algebraic
number field and $\mathcal{O}_{K}$ its ring of integers.

Let $\mathcal{X}_{g}$ be an irreducible algebraic curve with affine equation
$F(x,y)=0$ for $F(x,y)\in K[x,y]$. We define the height of the curve over $K$
to be

$H_{K}(\mathcal{X}_{g}):=\min\left\\{H_{K}(G)\,:H_{K}(G)\leq
H_{K}(F)\right\\}.$

where the curve $G(x,y)=0$ is isomorphic to $\mathcal{X}_{g}$ over $K$.

If we consider the equivalence over $\bar{K}$ then we get another height which
we denote it as $\overline{H}_{K}(\mathcal{X}_{g})$ and call it the height
over the algebraic closure. Namely,

$\overline{H}_{K}(\mathcal{X}_{g})=\min\\{H_{K}(G):H_{K}(G)\leq H_{K}(F)\\},$

where the curve $G(x,y)=0$ is isomorphic to $\mathcal{X}_{g}$ over
$\overline{K}$.

In the case that $K=\mathbb{Q}$ we do not write the subscript $K$ and use
$H(\mathcal{X}_{g})$ or $\overline{H}(\mathcal{X}_{g})$. Obviously, for any
algebraic curve $\mathcal{X}_{g}$ we have
$\overline{H}_{K}(\mathcal{X}_{g})\leq H_{K}(\mathcal{X}_{g})$. In [MR3525576]
is proved that given $K$ a number field such that $[K:\mathbb{Q}]=d$, the
height $H_{K}(\mathcal{X}_{g})$ and $\overline{H}_{K}(\mathcal{X}_{g})$ are
well defined.

######  Theorem 1 ([MR3525576]).

Let $K$ be a number field such that $[K:\mathbb{Q}]\leq d$. Given a constant
$\operatorname{h}_{0}\geq 1$ there are only finitely many curves such that
$H_{K}(\mathcal{X}_{g})\leq\operatorname{h}_{0}$.

As an immediate corollary we have the following

######  Corollary 1.

Let $\operatorname{h}_{0}\geq 1$ be a fixed integer, $K$ a number field, and
$\mathcal{O}_{K}$ its ring of integers. For any genus $g\geq 2$ curve
$\mathcal{X}_{g}$ defined over $\mathcal{O}_{K}$ with height
$\operatorname{h}(\mathcal{X}_{g})=\operatorname{h}_{0}$ there are only
finitely many twists of $\mathcal{X}_{g}$ with height $\operatorname{h}_{0}$.

Given a genus two curve $\mathcal{X}_{g}$ the following algorithm computes a
curve isomorphic over $K$ to $\mathcal{X}_{g}$ of minimum height

######  Algorithm 1.

Input: an algebraic curve $\mathcal{X}_{g}:F(x,y)=0$, where $F$ has degree $d$
and is defined over $K$

Output: an algebraic curve $\mathcal{X}_{g}^{\prime}:G(x,y)=0$ such that
$\mathcal{X}_{g}^{\prime}\cong_{K}\mathcal{X}_{g}$ and
$\mathcal{X}_{g}^{\prime}$ has minimum height.

Step 1: Compute $c_{0}=H_{K}(F)$

Step 2: List all points $P\in\mathbb{P}^{s}(K)$ such that $H_{K}(P)\leq
c_{0}$.

Note: $s$ is the number of terms of $F$ which is the number of monomials of
degree $d$ in $n$ variables, and this is equal to $\binom{d+n-1}{d}$. From
Thm. 1 there are only finitely many such points assume $P_{1},\dots,P_{r}$.

Step 3: for $i=1$ to $r$ do

Let $G_{i}(x,y)=p_{i}$;

if $g(G_{i}(x,y))=g(\mathcal{X}_{g})$ then

if $G_{i}(x,y)=0\cong_{K}F(x,y)=0$

then add $G_{i}$ to the list $L$

end if;

end if;

Step 4:  Return all entries of $L$ of minimum height , $L$ has curves
isomorphic over $K$ to $\mathcal{X}_{g}$ of minimum height.

Note that this algorithm is not very efficient if we start with an algebraic
curve of genus two and very big height. Hence, the question that can be raised
at this point is: how can we reduce the height of the curve? This is done
using reduction theory, see [MR3525574, SC] and others for more details, and
some elementary ways of reducing are given next. The following elementary
lemma is useful.

######  Lemma 1.

Let $\mathcal{X}$ be a superelliptic curve with Weierstrass equation
$y^{m}=\sum_{i=0}^{d}\,a_{i}x^{i}$, defined over $\mathbb{Z}$, and height
$\operatorname{h}(\mathcal{X})$. Let $p$ be a prime such that $p\,|\,a_{0}$
and $\mathfrak{v}_{p}(a_{i})=\alpha_{i}$, such that
$\alpha_{0}\geq\alpha_{i}$, for $i=0,\dots,d$. Choose $m$ to be the largest
nonnegative integer which satisfies

$m\leq\frac{\alpha_{0}-\alpha_{i}}{i},\quad i=1,\dots,d.$

Then, there is a twist $\mathcal{X}^{\prime}$ of $\mathcal{X}$ such that

$\mathfrak{v}_{p}\left(\operatorname{h}(\mathcal{X}^{\prime})\right)\leq\mathfrak{v}_{p}\left(\operatorname{h}(\mathcal{X})\right)-m.$

###### Proof.

Let $p$ be a prime such that

$a_{i}=p^{\alpha_{i}}\cdot b_{i},\;\;\textit{such that}\;\;(p,b_{i})=1$

In other words, $\mathfrak{v}_{p}(a_{i})=\alpha_{i}$, for $i=1,\dots,d$, as in
the assumptions of the theorem. Hence, the equation of the curve is

$y^{m}=\sum_{i=0}^{d}p^{\alpha_{i}}\,b_{i}\cdot x^{i}$

Choose $m$ as the largest nonnegative integer such that

$m\leq\frac{\alpha_{0}-\alpha_{i}}{i},\quad i=1,\dots,d.$

If $m=0$, then the curve can no further reduced by this method at the prime
$p$. If $m>0$, then we let

$(x,y)\mapsto\left(p^{m}\cdot x,\,y\right)$

which gives the curve

$y^{m}=\sum_{i=0}^{d}\;a_{i}^{\prime}\,x^{i}=\sum_{i=0}^{d}\;p^{\alpha_{i}+im}\,b_{i}\,x^{i}$

Then,

$\mathfrak{v}_{p}(a_{i}^{\prime})=\alpha_{i}+i\cdot m.$

Hence, to have reduction of
$\mathfrak{v}_{p}\left(\operatorname{h}(\mathcal{X})\right)$ we must have
$\alpha_{i}+im\leq\alpha_{0}$ for $i=1,\dots,d$. Thus,
$m\leq\frac{\alpha_{0}-\alpha_{i}}{i}$, for $i=1,\dots d$. Choosing the
largest such $m$ will result to the biggest possible reduction on $p$.
Dividing both sides by the content of the polynomial, which has the maximum
power of $p$ as a factor, gives a twist $\mathcal{X}^{\prime}$ of
$\mathcal{X}$ with height which has valuation at $p$,
$\mathfrak{v}_{p}\left(\operatorname{h}(\mathcal{X}^{\prime})\right)\leq\mathfrak{v}_{p}\left(\operatorname{h}(\mathcal{X})\right)-m$.
This completes the proof. ∎

######  Remark 1.

Notice that the curve $\mathcal{X}^{\sigma}$ could be in the same
$\Gamma$-orbit of $\mathcal{X}$ or could be a twist of $\mathcal{X}$,
depending on the values of $m$, where $\Gamma$ is the modular group.

######  Corollary 2.

Let $\mathcal{X}$ be a curve with $\operatorname{Aut}(\mathcal{X})\cong D_{6}$
and equation

$y^{2}=x^{6}+x^{3}+s$

where $s\in\mathbb{Z}$ such that it has a prime factorization

$s=p^{\alpha}\cdot s^{\prime},\quad\textit{where }\quad(s,s^{\prime})=1$

Then, we can reduce the height by the transformation $x\mapsto p^{m}\cdot x$,
where $m=\lfloor{\frac{\alpha}{6}}\rfloor$

######  Example 2.

Let us consider the curve from Ex. 4, namely

$y^{2}=x^{6}+x^{3}+2^{33}$

This curve has height $\operatorname{h}=2^{33}$. Then $m$ has to be the
largest nonnegative integer such that it is $\leq$ to

$\frac{33-0}{6},\frac{33-0}{5},$

which makes $m=5$.

Consider the transformation $x\mapsto 2^{5}\cdot x$. Then the curve becomes

$y^{2}=2^{15}\cdot x^{6}+x^{3}+2^{18}$

which is with height $\operatorname{h}=2^{18}$.

Next we will define the moduli height of genus $g$ curves.

###  3.1. Moduli height of curves

Let $g$ be an integer $g\geq 2$ and ${\mathcal{M}}_{g}$ denote the coarse
moduli space of smooth, irreducible algebraic curves of genus $g$. It is known
that ${\mathcal{M}}_{g}$ is a quasi projective variety of dimension $3g-3$.
Hence, ${\mathcal{M}}_{g}$ is embedded in $\mathbb{P}^{3g-2}$. Let
$\mathfrak{p}\in{\mathcal{M}}_{g}$. We call the moduli height
$\operatorname{\mathfrak{h}}(\mathfrak{p})$ the usual height $H(P)$ in the
projective space $\mathbb{P}^{3g-2}$. Obviously,
$\operatorname{\mathfrak{h}}(\mathfrak{p})$ is an invariant of the curve. In
[MR3525576] is proved the following result.

######  Theorem 2.

For any constant $c\geq 1$, degree $d\geq 1$, and genus $g\geq 2$ there are
finitely many superelliptic curves $\mathcal{X}_{g}$ defined over the ring of
integers $\mathcal{O}_{K}$ of an algebraic number field $K$ such that
$[K:\mathbb{Q}]\leq d$ and $\operatorname{\mathfrak{h}}(\mathcal{X}_{g})\leq
c$.

While the above theorem shows that the number of curves with bounded moduli
height is finite, determine this number seems to be a very difficult problem.

##  4\. Genus 2 curves over $\mathbb{C}$

In this section we give a quick overview of the basic setup for genus two
curves. The material is part of the folklore on the literature of genus 2
curves and we don’t mention all the possible references. While the main
definitions and results on what follows are valid for any $g\geq 2$ we only
state them for the case $g=2$. We mainly follow the approach of [Cl, Bo, Ig,
Ig-62, Ig-67, Me].

###  4.1. Periods and invariants

Let $\mathcal{X}$ be a genus $g=2$ algebraic curve. We choose a symplectic
homology basis for $\mathcal{X}$, say $\\{A_{1},A_{2},B_{1},B_{2}\\},$ such
that the intersection products $A_{i}\cdot A_{j}=B_{i}\cdot B_{j}=0$ and
$A_{i}\cdot B_{j}={\delta}_{ij},$ where ${\delta}_{ij}$ is the Kronecker
delta. We choose a basis $\\{w_{i}\\}$ for the space of holomorphic 1-forms
such that $\int_{A_{i}}w_{j}={\delta}_{ij}$. The matrix

$\Omega=\left[\int_{B_{i}}w_{j}\right]$

is the _period matrix_ of $\mathcal{X}$. The columns of the matrix $\left[I\
|\Omega\right]$ form a lattice $\Lambda$ in $\mathbb{C}^{g}$ and the Jacobian
of $\mathcal{X}$ is $\mbox{Jac }(\mathcal{X})=\mathbb{C}^{g}/\Lambda$. Let
$\mathbb{H}_{g}$ be the _Siegel upper-half space_ and $Sp_{4}(\mathbb{Z})$ is
the _symplectic group_. Then $\Omega\in\mathbb{H}_{g}$.

######  Proposition 1.

Two period matrices $\Omega,\Omega^{\prime}$ define isomorphic principally
polarized abelian varieties if and only if they are in the same orbit under
the action of $Sp_{4}(\mathbb{Z})$ on $\mathbb{H}_{g}$.

Hence, there is an injection

${\mathcal{M}}_{2}\hookrightarrow\mathbb{H}_{2}/Sp_{4}(\mathbb{Z})=:\mathcal{A}_{2}$

For any $z\in\mathbb{C}^{2}$ and $\tau\in\mathbb{H}_{2}$ _Riemann’s theta
function_ is defined as

${\theta}(z,\tau)=\sum_{u\in\mathbb{Z}^{2}}e^{\pi i(u^{t}\tau u+2u^{t}z)}$

where $u$ and $z$ are $2-$dimensional column vectors and the products involved
in the formula are matrix products. The fact that the imaginary part of $\tau$
is positive makes the series absolutely convergent over any compact sets.
Therefore, the function is analytic. The theta function is holomorphic on
$\mathbb{C}^{2}\times{\mathbb{H}}_{2}$ and satisfies

${\theta}(z+\tau u)={\theta}(z,\tau),\quad{\theta}(z+\tau,\tau u)=e^{-\pi
i(u^{t}\tau u+2z^{t}u)}\cdot{\theta}(z,\tau),$

where $u\in\mathbb{Z}^{2}$. Any point $e\in\mbox{Jac }(\mathcal{X})$ can be
written uniquely as $e=(b,a)\begin{pmatrix}1_{2}\\\ \Omega\end{pmatrix}$,
where $a,b\in\mathbb{R}^{2}$ are row vectors. We shall use the notation
$[e]=\begin{bmatrix}a\\\ b\\\ \end{bmatrix}$ for the characteristic of $e$.
For any $a,b\in\mathbb{Q}^{2}$, the theta function with rational
characteristics is defined as

${\theta}\begin{bmatrix}a\\\ b\\\
\end{bmatrix}(z,\tau)=\sum_{u\in\mathbb{Z}^{2}}e^{\pi
i((u+a)^{t}\tau(u+a)+2(u+a)^{t}(z+b))}.$

When the entries of column vectors $a^{t}$ and $b^{t}$ are from the set
$\\{0,\frac{1}{2}\\}$, then the characteristics $\begin{bmatrix}a\\\ b\\\
\end{bmatrix}$ are called the _half-integer characteristics_. The
corresponding theta functions with rational characteristics are called _theta
characteristics_. A scalar obtained by evaluating a theta characteristic at
$z=0$ is called a _theta constant_. Any half-integer characteristic is given
by

$\mathfrak{m}=\frac{1}{2}m=\frac{1}{2}\begin{pmatrix}m_{1}&m_{2}\\\
m_{1}^{\prime}&m_{2}^{\prime}\end{pmatrix}$

where $m_{i},m_{i}^{\prime}\in\mathbb{Z}.$ For
$\gamma=\begin{bmatrix}\gamma^{\prime}\\\ \gamma^{\prime\prime}\\\
\end{bmatrix}\in\frac{1}{2}\mathbb{Z}^{4}/\mathbb{Z}^{4}$ we define

$e_{*}(\gamma)=(-1)^{4(\gamma^{\prime})^{t}\gamma^{\prime\prime}}.$

Then,

${\theta}[\gamma](-z,\tau)=e_{*}(\gamma){\theta}[\gamma](z,\tau).$

We say that $\gamma$ is an even (resp. odd) characteristic if
$e_{*}(\gamma)=1$ (resp. $e_{*}(\gamma)=-1$).

For any genus 2 curve we have six odd theta characteristics and ten even theta
characteristics. The following are the sixteen theta characteristics, where
the first ten are even and the last six are odd. For simplicity, we denote
them by $\theta_{i}=\begin{bmatrix}a\\\ b\\\ \end{bmatrix}$ instead of
$\theta_{i}\begin{bmatrix}a\\\ b\\\ \end{bmatrix}(z,\tau)$ where
$i=1,\dots,10$ for the even theta functions.

$\begin{split}\theta_{1}=\begin{bmatrix}0&0\\\
0&0\end{bmatrix},\,\theta_{2}=\begin{bmatrix}0&0\\\
\frac{1}{2}&\frac{1}{2}\end{bmatrix},\theta_{3}=\begin{bmatrix}0&0\\\
\frac{1}{2}&0\end{bmatrix},\,\,\theta_{4}=\begin{bmatrix}0&0\\\
0&\frac{1}{2}\end{bmatrix},\,\,\theta_{5}=\begin{bmatrix}\frac{1}{2}&0\\\
0&0\end{bmatrix},\\\ \theta_{6}=\begin{bmatrix}\frac{1}{2}&0\\\
0&\frac{1}{2}\end{bmatrix},\,\,\theta_{7}=\begin{bmatrix}0&\frac{1}{2}\\\
0&0\end{bmatrix},\,\,\theta_{8}=\begin{bmatrix}\frac{1}{2}&\frac{1}{2}\\\
0&0\end{bmatrix},\,\,\theta_{9}=\begin{bmatrix}0&\frac{1}{2}\\\
\frac{1}{2}&0\end{bmatrix},\,\,\theta_{10}=\begin{bmatrix}\frac{1}{2}&\frac{1}{2}\\\
\frac{1}{2}&\frac{1}{2}\end{bmatrix},\\\ \end{split}$

and the odd theta functions correspond to the following characteristics

$\begin{bmatrix}0&\frac{1}{2}\\\
0&\frac{1}{2}\end{bmatrix},\,\begin{bmatrix}0&\frac{1}{2}\\\
\frac{1}{2}&\frac{1}{2}\end{bmatrix},\,\begin{bmatrix}\frac{1}{2}&0\\\
\frac{1}{2}&0\end{bmatrix},\,\,\begin{bmatrix}\frac{1}{2}&\frac{1}{2}\\\
\frac{1}{2}&0\end{bmatrix},\,\begin{bmatrix}\frac{1}{2}&0\\\
\frac{1}{2}&\frac{1}{2}\end{bmatrix},\,\begin{bmatrix}\frac{1}{2}&\frac{1}{2}\\\
0&\frac{1}{2}\end{bmatrix}$

The complete set of thetanulls above are not independent, their relations are
given via Frobenious relations. There are four theta constants which generate
all the others, namely fundamental theta constants
$\theta_{1},\,\theta_{2},\,\theta_{3},\,\theta_{4}$; see [theta-1] for
details. The following is Igusa’s result, which is valid for any $g\geq 2$. We
only state it for $g=2$.

######  Theorem 3.

The complete set of theta constants uniquely determine the isomorphism class
of a principally polarized abelian variety of dimension 2.

For curves of genus 2 this can be made more precise. Let a genus 2 curve in
Rosenheim form be given by

(2) $Y^{2}=X(X-1)(X-\lambda)(X-\mu)(X-\nu).$

By the sa called Picard’s lemma $\lambda,\mu,\nu$ can be written as follows:

(3)
$\lambda=\frac{\theta_{1}^{2}\theta_{3}^{2}}{\theta_{2}^{2}\theta_{4}^{2}},\quad\mu=\frac{\theta_{3}^{2}\theta_{8}^{2}}{\theta_{4}^{2}\theta_{10}^{2}},\quad\nu=\frac{\theta_{1}^{2}\theta_{8}^{2}}{\theta_{2}^{2}\theta_{10}^{2}}.$

We can determine an equation of the curve in terms of the fundamental thetas
as follows:

######  Proposition 2 ([theta-1]).

Every genus two curve can be written in the form:

(4)
$y^{2}=x\,(x-1)\,\left(x-\frac{\theta_{1}^{2}\theta_{3}^{2}}{\theta_{2}^{2}\theta_{4}^{2}}\right)\,\left(x^{2}\,-\frac{\theta_{2}^{2}\,\theta_{3}^{2}+\theta_{1}^{2}\,\theta_{4}^{2}}{\theta_{2}^{2}\,\theta_{4}^{2}}\cdot\alpha\,x+\frac{\theta_{1}^{2}\theta_{3}^{2}}{\theta_{2}^{2}\theta_{4}^{2}}\,\alpha^{2}\right),$

where $\alpha=\frac{\theta_{8}^{2}}{\theta_{10}^{2}}$ and in terms of
$\,\,\theta_{1},\dots,\theta_{4}$ is given by

$\alpha^{2}+\frac{\theta_{1}^{4}+\theta_{2}^{4}-\theta_{3}^{4}-\theta_{4}^{4}}{\theta_{1}^{2}\theta_{2}^{2}-\theta_{3}^{2}\theta_{4}^{2}}\,\alpha+1=0$

Furthermore, if $\alpha={\pm}1$ then $\mathcal{X}$ has an extra involution.

From the above we have that $\theta_{8}^{4}=\theta_{10}^{4}$ implies that
$\mathcal{X}$ has an extra involution. Hence, the Klein viergrouppe
$V_{4}\hookrightarrow Aut(\mathcal{X})$. The last part of the lemma above
shows that if $\theta_{8}^{4}=\theta_{10}^{4}$ then all coefficients of the
genus 2 curve are given as rational functions of the 4 fundamental theta
functions. Such fundamental theta functions determine the field of moduli of
the given curve. Hence, the curve is defined over its field of moduli.

######  Corollary 3.

Let $\mathcal{X}$ be a genus 2 curve which has an extra involution. Then
$\mathcal{X}$ is defined over its field of moduli.

We will revisit the curves defined over their field of moduli again in the
coming sections.

###  4.2. Siegel modular forms

Here we define Siegel modular forms $\psi_{4}$, $\psi_{6}$, $\chi_{10}$,
$\chi_{12}$, of degree 4, 6, 10, and 12, as in [Ig-67]*pg. 848.

(5) $\begin{split}2^{2}\cdot\psi_{4}&=\sum(\theta_{m})^{8}\\\
2^{2}\cdot\psi_{6}&=\sum_{syzygous}\pm\left(\theta_{m_{1}}\theta_{m_{2}}\theta_{m_{3}}\right)^{4}\\\
-2^{14}\cdot\chi_{10}&=\prod(\theta_{m})^{2}\\\ 2^{17}\cdot
3\cdot\chi_{12}&=\sum\left(\theta_{m_{1}}\theta_{m_{2}}\cdots\theta_{m_{6}}\right)^{4}\\\
2^{39}\cdot
5^{3}\cdot\chi_{35}&=\left(\prod\theta_{m}\right)\left(\sum_{azygous}\pm\left(\theta_{m_{1}}\theta_{m_{2}}\theta_{m_{3}}\right)^{20}\right).\\\
\end{split}$

In definition of $\chi_{12}$ the summation is taken over all Göpel systems as
explained in [theta-1], where all the Göpel systems are displayed.

Theta constants provide a complete system of invariants for isomorphism
classes of principally polarized varieties of dimension $g=2$. But there are
two main issues with this approach: First, these invariants are not
independent. This can be fixed via the fundamental theta constants as in Prop.
2, however computationally things get difficult when we try to express all the
results in terms of $\theta_{1},\theta_{2},\theta_{3},\theta_{4}$. Secondly,
and more importantly, they are defined analytically. Naturally, we would like
to have algebraically defined invariants.

##  5\. Algebraic invariants

Let $f(x,z)$ be a binary sextic defined over a field $k$, $\mbox{char }k=0$,
given by

(6) $\begin{split}f(x,z)&=a_{0}x^{6}+a_{1}x^{5}z+\dots+a_{6}z^{6}\\\
&=(z_{1}x-x_{1}z)(z_{2}x-x_{2}z)\dots(z_{6}x-x_{6}z)\end{split}$

A covariant $I$ of $f(x,z)$ is a homogenous polynomial in $x,z$ with
coefficients in $k[a_{0},\dots,a_{2g+2}]$. The order  of $I$ is the degree of
$I$ as a polynomial in $x,z$ and the degree of $I$ is the degree of $I$ as a
polynomial in $k[a_{0},\dots,a_{2g+2}]$. An invariant is a covariant of order
zero. The binary form $f(x,z)$ is a covariant of order $2g+2$ and degree 1.
Throughout this paper we will use as basic references [Cl], [Bo], and [Ig,
Ig-62, Ig-67].

###  5.1. Invariants and covariants via transvections

For any two binary forms $f$ and $g$ the symbol $(f,g)_{r}$ denotes the
$r$-transvection. Notice that the transvections are conveniently computed in
terms of the coefficients of the binary forms.

Let $f(x,z)$ be a binary sextic as in Eq. (6) and consider the following
covariants

(7) $\displaystyle\Delta=\left((f,f)_{4},(f,f)_{4}\right)_{2},$ $\displaystyle
Y_{1}=\left(f,(f,f)_{4}\right)_{4}$ $\displaystyle
Y_{2}=\left((f,f)_{4},Y_{1}\right)_{2},$ $\displaystyle
Y_{3}=\left((f,f)_{4},Y_{2}\right)_{2}$

The Clebsch invariants $A,B,C,D$ are defined as follows

(8) $\displaystyle A=(f,f)_{6},\;$ $\displaystyle
B=\left((f,f)_{4},(f,f)_{4}\right)_{4},$ $\displaystyle
C=\left((f,f)_{4},\Delta\right)_{4},\;$ $\displaystyle
D=\left(Y_{3},Y_{1}\right)_{2}$

see Clebsch [Cl] or Bolza [Bo]*Eq. (7), (8), pg. 51 for details.

We display the invariants $A,B,C,D$ in terms of the coefficients in the
Appendix. The following result is elementary but very important in our
computations.

###  5.2. Root differences

Let $f(x,z)$ be a binary sextic as above and set
$D_{ij}:=\begin{pmatrix}x_{i}&x_{j}\\\ z_{i}&z_{j}\end{pmatrix}$. For $\tau\in
SL_{2}(k)$, we have

$\tau(f)=(z_{1}^{{}^{\prime}}x-x_{1}^{{}^{\prime}}z)\dots(z_{6}^{{}^{\prime}}x-x_{6}^{{}^{\prime}}z),\quad\textit{
with }\quad\begin{pmatrix}x_{i}^{{}^{\prime}}\\\
z_{i}^{{}^{\prime}}\end{pmatrix}=\tau^{-1}\,\begin{pmatrix}x_{i}\\\
z_{i}\end{pmatrix}.$

Clearly $D_{ij}$ is invariant under this action of $SL_{2}(k)$ on
$\mathbb{P}^{1}$. Let $\\{i,j,k,l,m,n\\}=\\{1,2,3,4,5,6\\}$. Treating $a_{i}$
as variables, we construct the following elements in the ring of invariants
$\mathcal{R}_{6}$

(9)
$\begin{split}\mathfrak{A}&=a_{0}^{2}\,\prod_{fifteen}(12)^{2}(34)^{2}(56)^{2}=\displaystyle\sum_{\begin{subarray}{c}i<j,k<l,m<n\end{subarray}}D_{ij}^{2}D_{kl}^{2}D_{mn}^{2}\\\
&\\\
\mathfrak{B}&=a_{0}^{4}\,\prod_{ten}(12)^{2}(23)^{2}(31)^{2}(45)^{2}(56)^{2}(64)^{2}=\sum_{\begin{subarray}{c}i<j,j<k,\\\
l<m,m<n\end{subarray}}D_{ij}^{2}D_{jk}^{2}D_{ki}^{2}D_{lm}^{2}D_{mn}^{2}D_{nl}^{2}\\\
&\\\
\mathfrak{C}&=a_{0}^{6}\,\prod_{sixty}(12)^{2}(23)^{2}(31)^{2}(45)^{2}(56)^{2}(64)^{2}(14)^{2}(25)^{2}(36)^{2}\\\
&=\sum_{\begin{subarray}{c}i<j,j<k,l<m,m<n\\\
i<l^{\prime},j<m^{\prime},k<n^{\prime}\\\
l^{\prime},m^{\prime},n^{\prime}\in\\{l,m,n\\}\end{subarray}}D_{ij}^{2}D_{jk}^{2}D_{ki}^{2}D_{lm}^{2}D_{mn}^{2}D_{nl}^{2}D_{{il}^{{}^{\prime}}}^{2}D_{{jm}^{{}^{\prime}}}^{2}D_{{kn}^{{}^{\prime}}}^{2}\\\
&\\\ \mathfrak{D}&=a_{0}^{10}\prod_{i<j}(ij)^{2}\\\ \end{split}$

These invariants, sometimes called integral invariants, are defined in
[Ig]*pg. 620 where they are denoted by $A,B,C,D$. Incidentally even Clebsch
invariants which are defined next are also denoted by $A,B,C,D$ by many
authors.

To quote Igusa ”if we restrict to integral invariants, the discussion will
break down in characteristic 2 simply because Weierstrass points behave badly
under reduction modulo 2”; see [Ig]*pg. 621. Next we define invarints which
will work in every characteristic.

###  5.3. Igusa invariants

In [Ig]*pg. 622 Igusa defined what he called basic arithmetic invariants,
which are now commonly known as Igusa invariants

$\displaystyle J_{2}$ $\displaystyle=\frac{1}{2^{3}}\mathfrak{A},\;$
$\displaystyle J_{4}=\frac{1}{2^{5}\cdot 3}(4J_{2}^{2}-\mathfrak{B}),$
$\displaystyle J_{6}$ $\displaystyle=\frac{1}{2^{6}\cdot
3^{2}}(8J_{2}^{3}-160J_{2}J_{4}-\mathfrak{C}),\;$ $\displaystyle
J_{10}=\frac{1}{2^{12}}\mathfrak{D}$

While most of the current literature on genus 2 curves uses invariants
$\mathfrak{A},\mathfrak{B},\mathfrak{C},\mathfrak{D}$, which are now most
commonly labeled as $I_{2},I_{4},I_{6},I_{10}$, Igusa went to great lengths in
[Ig] to define $J_{2},J_{4},J_{6},J_{10}$ and to show that they also work in
characteristic 2.

######  Lemma 2.

$J_{2i}$ are homogeneous elements in $\mathcal{R}_{6}$ of degree $2i$, for $i$
= 1,2,3,5.

A degree $d\geq 2$ binary form $f(x,z)$ is called semistable if it has no root
of multiplicity $>\frac{d}{2}$.

######  Lemma 3.

A sextic has a root of multiplicity at least four if and only if the basic
invariants vanish simultaneously.

So a sextic for which all the basic invariants vanish simultaneously is not
semistable.

Throughout this paper, we will use these invariants
$J_{2},J_{4},J_{6},J_{10}$. For a binary form
$f(x,z)=\sum_{i=0}^{6}a_{i}\,x^{i}z^{6-i}$ as in Eq. (6) we display these
invariants in terms of the coefficients $a_{0},\dots,a_{6}$ in B.

Invariants $\,\,\\{J_{2i}\\}$ are homogeneous polynomials of degree $2i$ in
$k[a_{0},\dots,a_{6}]$, for $i=1,2,3,5$. These $J_{2i}$ are invariant under
the natural symmetries of the roots, namely the action of $S_{6}$ on the set
of the roots. They are also natural on the action of $SL_{2}(k)$ on the
sextic.

######  Lemma 4.

[Ig]*Cor. pg. 632 Two genus 2 curves $C$ and $C^{\prime}$ are isomorphic if
and only if there exists an $r\neq 0$ such that

$J_{2i}(C)=r^{2i}\cdot J_{2i}(C^{\prime}),\qquad\textit{ for }\quad
i=1,2,3,5.$

######  Remark 2.

There is a lot of confusion in the literature about the definitions of the
generators of $\mathcal{R}_{6}$. Many authors have used different names and
different notations for generators of $\mathcal{R}_{6}$. Moreover, many times
the symbols $\mathfrak{A},\mathfrak{B},\mathfrak{C},\mathfrak{D}$ are defined
by scaling with different constants by many authors. Hence, sometimes
equations in terms of these invariants might not be the same. To avoid any
confusion, we display the expressions of $j_{2},J_{4},J_{6},j_{10}$ in terms
of the coefficients of the binary sextic in Eq. (B).

Igusa invariants are expressed in terms of the Clebsch invariants as follows:

(10) $\begin{split}J_{2}=&\;-2^{3}\cdot 3\cdot 5\,A\;,\\\ J_{4}=&\;2^{3}\cdot
2^{5}\,(75B-8A^{2})\\\ J_{6}=&\;2^{2}\cdot 3^{3}\cdot
5\;\left(16\,A^{3}-200\,A\,B+375\,C\right)\\\ J_{10}=&\;2^{3}\cdot
4\,\left(-384\,A^{5}+6000\,A^{3}\,B+10000\,A^{2}\,C-18750\,A\,B^{2}\right.\\\
&\;\;\;\qquad\left.-37500\,B\,C-28125\,D\right)\\\ \end{split}$

Conversely, the invariants $(A,B,C,D)$ are polynomial expressions in the Igusa
invariants $(J_{2},J_{4},J_{6},J_{10})$ with rational coefficients are
displayed in Eq. B in the Appendix.

Thus, the invariants of a sextic define a point in a weighted projective space
$[J_{2}:J_{4}:J_{6}:J_{10}]\in\mathbb{WP}^{3}_{(2,4,6,10)}$. It was shown in
[vishi] that points in the projective variety
$\operatorname{Proj}\,\mathbb{C}[J_{2},J_{4},J_{6},J_{10}]$ which are not on
$J_{2}=0$ form the variety $\mathcal{U}_{6}$ of moduli of sextics.
Equivalently, points in the weighted projective space
$\\{[J_{2}:J_{4}:J_{6}:J_{10}]\in\mathbb{WP}^{3}_{(2,4,6,10)}:J_{10}\not=0\\}$
are in one-to-one correspondence with isomorphism classes of sextics.

###  5.4. Absolute invariants

Dividing any $SL_{2}(k)$ invariant by another one of the same degree gives an
invariant under $GL_{2}(k)$ action.

We follow [Ig-62]*pg. 181 and define the following absolute invariants

(11) $i_{1}:=144\frac{J_{4}}{J_{2}^{2}},\quad
i_{2}:=-1728\frac{J_{2}J_{4}-3J_{6}}{J_{2}^{3}},\quad
i_{3}:=486\frac{J_{10}}{J_{2}^{5}}$

for $J_{2}\neq 0$. Notice that Igusa denotes them by $x_{1},x_{2},x_{3}$.

######  Theorem 4.

[Ig-62]*Theorem 3 The three absolute invariants can be expressed by the four
modular forms in the form

(12) $i_{1}=\frac{\psi_{4}\chi_{10}^{2}}{\chi_{12}^{2}},\quad
i_{2}=\frac{\psi_{6}\,\chi_{10}^{3}}{\chi_{12}^{3}},\quad
i_{3}=\frac{\chi_{10}^{6}}{\chi_{12}^{5}}$

Hence, the theorem says that every modular form, in the degree 2 case, is
expressed in the Eisenstein series of weight four, six, ten, and twelve. It is
the main result of [Ig-62].

There is another set of absolute invariants

(13) $j_{1}:=\frac{\mathfrak{A}^{5}}{\mathfrak{D}},\quad
j_{2}:=\frac{\mathfrak{B}\mathfrak{A}^{3}}{\mathfrak{D}},\quad
j_{3}:=\frac{\mathfrak{C}\mathfrak{A}^{2}}{\mathfrak{D}},$

The popularity of invariants $j_{1},j_{2},j_{3}$ is due to P. Van Wamelen who
used them in [wamelen] and later implemented them in Magma. They have no
advantages over the invariants $i_{1},i_{2},i_{3}$ since both sets of
invariants are not defined for $J_{2}=0$ or equivalently $\mathfrak{A}=0$.
Indeed, we could not find a direct proof that such invariants generate the
field of invariants for $\mathcal{R}_{6}$, even though it is probably true.
The results of other authors who have worked with $j_{1},j_{2},j_{3}$ can be
converted into our results and vice-versa using the formulas

$i_{1}=144\,\frac{j_{2}}{j_{1}},\quad
i_{2}=-1728\,\frac{j_{2}-3j_{3}}{j_{1}},\quad i_{3}=486\frac{1}{j_{1}}$

and conversely

$j_{1}=486\,\frac{1}{i_{3}},\quad
j_{2}=\frac{27}{8}\,\frac{i_{1}}{i_{3}},\quad
j_{3}=\frac{3}{32}\,\frac{i_{2}+12i_{1}}{i_{3}}$

######  Remark 3.

It is to be noted that for computational purposes $i_{1},i_{2},i_{3}$ are much
better than $j_{1},j_{2},j_{3}$ since they are of lower degrees. Especially,
for our purposes it is very important that the moduli point
$\mathfrak{p}=(r,i_{1},i_{2},i_{3})$ (cf. next section) it is expressed in as
small numbers as possible.

There is another set of invariants defined by Igusa in [Ig],

$t_{1}=\frac{J_{2}^{5}}{J_{10}},\quad t_{2}=\frac{J_{4}^{5}}{J_{10}^{2}},\quad
t_{3}=\frac{J_{6}^{5}}{J_{10}^{3}}$

which are defined everywhere in the moduli space. Due to their high degrees,
they become difficult to use especially when someone wants to do symbolic
computations.

In the case $J_{2}=0$ we define

(14) $a_{1}:=\frac{J_{4}\cdot J_{6}}{J_{10}},\quad a_{2}:=\frac{J_{6}\cdot
J_{10}}{J_{4}^{4}}$

to determine genus two fields with $J_{2}=0$, $J_{4}\neq 0$, and $J_{6}\neq 0$
up to isomorphism. They were used in [Me, gaudry, deg2, ants] and others.
Moreover, when $J_{2}=0,J_{4}=0,J_{6}\neq 0$ we have
$\frac{J_{6}^{5}}{J_{10}^{3}}$ as an invariant and when
$J_{2}=0,J_{6}=0,J_{4}\neq 0$ we have $\frac{J_{4}^{5}}{J_{10}^{2}}$ as an
invariant.

All our computations are made in terms of absolute invariants
$i_{1},i_{2},i_{3}$ or the Igusa invariants $J_{2},J_{4},J_{6},J_{10}$. In the
Appendix, we provide formulas how to convert back and forth among all sets of
invariants. When $J_{2}=0$ we will use invariants $t_{1},t_{2},t_{3}$ instead,
in this case $t_{1}=0$, so this locus is determined by $t_{2}$ and $t_{3}$.

###  5.5. Representing a point in the moduli space

In creating a large database of genus two curves we need a way to identify
uniquely a point $\mathfrak{p}\in{\mathcal{M}}_{2}$. Such point would ideally
be defined everywhere, computationally feasable, and should not involve large
integers. Of course, the representation $\mathfrak{p}=(t_{1},t_{2},t_{3})$ is
unique, but these invariants are very large and extremely difficult to handle
in a large database. The set of invariants $(i_{1},i_{2},i_{3})$ is nicer, but
they are not defined for $J_{2}=0$.

To find a compromise and simplify the implementation in [data], for a given
genus 2 curve $\mathcal{X}$ we define the corresponding moduli point
$\mathfrak{p}=[\mathcal{X}]$ to be

(15) $\mathfrak{p}=\left\\{\begin{aligned}
&(i_{1},i_{2},i_{3})\;\;if\;J_{2}\neq 0\\\
&(t_{1},t_{2},t_{3})\;\;if\;J_{2}=0\\\ \end{aligned}\right.$

This moduli point $\mathfrak{p}$ together with $J_{2}$ will be the key in our
dictionary of genus 2 curves. It will be stored as a 4-tuple
$(r,\mathfrak{p})$ as explained before.

For $I_{2}\not=0$ we use the variables $i_{1},i_{2},i_{3}$ to write

$\begin{split}\Big{[}J_{2}:J_{4}:J_{6}:J_{10}\Big{]}=\left[1:\frac{1}{2^{4}\,3^{2}}\,i_{1}:\frac{1}{2^{6}\,3^{4}}\,i_{2}+\frac{1}{2^{4}\,3^{3}}\,i_{1}:\frac{1}{2\cdot
3^{5}}\,i_{3}\right]\in\mathbb{WP}^{3}_{(2,4,6,10)}\;,\end{split}$

see [MS-1] *pg. 13 for details. Notice that $i_{1},i_{2},i_{3}$ are denoted by
$\mathbf{x}_{1},\mathbf{x}_{2},\mathbf{x}_{3}$ and $J_{2},J_{4},J_{6},J_{10}$
respectively by $I_{2},I_{4},I_{6},I_{10}$ in [MS-1].

###  5.6. Relations among different sets of invariants

There are three sets of $SL_{2}(k)$ invariants most commonly used, namely
$(A,B,C,D)$, $(\mathfrak{A},\mathfrak{B},\mathfrak{C},\mathfrak{D})$, and
$(J_{2},J_{4},J_{6},J_{10})$. Many authors will use the notation
$(I_{2},I_{4},I_{6},I_{10})$ for
$(\mathfrak{A},\mathfrak{B},\mathfrak{C},\mathfrak{D})$ due to the Magma
notation (they are called Igusa Clebsch invariants in Magma) or some scaling
of them.

Notice that

(16) $J_{2i}=\frac{1}{2^{4i}}\;I_{2i},\qquad i=1,2,3,5.$

For a given genus two curve $\mathcal{X}$ with equation $y^{2}=f(x)$, the
relations between invariants
$\mathfrak{A},\mathfrak{B},\mathfrak{C},\mathfrak{D}$ and even Siegel modular
forms $\psi_{4},\psi_{6},\chi_{10},\chi_{12},\chi_{35}$ of $\mathcal{A}_{2}$
are given by Igusa in [Ig-67]*p. 848:

(17) $\displaystyle\mathfrak{A}(f)$ $\displaystyle=-2^{3}\cdot
3\,\dfrac{\chi_{12}(\tau)}{\chi_{10}(\tau)}\;,$
$\displaystyle\mathfrak{B}(f)=\phantom{-}2^{2}\,\psi_{4}(\tau)\;,$
$\displaystyle\mathfrak{C}(f)$
$\displaystyle=-\frac{2^{3}}{3}\,\psi_{6}(\tau)-2^{5}\,\dfrac{\psi_{4}(\tau)\,\chi_{12}(\tau)}{\chi_{10}(\tau)}\;,$
$\displaystyle\mathfrak{D}(f)=-2^{14}\,\chi_{10}(\tau)\;.$

In the Appendix are given the relations between $A,B,C,D$ and
$J_{2},J_{4},J_{6},J_{10}$. Using such relations we get the absolute
invariants $i_{1},i_{2},i_{3}$ in terms of $A,B,C,D$ are as follows:

$\displaystyle i_{1}$
$\displaystyle=\frac{9}{10}\cdot\frac{75B-8A^{2}}{A^{2}}$ $\displaystyle
i_{2}$ $\displaystyle=\frac{27}{50}\cdot\frac{112A^{3}-900AB-1125C}{A^{3}}$
$\displaystyle i_{3}$ $\displaystyle=\frac{81}{2^{13}\cdot
5^{5}}\cdot\frac{384A^{5}-6000A^{3}B-10000A^{2}C+18750AB^{2}+37500BC+28125CD}{A^{5}}$

Next we see how to construct the equation of a curve starting from a moduli
point $\mathfrak{p}\in{\mathcal{M}}_{2}$.

###  5.7. Recovering an equation of the curve from the moduli point

Some other invariants are as follows

$\begin{split}&A_{ij}=(Y_{i}Y_{j})_{2},\qquad\qquad\qquad\quad(1\leq i,j\leq
3)\\\ &H_{ijk}=(fY_{i})_{2}(fY_{j})_{2}(fY_{k})_{2}\qquad(1\leq i,j,k\leq
3)\\\ &R=-(Y_{1}Y_{2})(Y_{2}Y_{3})(Y_{3}Y_{1})\end{split}$

Note that $A_{ij}$ and $H_{ijk}$ can be expressed in terms of the Clebsch
invariants, see [Me]*pg. 318. Moreover, $R^{2}=\frac{1}{2}\det M$ where $M$ is
the Clebsch matrix defined as follows.

(18) $M=\begin{bmatrix}A_{11}&A_{12}&A_{13}\\\ A_{12}&A_{22}&A_{23}\\\
A_{13}&A_{23}&A_{33}\end{bmatrix}$

The following is an immediate consequence of classical invariant theory

Let $J_{30}$ denote the degree 30 invariant

$J_{30}:=\det M$

An expression of $J_{30}$ in terms of $J_{2},J_{4},J_{6},J_{10}$ is given in
[deg2]*Theorem 3, where it is also expressed in terms of the roots of the
sextic and its coefficients.

Let us now compute the determinant of the minor
$S=\begin{pmatrix}A_{1,1}&A_{1,2}\\\ A_{2,1}&A_{2,2}\end{pmatrix}$ from the
matrix $M=\left[A_{i,j}\right]$.

$\begin{split}\det S&=A_{1,1}A_{2,2}-A_{1,2}^{2}\\\ &=\frac{2^{16}}{3^{6}\cdot
5^{4}J_{2}^{8}}\left(15\,J_{2}^{3}J_{4}\,J_{6}-4\,J_{2}^{4}J_{4}^{2}-175\,J_{2}^{2}J_{4}^{3}+2430\,J_{10}\,J_{2}^{3}-9\,J_{2}^{2}J_{6}^{2}\right.\\\
&\left.+1488\,J_{2}\,J_{4}^{2}J_{6}-64\,J_{4}^{4}+113400\,J_{10}\,J_{2}\,J_{4}-2880\,J_{4}\,J_{6}^{2}-648000\,J_{10}\,J_{6}\right)\end{split}$

We define the new invariant

$\begin{split}J_{16}:=&15\,J_{2}^{3}J_{4}\,J_{6}-4\,J_{2}^{4}J_{4}^{2}-175\,J_{2}^{2}J_{4}^{3}+2430\,J_{10}\,J_{2}^{3}-9\,J_{2}^{2}J_{6}^{2}+1488\,J_{2}\,J_{4}^{2}J_{6}\\\
&-64\,J_{4}^{4}+113400\,J_{10}\,J_{2}\,J_{4}-2880\,J_{4}\,J_{6}^{2}-648000\,J_{10}\,J_{6}\\\
\end{split}$

which is a degree 16 invariant.

##  6\. Automorphisms

Let $\mathcal{X}$ be a genus 2 curve defined over an algebraically closed
field $k$. Let $K$ be the function field of $\mathcal{X}$. Then $K$ has
exactly one genus 0 subfield of degree 2, denote it $k(x)$. Since $k(x)$ is
the only genus 0 subfield of degree 2 of $K$, then $G=\operatorname{Aut}(K/k)$
(or equivalently $\operatorname{Aut}(\mathcal{X})$) fixes $k(x)$. It is the
fixed field of the hyperelliptic involution $\tau$ in $\operatorname{Aut}(K)$.
Thus $\tau$ is in the center of $\operatorname{Aut}(K)$ and
$\langle\tau\rangle\lhd\operatorname{Aut}(K)$. The quotient group
$\overline{\operatorname{Aut}}(K)=\operatorname{Aut}(K)/\langle\tau\rangle$ is
called the reduced automorphism group of $\mathcal{X}$ (or equivalently of
$K$).

The reduced automorphism group embeds as a finite subgroup of
$PGL(2,\mathbb{C})$, therefore it is isomorphic to
$C_{n},D_{n},A_{4},S_{4},A_{5}$. Hence, the full automorphism group
$\operatorname{Aut}(\mathcal{X})$ is a degree 2 central extension of
$\overline{\operatorname{Aut}}(\mathcal{X})$.

The extension $K/k(x)$ is ramified at exactly six distinct points, namely the
points $P=\\{\omega_{1},\cdots,\omega_{6}\\}$ in $\mathbb{P}^{1}$. The
corresponding places of this points in $K$ are called Weierstrass points of
$K$. The group $\operatorname{Aut}(K)$ permutes the 6 Weierstrass points and
the group $\overline{\operatorname{Aut}}(K)$ permutes accordingly
$\\{\omega_{1},\cdots,\omega_{6}\\}$ in its action on $\mathbb{P}^{1}$ as
subgroup of $PLG(2,\mathbb{C})$. This yields an embedding
$\overline{\operatorname{Aut}}(K)\hookrightarrow S_{6}$, so all elements of
$\overline{\operatorname{Aut}}(K)$ have order $\leq 6$.

In any characteristic different from 2, the automorphism group
$\operatorname{Aut}(\mathcal{X})$ is isomorphic to one of the groups given by
the following theorem from [deg2].

######  Theorem 5 ([deg2]).

The automorphism group $G$ of a genus 2 curve $\mathcal{X}$ in characteristic
$\neq 2$ is isomorphic to $\mathbb{Z}_{2}$, $\mathbb{Z}_{10}$, $V_{4}$,
$D_{8}$, $D_{12}$, $SL_{2}(3)$, $GL_{2}(3)$, or $2^{+}S_{5}$. The case when
$G\cong 2^{+}S_{5}$ occurs only in characteristic 5. If $G\cong SL_{2}(3)$
(resp., $GL_{2}(3)$) then $\mathcal{X}$ has equation $Y^{2}=X^{6}-1$ (resp.,
$Y^{2}=X(X^{4}-1)$). If $G\cong\mathbb{Z}_{10}$ then $\mathcal{X}$ has
equation $Y^{2}=X^{6}-X$.

An elliptic involution of $K$ is an involution in $G$ which is different from
$z_{0}$ (the hyperelliptic involution). Thus the elliptic involutions of $G$
are in 1-1 correspondence with the elliptic subfields of $K$ of degree 2 (by
the Riemann-Hurwitz formula). For the number of elliptic involutions of a
genus 2 curve see [deg2].

###  6.1. Automorphisms in terms of theta functions

The locus $\mathcal{L}_{2}$ of genus 2 curves $\mathcal{X}$ which have an
elliptic involution is a closed subvariety of $\mathcal{M}_{2}$.

######  Lemma 5.

Let $\mathcal{X}$ be a genus 2 curve. Then $Aut(\mathcal{X})\cong V_{4}$ if
and only if the theta functions of $\mathcal{X}$ satisfy

(19)
$\begin{split}(\theta_{1}^{4}-\theta_{2}^{4})(\theta_{3}^{4}-\theta_{4}^{4})(\theta_{8}^{4}-\theta_{10}^{4})(-\theta_{1}^{2}\theta_{3}^{2}\theta_{8}^{2}\theta_{2}^{2}-\theta_{1}^{2}\theta_{2}^{2}\theta_{4}^{2}\theta_{10}^{2}+\theta_{1}^{4}\theta_{3}^{2}\theta_{10}^{2}+\theta_{3}^{2}\theta_{2}^{4}\theta_{10}^{2})\\\
(\theta_{3}^{2}\theta_{8}^{2}\theta_{2}^{2}\theta_{4}^{2}-\theta_{2}^{2}\theta_{4}^{4}\theta_{10}^{2}+\theta_{1}^{2}\theta_{3}^{2}\theta_{4}^{2}\theta_{10}^{2}-\theta_{3}^{4}\theta_{2}^{2}\theta_{10}^{2})(-\theta_{8}^{4}\theta_{3}^{2}\theta_{2}^{2}+\theta_{8}^{2}\theta_{2}^{2}\theta_{10}^{2}\theta_{4}^{2}+\theta_{1}^{2}\theta_{3}^{2}\theta_{8}^{2}\theta_{10}^{2}-\theta_{3}^{2}\theta_{2}^{2}\theta_{10}^{4})\\\
(-\theta_{1}^{2}\theta_{8}^{4}\theta_{4}^{2}-\theta_{1}^{2}\theta_{10}^{4}\theta_{4}^{2}+\theta_{8}^{2}\theta_{2}^{2}\theta_{10}^{2}\theta_{4}^{2}+\theta_{1}^{2}\theta_{3}^{2}\theta_{8}^{2}\theta_{10}^{2})(-\theta_{1}^{2}\theta_{8}^{2}\theta_{3}^{2}\theta_{4}^{2}+\theta_{1}^{2}\theta_{10}^{2}\theta_{4}^{4}+\theta_{1}^{2}\theta_{3}^{4}\theta_{10}^{2}-\theta_{3}^{2}\theta_{2}^{2}\theta_{10}^{2}\theta_{4}^{2})\\\
(-\theta_{1}^{2}\theta_{8}^{2}\theta_{2}^{2}\theta_{4}^{2}+\theta_{1}^{4}\theta_{10}^{2}\theta_{4}^{2}-\theta_{1}^{2}\theta_{3}^{2}\theta_{2}^{2}\theta_{10}^{2}+\theta_{2}^{4}\theta_{4}^{2}\theta_{10}^{2})(-\theta_{8}^{4}\theta_{2}^{2}\theta_{4}^{2}+\theta_{1}^{2}\theta_{8}^{2}\theta_{10}^{2}\theta_{4}^{2}-\theta_{2}^{2}\theta_{10}^{4}\theta_{4}^{2}+\theta_{3}^{2}\theta_{8}^{2}\theta_{2}^{2}\theta_{10}^{2})\\\
(\theta_{1}^{4}\theta_{8}^{2}\theta_{4}^{2}-\theta_{1}^{2}\theta_{2}^{2}\theta_{4}^{2}\theta_{10}^{2}-\theta_{1}^{2}\theta_{3}^{2}\theta_{8}^{2}\theta_{2}^{2}+\theta_{8}^{2}\theta_{2}^{4}\theta_{4}^{2})(\theta_{1}^{4}\theta_{3}^{2}\theta_{8}^{2}-\theta_{1}^{2}\theta_{8}^{2}\theta_{2}^{2}\theta_{4}^{2}-\theta_{1}^{2}\theta_{3}^{2}\theta_{2}^{2}\theta_{10}^{2}+\theta_{3}^{2}\theta_{8}^{2}\theta_{2}^{4})\\\
(\theta_{1}^{2}\theta_{8}^{4}\theta_{3}^{2}-\theta_{1}^{2}\theta_{8}^{2}\theta_{10}^{2}\theta_{4}^{2}+\theta_{1}^{2}\theta_{3}^{2}\theta_{10}^{4}-\theta_{3}^{2}\theta_{8}^{2}\theta_{2}^{2}\theta_{10}^{2})(\theta_{1}^{2}\theta_{8}^{2}\theta_{4}^{4}-\theta_{1}^{2}\theta_{3}^{2}\theta_{4}^{2}\theta_{10}^{2}+\theta_{1}^{2}\theta_{3}^{4}\theta_{8}^{2}-\theta_{3}^{2}\theta_{8}^{2}\theta_{2}^{2}\theta_{4}^{2})&=0\end{split}$

This was done in [theta-1]. We would like to express the conditions of the
previous lemma in terms of the fundamental theta constants only.

######  Lemma 6.

Let $\mathcal{X}$ be a genus 2 curve. Then we have the following:

i:

$V_{4}\hookrightarrow Aut(\mathcal{X})$ if and only if the fundamental theta
constants of $\mathcal{X}$ satisfy

(20)
$\begin{split}\left(\theta_{3}^{4}-\theta_{4}^{4}\right)\left(\theta_{1}^{4}-\theta_{3}^{4}\right)\left(\theta_{2}^{4}-\theta_{4}^{4}\right)\left(\theta_{1}^{4}-\theta_{4}^{4}\right)\left(\theta_{3}^{4}-\theta_{2}^{4}\right)\left(\theta_{1}^{4}-\theta_{2}^{4}\right)\\\
\left(-\theta_{4}^{2}+\theta_{3}^{2}+\theta_{1}^{2}-\theta_{2}^{2}\right)\left(\theta_{4}^{2}-\theta_{3}^{2}+\theta_{1}^{2}-\theta_{2}^{2}\right)\left(-\theta_{4}^{2}-\theta_{3}^{2}+\theta_{2}^{2}+\theta_{{1}}^{2}\right)\left(\theta_{4}^{2}+\theta_{3}^{2}+\theta_{2}^{2}+\theta_{1}^{2}\right)\\\
\left({\theta_{1}}^{4}{\theta_{2}}^{4}+{\theta_{3}}^{4}{\theta_{2}}^{4}+{\theta_{1}}^{4}{\theta_{3}}^{4}-2\,\theta_{1}^{2}\theta_{2}^{2}\theta_{3}^{2}\theta_{4}^{2}\right)\left(-{\theta_{3}}^{4}{\theta_{2}}^{4}-{\theta_{2}}^{4}{\theta_{4}}^{4}-{\theta_{3}}^{4}{\theta_{4}}^{4}+2\,\theta_{1}^{2}\theta_{2}^{2}\theta_{3}^{2}\theta_{4}^{2}\right)\\\
\left({\theta_{2}}^{4}{\theta_{4}}^{4}+{\theta_{1}}^{4}{\theta_{2}}^{4}+{\theta_{1}}^{4}{\theta_{4}}^{4}-2\,\theta_{1}^{2}\theta_{2}^{2}\theta_{3}^{2}\theta_{4}^{2}\right)\left({\theta_{1}}^{4}{\theta_{4}}^{4}+{\theta_{3}}^{4}{\theta_{4}}^{4}+{\theta_{1}}^{4}{\theta_{3}}^{4}-2\,\theta_{{1}}^{2}\theta_{2}^{2}\theta_{3}^{2}\theta_{4}^{2}\right)=&0\\\
\end{split}$

ii:

$D_{8}\hookrightarrow Aut(\mathcal{X})$ if and only if the fundamental theta
constants of $\mathcal{X}$ satisfy Eq. (3) in [theta-1]

iii:

$D_{6}\hookrightarrow Aut(\mathcal{X})$ if and only if the fundamental theta
constants of $\mathcal{X}$ satisfy Eq. (4) in [theta-1]

Hence, one can determine very easily the automorphism group of the curve once
its thetanulls are known. However, this is not quite easy since thetanulls are
defined in terms of the complex integrals. However, since the Siegel modular
forms are expressed in terms of the thetanulls as in Eq. (5), we should be
able to express such loci in terms of the modular forms.

Instead, it is much easier to determine the automorphism groups in terms of
the algebraic invariants which are given in terms of the coefficients of the
curve, but are invariant of the equation of the curve. In the next section we
will express such loci in terms of the absolute invariants
$i_{1},i_{2},i_{3}$. Then, using formulas in Eq.(12) we can always express
these equations in terms of the modular forms.

###  6.2. Automorphisms in terms of algebraic invariants

The locus of genus 2 curves with an extra involution is computed in
[deg2]*Theorem 3 in terms of $J_{2},J_{4},J_{6},J_{10}$. It corresponds
precisely to the locus $\det M=0$.

######  Proposition 3.

Let $\mathfrak{p}\in{\mathcal{M}}_{2}$. The following hold:

  1. (1)

If $J_{30}(\mathfrak{p})\neq 0$, then $|\operatorname{Aut}(\mathfrak{p})|=2$
or $\mathfrak{p}$ correspond to the curve $y^{2}=x(x^{5}-1)$. Moreover,
$J_{30}(\mathfrak{p})=0$ if and only if
$V_{4}\hookrightarrow\operatorname{Aut}(\mathfrak{p})$.

  2. (2)

If $J_{30}(\mathfrak{p})=0$ and $J_{16}(\mathfrak{p})=0$, then
$V_{4}\hookrightarrow\operatorname{\overline{Aut}}(\mathfrak{p})$.

Part i) was known to Clebsch and proved independently in [deg2]*Theorem 3. The
proof of 2) can be found in [deg2]*Lemma 3. In the following proposition we
give conditions on the absolute invariants $i_{1},i_{2}$ and $i_{3}$ for each
automorphism to happen.

######  Proposition 4.

Let
$\mathfrak{p}=(i_{1},i_{2},i_{3})\in{\mathcal{M}}_{2}\setminus\\{J_{2}=0\\}$.
Then the following hold:

  * i)

If $\mathfrak{p}=(0,0,0)$ then $\operatorname{Aut}(\mathfrak{p})\cong C_{10}$

  * ii)

If $\mathfrak{p}=(\frac{81}{20},-\frac{729}{200},\frac{729}{25600000})$, then
$\operatorname{Aut}(\mathfrak{p})\cong SL_{2}(3)$

  * iii)

If $\mathfrak{p}=(-\frac{36}{5},\frac{1512}{25},\frac{243}{200000})$, then
$\operatorname{Aut}(\mathfrak{p})\cong GL_{2}(3)$

  * iv)

If the following conditions are satisfied

(21)
$\begin{split}27i_{2}^{4}-9i_{2}^{4}i_{1}+9459597312000i_{3}^{2}i_{1}^{2}-111451255603200i_{3}^{2}i_{1}+55240704i_{3}i_{1}^{4}\\\
-161243136i_{3}i_{1}^{3}+27i_{1}^{6}+240734712102912i_{3}^{2}+264180754022400000i_{3}^{3}\\\
-9i_{1}^{7}+18i_{2}^{2}i_{1}^{4}-54i_{1}^{3}i_{2}^{2}-161243136i_{3}i_{2}^{2}+12441600i_{3}i_{2}^{3}-107495424i_{3}i_{2}i_{1}^{2}\\\
+52254720i_{3}i_{2}^{2}i_{1}-2i_{2}i_{1}^{6}+4i_{2}^{3}i_{1}^{3}-20639121408000i_{3}^{2}i_{2}+8294400i_{3}i_{2}^{2}i_{1}^{2}\\\
-331776i_{3}i_{1}^{5}+2866544640000i_{3}^{2}i_{1}i_{2}+47278080i_{3}i_{1}^{3}i_{2}-2i_{2}^{5}&=0\\\
-243i_{1}^{2}+80i_{1}^{3}-1458i_{2}+540i_{2}i_{1}+100i_{2}^{2}&=0\\\
\end{split}$

and $\mathfrak{p}$ is not one of the cases ii) and iii), then
$\operatorname{Aut}(\mathfrak{p})\cong D_{4}$

  * v)

If the following conditions are satisfied

(22)
$\begin{split}3888i_{1}+432i_{2}-1188i_{1}^{2}+5i_{1}^{3}-25i_{2}^{2}-360i_{2}i_{1}&=0\\\
26873856i_{3}+5184000i_{3}i_{1}^{2}-9331200i_{3}i_{1}-149299200000i_{3}^{2}-729i_{1}^{2}-\\\
27i_{1}^{4}+243i_{1}^{3}+i_{1}^{5}&=0\\\ \end{split}$

and $\mathfrak{p}$ is not one of the cases ii) and iii), then
$\operatorname{Aut}(\mathfrak{p})\cong D_{6}$

  * vi)

If the following is satisfied

(23)
$\begin{split}-27i_{1}^{6}-9459597312000i_{3}^{2}i_{1}^{2}+20639121408000i_{3}^{2}i_{2}+111451255603200i_{3}^{2}i_{1}\\\
-240734712102912i_{3}^{2}-55240704i_{3}i_{1}^{4}-18i_{1}^{4}i_{2}^{2}-8294400i_{3}i_{2}^{2}i_{1}^{2}-47278080i_{3}i_{2}i_{1}^{3}\\\
-264180754022400000i_{3}^{3}-2866544640000i_{3}^{2}i_{2}i_{1}+2i_{1}^{6}i_{2}-4i_{1}^{3}i_{2}^{3}+9i_{1}^{7}+331776i_{3}i_{1}^{5}\\\
+107495424i_{3}i_{2}i_{1}^{2}-27i_{2}^{4}+9i_{1}i_{2}^{4}-52254720i_{3}i_{2}^{2}i_{1}+2i_{2}^{5}+161243136i_{3}i_{2}^{2}\\\
+161243136i_{3}i_{1}^{3}-12441600i_{3}i_{2}^{3}+54i_{1}^{3}i_{2}^{2}&=0\end{split}$

and $\mathfrak{p}$ is not one of the above cases, then
$\operatorname{Aut}(\mathfrak{p})\cong V_{4}$.

###### Proof.

The proof is computational and straightforward. The reader can check
[MR2701993] or [deg2] for details. ∎

If $\mathcal{X}$ has extra involutions (i.e. it has automorphisms other then
the hyperelliptic involution), then $X$ is isomorphic to a curve given by

(24) $y^{2}=x^{6}+ax^{4}+bx^{2}+1$

for appropriate values of $a,b$ (i.e. the discriminant is nonzero). Such form
of genus 2 curves with automorphisms was know to XIX century mathematicians,
but it has been used in the last decade quite often mostly due to Shaska and
Völklein paper [deg2] which first appeared in 2000.

In [deg2] for curves with automorphism were defined the dihedral invariants

(25) $u:=ab,\qquad v:=a^{3}+b^{3}$

which give a birational parametrization of this locus $\mathcal{L}_{2}$ which
is a 2-dimensional subvariety of ${\mathcal{M}}_{2}$. It was the first time
that such parametrization was discovered and since it has become the common
way of computing with genus 2 curves with automorphisms. We can express $u$,
and $v$ in terms of the absolute invariants $i_{1},i_{2},i_{3}$ as in [deg2].
Moreover, the invariant $J_{16}$ is expressed in terms of $u$, and $v$ as
follows

$J_{16}=\left(4u^{3}-v^{2}\right)\left({u}^{2}-110\,u-4\,v+1125\right)^{2}$

We have the following:

######  Proposition 5.

Let $\mathfrak{p}$ be a genus 2 curve such that
$G:=\operatorname{Aut}(\mathfrak{p})$ has an elliptic involution. Then,

a) $G\cong SL_{2}(3)$ if and only if $(u,v)=(0,0)$ or $(u,v)=(225,6750)$.

b) $G\cong GL_{2}(3)$ if and only if $u=25$ and $v=-250$.

c) $G\cong D_{6}$ if and only if $4v-u^{2}+110u-1125=0$, for $u\neq
9,70+30\sqrt{5},25$.

d) $G\cong D_{4}$ if and only if $v^{2}-4u^{3}=0$, for $u\neq 1,9,0,25,225$.
Cases $u=0,225$ and $u=25$ are reduced to cases a),and b) respectively.

The $V_{4}$-locus (i.e., the locus $J_{30}=0$) is birationally parametrized by
dihedral invariants $u,v$ as in [deg2].

###  6.3. The locus $J_{2}=0$

If a point $p\in{\mathcal{M}}_{2}$ is in the locus $J_{2}=0$ then the
following result holds true:

######  Proposition 6.

Let $\mathfrak{p}=(0,t_{1},t_{2},t_{3})\in{\mathcal{M}}_{2}$. Then
$|\operatorname{Aut}(\mathfrak{p})|>2$ if and only if the invariants
$t_{2},t_{3}$ satisfy the Eq. (34) in the Appendix. Moreover,
$\operatorname{Aut}(\mathfrak{p})\cong D_{4}$ if and only if

$y^{2}=x^{5}+x^{3}-\frac{3}{20}x$

and $\operatorname{Aut}(\mathfrak{p})\cong D_{6}$ if and only if

$y^{2}=x^{6}+x^{3}+-\frac{1}{40}$

###### Proof.

We let $\mathfrak{p}$ be a curve with extra automorphisms. Then its equation
can be written as

$y^{2}=x^{6}+ax^{4}+bx^{2}+1$

Compute $t_{2},t_{3}$ in terms of the dihedral invariants $u,v$. Since
$J_{2}=0$, then $u=0$. Hence, $t_{2}$ and $t_{3}$ are expressed only in terms
of $v$. Eliminate $v$ and we get the equation in Eq. (34).

Assume that $\operatorname{Aut}(\mathfrak{p})\cong D_{4}$. Then, the curve has
equation

$y^{2}=x^{5}+x^{3}+sx,$

cf. section 8.2. We then have

$J_{2}=6+40s=0$

Hence, $s=-\frac{3}{20}$.

If $\operatorname{Aut}(\mathfrak{p})\cong D_{4}$, then the equation of the
curve is

$y^{2}=x^{6}+x^{3}+w,$

cf. section 8.3. We then have

$J_{2}=6-240w=0$

Hence, $w=-\frac{1}{40}$.

∎

In the next section we will focus on genus two curves over $\mathbb{Q}$.

##  7\. Genus 2 curves defined over $\mathbb{Q}$

Let $\mathcal{X}$ be a genus two curve defined over $\mathbb{Q}$. The moduli
point in ${\mathcal{M}}_{2}$ corresponding to $\mathcal{X}$ is given by
$\mathfrak{p}=(i_{1},i_{2},i_{3})$. Since $i_{1},i_{2},i_{3}$ are rational
functions in terms of the coeffiecients of $\mathcal{X}$, then
$i_{1},i_{2},i_{3}\in\mathbb{Q}$.

The converse isn’t necessarily true. Let
$\mathfrak{p}=(i_{1},i_{2},i_{3})\in{\mathcal{M}}_{2}(\mathbb{Q})$. The
universal equation of a genus 2 curve corresponding to $\mathfrak{p}$ is
determined in [MS-1], which is defined over a quadratic number field $K$. The
main questions we want to consider is what percentage of the rational moduli
points are defined over $\mathbb{Q}$? How can we determine a minimal equation
for such curves?

The quotient space $C:=\mathcal{X}/\operatorname{Aut}(\mathcal{X})$ is a genus
zero curve, i.e. is isomorphic to a conic which is determined as follows. Let
$\mathbf{x}=(x_{1},x_{2},x_{3})$ be a coordinate in $\mathbb{P}^{2}$ and

(26) $C:\,\,\psi(x_{1},x_{2},x_{3}):=\mathbf{x}^{t}\,M\,\mathbf{x}$

be the corresponding conic for the symmetric matrix $M$ given as in Eq. (18).

There is also a cubic $L$ given by the equation

(27) $L:\;\sum_{1\leq j,k,l\leq 3}a_{jkl}x_{j}x_{k}x_{l}=0$

where the coefficients $a_{jkl}$ are given explicitly in terms of the
invariants in [MS-1].

######  Lemma 7 ([Me]).

The genus 2 curve $\mathcal{X}$ is the intersection of the conic $C$ and the
cubic $L$.

Mestre’s algorithm has been implemented in Magma, Maple, Sage. For an
implementation in Sage see Bouyer/Streng and their paper [streng]. However,
all these implementations have had issues. For example, the implementation in
Sage will not work for curves with $J_{2}=0$.

######  Example 3.

Let $\mathcal{X}$ be the genus 2 curve given by the equation

$y^{2}=x^{6}+x^{5}+x+1/6.$

One can check that for this curve the Igusa-Clebsch invariants are

$[\,0,\,-32000,\,5120000/3,\,295116800000/81\,]$

and the algorithm fails in this case.

######  Theorem 6.

[MS-1]*Thm. 2 For every point
$\mathfrak{p}\in{\mathcal{M}}_{2}\setminus\\{D=0\\}$ such that
$\mathfrak{p}\in{\mathcal{M}}_{2}(k)$, for some number field $K$, there is a
pair of genus-two curves $\mathcal{C}^{\pm}$ given by

$\mathcal{C}^{\pm}:\quad y^{2}=\sum_{i=0}^{6}\,a_{6-i}^{\pm}\,x^{i}\;,$

corresponding to $\mathfrak{p}$, such that $a^{\pm}_{i}\in K(d)$,
$i=0,\dots,6$ as given explicitly in [MS-1]*Eq. 41.

###### Proof.

The proof is exactly the same as [MS-1]*Thm. 4, other then the fact that the
coefficients are expressed in terms of $i_{1},i_{2},i_{3}$. We can take
$J_{2},\dots,J_{10}$ as in Eq. (5.5). Substituting in formulas Eq. (B) we get,

(28) $\begin{split}A&=-\frac{1}{120}\\\ B&=\frac{1}{2^{5}\cdot 3^{5}\cdot
5^{4}}\,(36+5i_{1})\\\ C&=\frac{1}{2^{8}\cdot 3^{8}\cdot
5^{6}}\,(25i_{2}+180i_{1}-216)\\\ D&=\frac{1}{2^{12}\cdot 3^{14}\cdot
5^{10}}\,(2700i_{2}-675i_{1}^{2}-250i_{1}i_{2}-86400000i_{3}+13500i_{1}-34992)\\\
\end{split}$

Then $d^{2}$ in [MS-1] becomes

(29) $\begin{split}d^{2}=&-\frac{1}{2^{50}\cdot 3^{56}\cdot
5^{30}}\,(9i_{1}^{7}+2i_{1}^{6}i_{2}-27i_{1}^{6}-18i_{1}^{4}i_{2}^{2}-4i_{1}^{3}i_{2}^{3}+331776i_{1}^{5}i_{3}+54i_{1}^{3}i_{2}^{2}+9i_{1}i_{2}^{4}\\\
&+2i_{2}^{5}-55240704i_{1}^{4}i_{3}-47278080i_{1}^{3}i_{2}i_{3}-8294400i_{1}^{2}i_{2}^{2}i_{3}-27i_{2}^{4}+161243136i_{1}^{3}i_{3}\\\
&+107495424i_{1}^{2}i_{2}i_{3}-52254720i_{1}i_{2}^{2}i_{3}-12441600i_{2}^{3}i_{3}-9459597312000i_{1}^{2}i_{3}^{2}\\\
&-2866544640000i_{1}i_{2}i_{3}^{2}+161243136i_{2}^{2}i_{3}+111451255603200i_{1}i_{3}^{2}\\\
&+20639121408000i_{2}i_{3}^{2}-264180754022400000i_{3}^{3}-240734712102912i_{3}^{2})\cdot\\\
&\cdot(675i_{1}^{2}+250i_{1}i_{2}-13500i_{1}-2700i_{2}+86400000i_{3}+34992)\end{split}$

Notice that $d^{2}$ has two significant factors: one is $J_{30}$ which
correspond exactly to the locus of the curves with extra involutions, and the
other one is the Clebsch invariant $D$ expressed in terms of
$i_{1},i_{2},i_{3}$. Substituting $A,B,C,D$ for the expressions in Eq. (28) we
get $a_{0},\dots,a_{6}$ in terms of $i_{1},i_{2},i_{3}$ as in the Appendix. ∎

######  Remark 4.

Notice that the above theorem does not provide an equation for the curve if
the Clebsch invariant $D=0$.

###  7.1. Genus 2 curve with $\operatorname{Aut}(\mathcal{X})\cong V_{4}$

The above theorem is valid for any point $\mathfrak{p}\in{\mathcal{M}}_{2}$,
including the points $\mathfrak{p}\in{\mathcal{M}}_{2}$ with
$V_{4}\hookrightarrow\operatorname{Aut}(\mathfrak{p})$. On contrary, Mestre’s
approach which has been implemented in many computer algebra packages is valid
only for $\mathfrak{p}\in{\mathcal{M}}_{2}$ with
$\operatorname{Aut}(\mathfrak{p})\cong C_{2}$.

Let $\mathcal{X}$ be a genus 2 curve with
$\operatorname{Aut}(\mathcal{X})\cong V_{4}$. The $V_{4}$ locus has dimension
two and is parametrized by the parameters $u,$ and $v$ as explained above. The
rational model of a genus 2 curve with automorphism group $V_{4}$ given in
terms of this parameters $u$, and $v$ has equation given as follows

(30)
$\begin{split}f(x,z)=&\left({v}^{2}+{u}^{2}v-2\,{u}^{3}\right){x}^{6}+2\,\left({u}^{2}+3\,v\right)\left({v}^{2}-4\,{u}^{3}\right){x}^{5}z+\\\
&+\left(15\,{v}^{2}-{u}^{2}v-30\,{u}^{3}\right)\left({v}^{2}-4\,{u}^{3}\right){x}^{4}z^{2}+4\,\left(5\,v-{u}^{2}\right)\left({v}^{2}-4\,{u}^{3}\right)^{2}{x}^{3}z^{3}+\\\
&+\left({v}^{2}-4\,{u}^{3}\right)^{2}\left(15\,{v}^{2}-{u}^{2}v-30\,{u}^{3}\right){x}^{2}z^{4}+2\,\left({v}^{2}-4\,{u}^{3}\right)^{3}\left({u}^{2}+3\,v\right)xz^{5}+\\\
&+\left({v}^{2}-4\,{u}^{3}\right)^{3}\left({v}^{2}+{u}^{2}v-2\,{u}^{3}\right)z^{6}\end{split}$

where $u$, and $v$ are expressed in terms of the absolute invariants
$(i_{1},i_{2},i_{3})$ as in [deg2]. The discriminant of the form is

$\Delta_{f}=2^{36}\,\,\left({u}^{2}+18\,u-4\,v-27\right)^{2}\left(4\,{u}^{3}-{v}^{2}\right)^{15}{u}^{30}.$

The case $\Delta_{f}=0$ correspond exactly to the cases when
$|\operatorname{Aut}(\mathfrak{p})|>4$ which is treated below. Notice that the
factors $2$ and $u$ have exponents $\geq 30$.

###  7.2. Genus 2 curve with $\operatorname{Aut}(\mathcal{X})\cong D_{4}$

Let $\mathcal{X}$ be a genus 2 curve with
$\operatorname{Aut}(\mathcal{X})\cong D_{4}$. The $D_{4}$ locus is a dimension
one locus parametrized by the parameter $s$, as proved in [ants]*Lemma 3ants.
The rational model of a genus 2 curve with automorphism group $D_{4}$ given in
terms of this parameter $s$ is as follows

$y^{2}={x}^{5}+{x}^{3}+s\,x$

where $s$ can be expressed in terms of the absolute invariants
$(i_{1},i_{2},i_{3})$ as follows

$s=-\frac{3}{4}\frac{345\,i_{1}^{2}+50\,i_{1}i_{2}-1296i_{1}-90i_{2}}{2925i_{1}^{2}+250\
i_{1}i_{2}-54000i_{1}-9450i_{2}+139968}.$

The discriminant is $\Delta_{f}=2^{4}\cdot\,{s}^{3}\left(4\,s-1\right)^{2}$.

###  7.3. Genus 2 curve with $\operatorname{Aut}(\mathcal{X})\cong D_{6}$

Now let us consider the case when $\mathcal{X}$ is a genus 2 curve with
$\operatorname{Aut}(\mathcal{X})\cong D_{6}$. The $D_{6}$ locus is a dimension
one locus parametrized by the parameter $w$ as proved in [ants]*Lemma 3. The
rational model of a genus 2 curve with automorphism group $D_{6}$ given in
terms of this parameter $w$ is as follows

$y^{2}={x}^{6}+{x}^{3}+w$

where the parameter $w$ given in terms of the absolute invariants
$(i_{1},i_{2},i_{3})$ is

$w=\frac{1}{4}\frac{540i_{1}^{2}+100i_{1}i_{2}-1728i_{1}+45i_{2}}{2700i_{1}^{2}+1000i_{1}i_{2}+204525i_{1}+40950i_{2}-708588}$

The discriminant of the form is
$\Delta_{f}=-3^{6}\cdot{w}^{2}\left(4\,w-1\right)^{3}$.

Hence, for every point $\mathfrak{p}\in{\mathcal{M}}_{2}$ we can find an
equation of a curve $C$ corresponding to $\mathfrak{p}$. Below we give an
example to illustrate how this work in our code in Sage.

######  Example 4.

Consider the curve

$\begin{split}y^{2}=&\,\,4294967297\,t^{6}+77309411328\,t^{5}+579820584969\,t^{4}+2319282339816\,t^{3}+5218385264643\,t^{2}\\\
&+6262062317592\,t+3131031158771\\\ \end{split}$

By using the functions above we find that this curve has automorphism group
$D_{6}$. Hence, we compute $w=2^{33}$. The equation of a curve over
$\mathbb{Q}$ isomorphic (over $\mathbb{C}$) to our curve is

$y^{2}=x^{6}+x^{3}+2^{33}$

In the next section we will see if we can transform this curve over
$\mathbb{Q}$ to another curve with smaller coefficients.

In the next section we will see how to minimize the discriminant of a genus
two curve when its equation is given in Weierstrass form.

##  8\. Minimal discriminant for Weierstrass equations

Let $K$ be a field with a discrete valuation $\mathfrak{v}$ and ring of
integers $\mathcal{O}_{K}$ and $C$ an irreducible, smooth, algebraic curve of
genus $g\geq 1$ defined over $K$ and function field $K(C)$. The discriminant
$\mathfrak{D}_{C/K}$ is an important invariant of the function field of the
curve and therefore of the curve. Since the discriminant is a polynomial given
in terms of the coefficients of the curve, then it is an ideal in the ring of
integers $\mathcal{O}_{K}$ of $K$. The valuation of this ideal is a positive
integer. A classical question is to find an equation of the curve such that
this valuation is minimal, in other words the discriminant is minimal.

When $g=1$, so that $C$ is an elliptic curve, there is an extensive theory of
the minimal discriminant ideal $\mathfrak{D}_{C/K}$. Tate [ta-75] devised an
algorithm how to determine the Weierstrass equation of an elliptic curve with
minimal discriminant as part of his larger project of determining Neron models
for elliptic curves. Such ideas were extended for genus 2 in [liu]. Here we
mostly follow [rachel].

For a binary form $f(x,z)$ and a matrix $M=\begin{bmatrix}a,b\\\
c,d\end{bmatrix}$, such that $M\in GL_{2}(k)$, we have that
$f^{M}:=f(aX+bZ,cX+dZ)$ has discriminant

$\Delta(f^{M})=(\det M)^{\frac{d(d-1)}{2}}\cdot\Delta(f)$

This property of the discriminant is crucial in our algorithm which is
explained later.

###  8.1. Discriminant of a curve

The concept of a minimal discriminant for elliptic curves was defined by Tate
and others in the 1970-s; see [ta-75]. Such definitions and results were
generalized by Lockhart in [lockhart] for hyperelliptic curves.

Let us assume now that $K$ is an algebraic number field with field of integers
$\mathcal{O}_{K}$. Let $M_{K}$ be the set of all inequivalent absolute values
on $K$ and $M_{K}^{0}$ the set of all non-archimedean absolute values in
$M_{K}$. We denote by $K_{\mathfrak{v}}$ the completion of $K$ for each
$\mathfrak{v}\in M_{K}^{0}$ and by $\mathcal{O}_{\mathfrak{v}}$ the valuation
ring in $K_{\mathfrak{v}}$. Let $\mathfrak{p}_{v}$ be the prime ideal in
$\mathcal{O}_{K}$ and $\mathfrak{m}_{v}$ the corresponding maximal ideal in
$K_{\mathfrak{v}}$. Let $(\mathcal{X},P)$ be a superelliptic curve of genus
$g\geq 2$ over $K$.

If $\mathfrak{v}\in M_{K}^{0}$ we say that $\mathcal{X}$ is integral at
$\mathfrak{v}$ if $\mathcal{X}$ is integral when viewed as a curve over
$K_{\mathfrak{v}}$. We say that $\mathcal{X}$ is minimal at $\mathfrak{v}$
when it is minimal over $K_{\mathfrak{v}}$.

An equation of $\mathcal{X}$ over $K$ is called integral (resp. minimal) over
$K$ if it is integral (resp. minimal) over $K_{\mathfrak{v}}$, for each
$\mathfrak{v}\in M_{K}^{0}$.

Next we will define the minimal discriminant over $K$ to be the product of all
the local minimal discriminants. For each $\mathfrak{v}\in M_{K}^{0}$ we
denote by $\Delta_{\mathfrak{v}}$ the minimal discriminant for
$(\mathcal{X},P)$ over $K_{\mathfrak{v}}$. The minimal discriminant of
$(\mathcal{X},P)$ over $K$ is the ideal

$\Delta_{\mathcal{X}/K}=\prod_{\mathfrak{v}\in
M_{K}^{0}}\mathfrak{m}_{\mathfrak{v}}^{\mathfrak{v}(\Delta_{\mathfrak{v}})}$

We denote by $\mathfrak{a}_{\mathcal{X}}$ the ideal
$\mathfrak{a}_{\mathcal{X}}=\prod_{\mathfrak{v}\in
M_{K}^{0}}\mathfrak{p}_{\mathfrak{v}}^{\mathfrak{v}(\Delta_{\mathfrak{v}})}$.

Let $\mathcal{X}$ be a genus two curve with equation $y^{2}=f(x)$. The
discriminant of $\mathcal{X}$ is the discriminant of $f(x)$, hence
$J_{10}(f)$, a degree 10 polynomial in terms of the coefficients of $f(x)$.

Let $M\in GL_{2}(K)$ such that $\det M=\lambda$. Then,
$\Delta(f^{M})=\lambda^{30}\,\Delta(f)$. Assume $\Delta(f)=p^{\alpha}\cdot N$,
where $\alpha>30$, for some prime $p$ and some integer $N$ such that
$(p,N)=1$. We perform the coordinate change

$x\to\frac{1}{p}x$

on $f(x)$. Then, the new discriminant is
$\Delta^{\prime}=\frac{1}{p^{30}}\,\cdot p^{\alpha}\cdot N=p^{\alpha-30}\cdot
N$. Hence, we have the following

######  Lemma 8.

A genus 2 curve $\mathcal{X}_{g}$ with integral equation

$y^{2}=a_{6}x^{6}+\cdots a_{1}x+a_{0}$

has minimal discriminant if $\mathfrak{v}(\Delta)<30$.

Thus, we factor $\Delta$ as a product of primes, say
$\Delta=p_{1}^{\alpha_{1}}\cdots p_{r}^{\alpha_{r}}$, and take $u$ to be the
product of those powers of primes with exponents $\alpha_{i}\geq 30$. Then,
the transformation $x\mapsto\frac{1}{p^{u}}\cdot x$ will reduce the
discriminant.

###  8.2. Twists

Of course we can make the discriminant as small as we want it if we allow
twists.

Let $\mathcal{X}$ be hyperelliptic curve with an extra automorphism of order
$n\geq 2$. Then, from [sh-issac] we know that the equation of $\mathcal{X}$
can be written as

$y^{2}=f(x^{n})$

If $\mathcal{X}$ is given with such equation over $\mathbb{Q}$ and
discriminant

$\Delta=u\,N,\quad\textit{such that }\quad(u,N)=1$

then for any transformation $\tau:x\mapsto{u^{-\frac{n}{30}}}\cdot x$ would
lead to $\mathcal{X}^{\tau}$ defined over $\mathbb{Q}$ and isomorphic to
$\mathcal{X}$ over $\mathbb{C}$. Hence, $\mathcal{X}^{\tau}$ is a twist of
$\mathcal{X}$ with discriminant

$\Delta^{\prime}=\frac{1}{u}\cdot u\,N=N,$

Hence, in this case we can reduce the discriminant even further.

######  Lemma 9.

Let $\mathcal{X}$ be a genus 2 curve with
$\operatorname{Aut}(\mathcal{X})\cong V_{4}$. Assume that $\mathcal{X}$ has
equation over $\mathbb{Q}$ as $y^{2}=f(x^{2})$. Then, $\mathcal{X}$ has
minimal discriminant $\Delta$ if $\mathfrak{v}(\Delta)<15$.

If $\operatorname{Aut}(\mathcal{X})\cong D_{6}$ and $\mathcal{X}$ has equation
over $\mathbb{Q}$ as $y^{2}=f(x^{3})$. Then, $\mathcal{X}$ has minimal
discriminant $\Delta$ if $\mathfrak{v}(\Delta)<10$.

###### Proof.

The proof is an immediate consequence of the above remarks. ∎

The following gives a universal curve with minimal discriminant for the
$V_{4}$-locus.

######  Lemma 10.

Let $\mathcal{X}$ be a genus 2 curve such that
$\operatorname{Aut}(\mathcal{X})\cong V_{4}$ and $(u,v)$ the corresponding
dihedral invariants. Then, $\mathcal{X}$ has minimal discriminant and is
defined over its field of moduli when given by the equation

(31) $y^{2}=b_{6}x^{6}+\cdots+b_{1}x+b_{0}$

where

$\begin{split}b_{6}=&-{\frac{\left(2\,{u}^{3}-{u}^{2}v-{v}^{2}\right)}{2^{6}{u}^{6}}}\\\
b_{5}=&-2{\frac{\left({u}^{2}+3\,v\right)\left(4\,{u}^{3}-{v}^{2}\right)}{2^{5}{u}^{5}}}\\\
b_{4}=&{\frac{\left(30\,{u}^{3}+{u}^{2}v-15\,{v}^{2}\right)\left(4\,{u}^{3}-{v}^{2}\right)}{2^{4}{u}^{4}}}\\\
b_{3}=&-4\,{\frac{\left({u}^{2}-5\,v\right)\left(4\,{u}^{3}-{v}^{2}\right)^{2}}{2^{3}{u}^{3}}}\\\
b_{2}=&-{\frac{\left(30\,{u}^{3}+{u}^{2}v-15\,{v}^{2}\right)\left(4\,{u}^{3}-{v}^{2}\right)^{2}}{2^{2}{u}^{2}}}\\\
b_{1}=&-{\frac{\left({u}^{2}+3\,v\right)\left(4\,{u}^{3}-{v}^{2}\right)^{3}}{u}}\\\
b_{0}=&\left(2\,{u}^{3}-{u}^{2}v-{v}^{2}\right)\left(4\,{u}^{3}-{v}^{2}\right)^{3}\\\
\end{split}$

###### Proof.

We know that the equation of the curve in Eq. (30) is defined over its field
of moduli. Let

$\sigma:(x,y)\mapsto\left(\frac{1}{2u}\,x,\frac{1}{(2u)^{6}}\,y\right).$

Then the discriminant of the new curve $\mathcal{X}^{\sigma}$ is

$\Delta_{f}=2^{6}\,\,\left({u}^{2}+18\,u-4\,v-27\right)^{2}\,\,\left(4u^{3}-v^{2}\right)^{15}.$

Since all exponents are $<15$, this discriminant can not be further reduced
over $\mathbb{Q}$. The equation of the curve in this case becomes as in Eq.
(31).

∎

Notice that for any curve $\mathcal{X}$ with
$\operatorname{Aut}(\mathcal{X})\cong V_{4}$, we have $u\neq 0$, so the curve
in Eq. (31) is defined everywhere.

In the next section we will see that smaller discriminant doesn’t necessarily
mean small coefficients. We illustrate below with an example.

######  Example 5.

Let $u=35$ and $v=6$. Using the function

    RatModSha_uv (35, 6)

we get

    (-4)*(19591*t^6+106564876*t^5-55428309960*t^4+35132884438720*t^3
    +9503959738981440*t^2+3132997049150231296*t-98758721142240034304)

Notice that this curve has large coefficients. Especially, when we have that
for $u=35$ and $v=6$ we get corresponding $(a,b)=(2,3)$ and therefore a curve

$y^{2}=x^{6}+2x^{4}+3x^{3}+1$

As it is shown in [MR3525576], it is almost always true that the curve
$y^{2}=x^{6}+ax^{4}+bx^{2}+1$ has smaller coefficients when such
$a,b\in\mathbb{Q}$ do exist.

##  9\. Constructing the databases

Now that we have all the necessary results we are ready to construct all three
dictionaries.

###  9.1. Curves with height $\leq 10$: the dictionary $\mathcal{L}_{1}$

In order to create the list $\mathcal{L}_{1}$ we first compile a list of all
7-tuples $(a_{0},\dots,a_{6})$ and from this list eliminate all the tuples
with $J_{10}=0$. Then, for each tuple we compute the moduli point
$\mathfrak{p}=(i_{1},i_{2},i_{3})$ and all the other invariants as described
in the previous sections.

Table 1. The number of curves for a given height $\operatorname{h}$ | $\mathbb{P}^{6}$ |  $\\#$ of curves | $V_{4}$ | $D_{4}$ | $D_{6}$
---|---|---|---|---|---
1 | 1 093 | 230 | 28 | 11 | 2
2 | 37 969 | 8 593 | 230 | 40 | 7
3 | 409 585 | 88 836 | 1054 | 112 | 26

######  Remark 5.

The bound for the number of genus 2 curves of height $\operatorname{h}$ is
$<(2h+1)^{7}$. For example, for $\operatorname{h}=1$ this bound is
$<3^{7}=2187$. In fact, as shown in Table 1 the number of such curves is 230,
all of which are listed in [MR3525576]*Table 2, 3.

In the Table 1 we display the number of curves with extra automorphism for a
given height $\operatorname{h}$.

###  9.2. Curve with extra involutions: the dictionary $\mathcal{L}_{2}$

The list $\mathcal{L}_{2}$ was computed in [MR3525576] for
$1\leq\operatorname{h}\leq 100$. Since every genus 2 curve with extra
automorphisms can be written (over $\mathbb{C})$ as

(32) $y^{2}=x^{6}+ax^{4}+bx^{2}+1$

we create the list of tuples $(1,0,b,0,a,0,1)$ for $a,b\leq\operatorname{h}$.
We go through the lists and delete the ones which have $J_{10}=0$. This number
is displayed in [MR3525576]*Table 1 for $1\leq\mathfrak{h}\leq 100$, including
information on how many points from $\mathcal{L}_{2}$ are already in
$\mathcal{L}_{1}$, how many of these curves have automorphism group $D_{4}$
and how many have automorphism group $D_{6}$.

Table 2. Curves with extra involutions and height $\operatorname{h}\leq 101$ $\operatorname{h}$ |  | new pts. | $D_{4}$ | $D_{6}$ | Total pts
---|---|---|---|---|---
| $J_{10}\neq 0$ | in ${\mathcal{M}}_{2}$ |  |  |
1 | 8 | 4 | 1 | 0 | 5
2 | 24 | 9 | 3 | 0 | 14
3 | 47 | 12 | 4 | 0 | 26
4 | 79 | 17 | 6 | 0 | 43
5 | 119 | 20 | 7 | 0 | 63
6 | 167 | 25 | 9 | 0 | 88
7 | 223 | 28 | 11 | 0 | 116
8 | 287 | 33 | 13 | 0 | 149
9 | 359 | 36 | 15 | 0 | 185
10 | 439 | 41 | 17 | 0 | 226

In [MR3525576] it is discussed when such curves have minimal height and how a
reduction as in [MR3525574] is easier to perform in this case. In Table 2 we
display only the first ten rows of Table 1 from [MR3525576].

There are 20 697 curves in $\mathcal{L}_{2}$, such that for each
$\operatorname{h}$ we have roughly $4\operatorname{h}$ curves. So it is
expected that the number of curves of height $\leq h$, with equation Eq. (32),
defined over $\mathbb{Q}$ is $\leq
4\frac{\operatorname{h}(\operatorname{h}+1)}{2}$; see Beshaj [MR3525576] for
more details.

The first curve with automorphism group isomorphic to $D_{4}$ occurs for
$\operatorname{h}=1$. It is only one such curve, namely the curve with
equation

$y^{2}=x^{6}+x^{4}+x^{2}+1.$

The are only two curves with automorphism group isomorphic to $D_{6}$, which
occur for $\operatorname{h}=79$ and $\operatorname{h}=83$. namely the curve

$y^{2}=x^{6}+79x^{4}-17x^{2}+1$

and the curve

$y^{2}=x^{6}+83x^{6}+19x^{2}+1.$

There are 195 curves with automorphism group isomorphic to $D_{4}$. For the
largest height $\operatorname{h}=101$ there are 405 curves two of which have
automorphism group isomorphic to $D_{4}$.

In [MR3525574] is proved the following theorem:

######  Theorem 7 ([MR3525574]).

Let ${\mathfrak{p}}\in\mathcal{M}_{2}(\mathbb{Q})$ be such that
$\operatorname{Aut}({\mathfrak{p}})\cong V_{4}$. There is a genus 2 curve
$\mathcal{X}$ corresponding to $\mathfrak{p}$ with equation
$y^{2}\,z^{4}=f(x^{2},z^{2})$, where

(33) $f(x,z)=x^{6}-s_{1}x^{4}z^{2}+s_{2}x^{2}z^{4}-z^{6}.$

If $f\in\mathbb{Z}[x,z]$, then $f(x,z)$ or $f(-z,x)$ is a reduced binary form.

It is interesting to not that from 20 292 such curves we found only 57 which
do not have minimal absolute height. For more details see the discussion in
the last section of [MR3525574].

###  9.3. Curves with small moduli height: the dictionary $\mathcal{L}_{3}$

For the last dictionary $\mathcal{L}_{3}$ we create a list of all points
$[x_{0}:x_{1}:x_{2}:x_{3}]$ of projective height $\leq\mathfrak{h}$ in
$\mathbb{P}^{3}(\mathbb{Q})$, for some integer $\mathfrak{h}\geq 1$. Each such
point correspond to the point
$[J_{2}^{5}:J_{4}J_{2}^{3}:J_{6}J_{2}^{2}:J_{10}]$. The number of such tuples
(up to equivalence in $\mathbb{P}^{3}$) is given in column 2 of Table 3. Not
all such points correspond to a genus 2 curve. We delete all the points such
that $x_{3}=0$ since they correspond to the cases when $J_{10}=0$. What is
left on the list is the number of points in the moduli space
${\mathcal{M}}_{2}$ with moduli height $\leq\mathfrak{h}$. This number is
given in the third column in the Table 3.

For each given point $\mathfrak{p}=(r,i_{1},i_{2},i_{3})$ in this list we find
the equation of the curves as described above. As we already know we don’t get
a curve defined over $\mathbb{Q}$ in each case. The number of points which are
defined over $\mathbb{Q}$ is given in column four. From those points
$\mathfrak{p}\in{\mathcal{M}}_{2}$ such that the field of definition is
$\mathbb{Q}$, the number of points with automorphisms is given in column five.

Table 3. Genus 2 curves with bounded moduli height $\mathfrak{h}$ | $\mathbb{P}^{3}(\mathbb{Q})$ | $\mathfrak{p}\in{\mathcal{M}}_{2}(\mathbb{Q})$ | $M_{\mathfrak{p}}=F_{\mathfrak{p}}$ | $|\operatorname{Aut}(\mathfrak{p})|>2$ | ratio
---|---|---|---|---|---
| $n_{1}$ | $n_{2}$ | $n_{3}$ | $n_{4}$ |
1 | 40 | 27 | 20 | 15 | 0.25
2 | 272 | 223 | 124 | 75 | 0.4
3 | 1120 | 975 | 514 | 243 | 0.53
4 | 2928 | 2639 | 1311 | 507 | 0.61
5 | 6928 | 6351 | 3056 | 1035 | 0.66
6 | 12768 | 11903 | 5561 | 1587 | 0.71
7 | 23760 | 22319 | 9963 | 2667 | 0.73
8 | 38128 | 36111 | 15648 | 3771 | 0.76
9 | 60640 | 57759 | 24214 | 5427 | 0.78
10 | 88448 | 84703 | 34936 | 7107 | 0.8
11 | 131088 | 125903 | 50630 | 9867 | 0.81
12 | 177712 | 171375 | 68046 | 12123 | 0.82
13 | 248080 | 239727 | 93229 | 16011 | 0.83
14 | 324736 | 314655 | 120358 | 19395 | 0.84
15 | 427968 | 415583 | 157569 | 23907 | 0.85
16 | 542720 | 528031 | 199136 | 28419 | 0.86
17 | 700032 | 681887 | 252947 | 35139 | 0.86
18 | 857328 | 836591 | 307964 | 40251 | 0.87
19 | 1076928 | 1051871 | 381219 | 48675 | 0.87

The following question is natural: What percentage of rational points
$\mathfrak{p}\in{\mathcal{M}}_{2}(\mathbb{Q})$ with a fixed moduli height
$\mathfrak{h}$ have $\mathbb{Q}$ as a field of definition, when $\mathfrak{h}$
becomes arbitrarily large? In other words, what is the limit

$\lim_{\mathfrak{h}\to\infty}\frac{n_{3}}{n_{2}}?$

######  Lemma 11.

For large enough moduli height $\mathfrak{h}\in{\mathcal{M}}_{2}(\mathbb{Q})$,
the majority of genus 2 curves with moduli height $\mathfrak{h}$ are not
defined over $\mathbb{Q}$.

###### Proof.

Let $\mathfrak{p}\in{\mathcal{M}}_{2}(\mathbb{Q})$ with moduli height
$\mathfrak{h}_{0}$. Then, $\mathfrak{p}=(i_{1},i_{2},i_{3})$ for
$i_{1},i_{2},i_{3}\in\mathbb{Q}$. The equation of a curve $\mathcal{X}$
corresponding to $\mathfrak{p}$ is the intersection of the conic
$C:\;\psi(x_{1},x_{2},x_{3})=0$ and the cubic $L:\;\phi(x_{1},x_{2},x_{3})=0$
which are both defined over $\mathbb{Q}$, since their equations are given in
terms of $i_{1},i_{2},i_{3}$. The intersection $C\cap L$ is obtained as the
resultant of the corresponding equations with respect to one of the variables
$x_{1},x_{2},x_{3}$. This resultant contains a square root which is given in
terms of $i_{1},i_{2},i_{3}$; see [MS-1]. $\mathcal{X}$ is defined over
$\mathbb{Q}$ if and only if the expression inside the square root is a
complete square. Since this is a surface in $\mathbb{Q}^{3}$, the set of
points of fixed projective height which make it a complete square has measure
zero. This completes the proof. ∎

Next we will consider a similar question. What percentage of curves defined
over $\mathbb{Q}$ don’t have extra automorphisms when $\mathfrak{h}$ becomes
arbitrary large? In other words, what it the limit

$\lim_{\mathfrak{h}\to\infty}\frac{n_{3}-n_{4}}{n_{3}}?$

Thus, we want to determine the ratio of points of height $\mathfrak{h}$ in
${\mathcal{M}}_{2}(\mathbb{Q})$ for which $M_{\mathfrak{p}}\neq
F_{\mathfrak{p}}$ over the total number of points of height $\mathfrak{h}$ in
${\mathcal{M}}_{2}(\mathbb{Q})$ as $\mathfrak{h}\to\infty$.

######  Lemma 12.

For large enough moduli height $\mathfrak{h}$, the majority of genus 2 curves
with moduli height $\mathfrak{h}$ don’t have extra automorphisms.

###### Proof.

The number of points with height $\mathfrak{h}$ in the projective space
increases as $\mathfrak{h}$ increases. However, such points
$\mathfrak{p}=(i_{1},i_{2},i_{3})$ have to satisfy the equation $J_{30}=0$
when they have automorphisms. ∎

There is another database of genus 2 curves described in [Booker:2016aa] which
is has all the curves with discriminant $<1000$. We checked our database to
see how many of our curves would be with small discriminant. We have to warn
the reader that our definition of the discriminant is just $J_{10}$ and
slightly different from what is used in [Booker:2016aa]. The majority of
curves in the third dictionary have small discriminant, but this is not a
surprise since small moduli height forces $J_{10}$ to be small. Only one curve
in the second dictionary has $|J_{10}|<1000$. Six curves in the first
dictionary have $|J_{10}|<10000$. It would be interesting to see exactly how
the database in [data] and [Booker:2016aa] intersect.

##  Appendix A Functions of the genus 2 package

## Genus 2 package

Input: a sextic polynomial $f(t)$

`J2` |  Igusa invariant $J_{2}$
---|---
`J4` |  Igusa invariant $J_{4}$
`J6` |  Igusa invariant $j_{6}$
`J10` |  Igusa invariant $J_{10}$
`J30` |  $J_{30}$: the $V_{4}$-locus
`Igusa` |  Igusa invariants $[J_{2},J_{4},J_{6},J_{10}]$
`Clebsch` | Invariants [A, B, C, D]
`i_1` |  absolute invariant $i_{1}$
`i_2` |  absolute invariant $i_{2}$
`i_3` |  absolute invariant $i_{3}$
`j1` |  Igusa function $j_{1}$
`j2` |  Igusa function $j_{2}$
`j3` |  Igusa function $j_{3}$
`RatMod` | Rational model of the curve over when such model exists.
`RatModSha` |  Rational model of the curve over its minimal field of definition as in Shaska [deg2]
`RatModMe` |  Rational model over $\mathbb{Q}$, when such model exists, as in Mestre [Me]
`height` | Height of the sextic
`EquivBin` | Checks if sextics are equivalent
`RatModTable` | Rational Model from the Table of minimal models
`MinField` | Minimal field of definition
`Info` |  Displays information about the curve $y^{2}=f(t)$
`RatForm` |  Rational Model from Malmendier/Shaska [MS-1]

Input: the moduli point $(i_{1},i_{2},i_{3})$

`J30_j` |  $J_{30}$ in terms of $i_{1},i_{2},i_{3}$
---|---
`Igusa_i` | $J_{2},\dots,J_{10}$
`Clebsch_i` | Invariants [A, B, C, D]
`ClebschMatrix` |  ClebschMatrix in terms of invariants $i_{1},i_{2},i_{3}$
`Cubic` |  Cubic defined in Eq. (27)
`Conic` |  Conic defined in Eq. (26)
`L_D4` |  Locus of curves with group $D_{4}$
`L_D6` |  Locus of curves with group $D_{6}$
`AutGroup` | Automorphism group of the curve
`Sh_u_v` |  Shaska invariants $u,v$
`ModHeight` | Modular height

## Moduli Space

`curves_moduli` |  Computes the number of rational points of height $\mathfrak{h}$ in the moduli space and
---|---
| how many of those have a rational model
`NumbCurvMod` |  This function computes the total number of rational points of moduli height $\mathfrak{h}$,
|  how many of them have a rational model over $\mathbb{Q}$, how many of them
have
| automorphisms and the ratio=obstruction/ fine points
`moduli_points` |  Computes the number of rational points of height $\mathfrak{h}$ in the moduli space
`MoPtsCurvAut` | Moduli points with automorphisms
`` |

## Creating the databases

`Curves(h, L)` |  Creates the dictionary $\mathcal{L}_{1}$ of curves with height $h$
---|---
`CurvesAut(h, L)` |  Creates the dictionary $\mathcal{L}_{2}$ of curves with automorphisms
`CurvHe` |  Number of curves with height $h$
`SelfRec(f)` | Checks if a sextic is self reciprocal
`CurvHeW(h, w)` |  Number of curves with height $h$ and $w$
`NCWT(h, w)` |  Number of curves with height $\operatorname{h}$ and twists $w$
`CurvesTabOverQ(h, w)` |  Counts the number of curves over $\mathbb{Q}$, including twist, for given height.

##  Appendix B Basic Invariants and relations among them

Clebsch invariants $A,B,C$ are in terms of the coefficients

$\begin{split}A=&\;2\,a_{6}\,a_{0}-\frac{1}{3}\,a_{5}\,a_{1}+\frac{2}{15}\,a_{4}\,a_{2}-\frac{1}{20}\,a_{3}^{2}\\\
B=&-\left(\frac{4}{225}\,a_{6}\,a_{2}\,a_{4}\,a_{0}-\frac{2}{75}\,a_{6}\,a_{2}\,a_{3}\,a_{1}+\frac{8}{1125}\,a_{6}\,a_{2}^{3}-\frac{2}{75}\,a_{5}\,a_{3}\,a_{4}\,a_{0}\frac{2}{225}\,a_{5}a_{3}^{2}\,a_{1}\right.\\\
&-\frac{2}{1125}\,a_{5}\,a_{3}\,a_{2}^{2}+\frac{8}{1125}\,a_{4}^{3}\,a_{0}-\frac{2}{1125}\,a_{4}^{2}\,a_{3}\,a_{1}+\frac{14}{5625}\,a_{4}^{2}\,a_{2}^{2}-\frac{2}{225}\,a_{5}\,a_{2}\,a_{4}\,a_{1}\\\
&+\frac{2}{45}\,a_{5}^{2}\,a_{2}\,a_{0}+\frac{2}{45}\,a_{6}\,a_{1}^{2}\,a_{4}-\frac{2}{9}\,a_{6}\,a_{1}\,a_{5}\,a_{0}-\frac{8}{5625}\,a_{4}\,a_{3}^{2}\,a_{2}+\frac{2}{3}\,a_{6}^{2}\,a_{0}^{2}\\\
&\left.+\frac{2}{75}\,a_{6}\,a_{0}\,a_{3}^{2}+\frac{1}{3750}\,a_{3}^{4}\right)\\\
C=&-\frac{2}{46875}\,a_{4}^{3}\,a_{2}^{3}-\frac{1}{5625}\,a_{4}^{4}\,a_{1}^{2}-\frac{1}{5625}\,a_{5}^{2}\,a_{2}^{4}-\frac{2}{9}\,a_{6}^{3}\,a_{0}^{3}+\frac{11}{5625}\,a_{6}\,a_{2}\,a_{3}^{2}\,a_{4}\,a_{0}\\\
&-\frac{7}{1125}\,a_{6}\,a_{2}^{2}\,a_{5}\,a_{0}\,a_{3}+\frac{4}{225}\,a_{6}\,a_{2}\,a_{4}\,a_{1}\,a_{5}\,a_{0}-\frac{1}{562500}\,a_{3}^{6}+\frac{8}{1125}\,a_{6}^{2}\,a_{2}^{3}\,a_{0}\\\
&-\frac{8}{28125}\,a_{6}\,a_{2}^{4}\,a_{4}+\frac{1}{9375}\,a_{6}\,a_{2}^{3}\,a_{3}^{2}+\frac{1}{5625}\,a_{5}\,a_{3}^{4}\,a_{1}+\frac{1}{90}\,a_{5}^{3}\,a_{3}\,a_{0}^{2}-\frac{2}{28125}\,a_{5}\,a_{3}^{3}\,a_{2}^{2}\\\
&-\frac{8}{28125}\,a_{4}^{4}\,a_{2}\,a_{0}+\frac{8}{1125}\,a_{4}^{3}\,a_{6}\,a_{0}^{2}+\frac{1}{9375}\,a_{4}^{3}\,a_{3}^{2}\,a_{0}-\frac{2}{28125}\,a_{4}^{2}\,a_{3}^{3}\,a_{1}-\frac{1}{225}\,a_{4}^{2}\,a_{5}^{2}\,a_{0}^{2}\\\
&-\frac{2}{140625}\,a_{4}^{2}\,a_{3}^{2}\,a_{2}^{2}+\frac{1}{90}\,a_{6}^{2}\,a_{1}^{3}\,a_{3}-\frac{1}{225}\,a_{6}^{2}\,a_{1}^{2}\,a_{2}^{2}+\frac{2}{140625}\,a_{4}\,a_{3}^{4}\,a_{2}-\frac{1}{3750}\,a_{6}\,a_{0}\,a_{3}^{4}\\\
&-\frac{1}{75}\,a_{6}^{2}\,a_{0}^{2}\,a_{3}^{2}+\frac{14}{225}\,a_{6}^{2}\,a_{2}\,a_{0}^{2}\,a_{4}-\frac{1}{3750}\,a_{6}\,a_{2}\,a_{3}^{3}\,a_{1}-\frac{2}{45}\,a_{6}\,a_{2}\,a_{5}^{2}\,a_{0}^{2}-\frac{1}{3750}\,a_{5}\,a_{3}^{3}\,a_{4}\,a_{0}\\\
&+\frac{1}{2250}\,a_{5}^{2}\,a_{3}^{2}\,a_{0}\,a_{2}+\frac{7}{28125}\,a_{5}\,a_{3}\,a_{4}\,a_{2}^{3}+\frac{1}{2250}\,a_{5}\,a_{3}\,a_{4}^{2}\,a_{1}^{2}+\frac{7}{28125}\,a_{4}^{3}\,a_{2}\,a_{3}\,a_{1}\\\
&+\frac{2}{1125}\,a_{4}^{3}\,a_{1}\,a_{5}\,a_{0}-\frac{1}{5625}\,a_{5}\,a_{2}^{2}\,a_{4}^{2}\,a_{1}+\frac{2}{1125}\,a_{5}\,a_{2}^{3}\,a_{6}\,a_{1}+\frac{1}{2250}\,a_{5}^{2}\,a_{2}^{2}\,a_{3}\,a_{1}\\\
&+\frac{1}{9}\,a_{6}^{2}\,a_{1}\,a_{0}^{2}\,a_{5}+\frac{1}{2250}\,a_{6}\,a_{1}^{2}\,a_{3}^{2}\,a_{4}-\frac{2}{45}\,a_{6}^{2}\,a_{1}^{2}\,a_{0}\,a_{4}-\frac{2}{1875}\,a_{6}\,a_{2}^{2}\,a_{4}^{2}\,a_{0}\\\
&+\frac{1}{1875}\,a_{6}\,a_{2}^{2}\,a_{4}\,a_{3}\,a_{1}-\frac{1}{75}\,a_{6}^{2}\,a_{2}\,a_{0}\,a_{3}\,a_{1}+\frac{1}{1875}\,a_{5}\,a_{3}\,a_{4}^{2}\,a_{2}\,a_{0}-\frac{1}{75}\,a_{5}\,a_{3}\,a_{6}\,a_{0}^{2}\,a_{4}\\\
&-\frac{7}{11250}\,a_{5}\,a_{3}^{2}\,a_{4}\,a_{2}\,a_{1}+\frac{1}{90}\,a_{5}\,a_{3}^{2}\,a_{6}\,a_{0}\,a_{1}-\frac{1}{225}\,a_{5}^{2}\,a_{3}\,a_{4}\,a_{1}\,a_{0}\\\
&-\frac{7}{1125}\,a_{4}^{2}\,a_{6}\,a_{0}\,a_{3}\,a_{1}-\frac{1}{225}\,a_{5}\,a_{2}\,a_{6}\,a_{1}^{2}\,a_{3}\end{split}$

Clebsch invariants $A,B,C,D$ in terms of the Igusa invariants
$J_{2},J_{4},J_{6},J_{10}$

$\begin{split}A=&\;-\frac{1}{2^{3}3\cdot 5}\,J_{2}\;,\\\
B=&\;\phantom{-}\frac{1}{2^{3}3^{3}5^{4}}\left(J_{2}^{2}+20\,J_{4}\right)\;,\\\
C=&\;-\frac{1}{2^{5}3^{5}5^{6}}\left(J_{2}^{3}+80\,J_{2}J_{4}-600\,J_{6}\right)\;,\\\
D=&\;-\frac{1}{2^{8}3^{9}5^{10}}\left(9\,J_{2}^{5}+700\,J_{2}^{3}J_{4}-3600\,J_{2}^{2}J_{6}-12400\,J_{2}J_{4}^{2}+48000\,J_{4}\,J_{6}+10800000\,J_{10}\right)\\\
\end{split}$

The entries of the Clebsch matrix $M$.

$\begin{split}A_{11}&=2\,C+\frac{1}{3}\,A\,B\;,\\\ A_{22}&=A_{13}=\,D\;,\\\
A_{33}&=\frac{1}{2}\,B\,D+\frac{2}{9}\,C\,(B^{2}+A\,C)\;,\\\
A_{23}&=\frac{1}{3}\,B\,(B^{2}+A\,C)+\frac{1}{3}\,C\,(2\,C+\frac{1}{3}\,A\,B)\;,\\\
A_{12}&=\frac{2}{3}\,(B^{2}+A\,C)\\\ \end{split}$

Igusa invariants $J_{2},J_{4},J_{6},J_{10}$ we display them below in terms of
the coefficients

$\begin{split}J_{2}=&\;6a_{3}^{2}-240a_{0}a_{6}+40a_{1}a_{5}-16a_{2}a_{4}\\\
J_{4}=&\;48a_{0}a_{4}^{3}+48a_{2}^{3}a_{6}+4a_{2}^{2}a_{4}^{2}+1620a_{0}^{2}a_{6}^{2}+36a_{1}a_{3}^{2}a_{5}-12a_{1}a_{3}a_{4}^{2}-12a_{2}^{2}a_{3}a_{5}+300a_{1}^{2}a_{4}a_{6}\\\
&+300a_{0}a_{5}^{2}a_{2}+324a_{0}a_{6}a_{3}^{2}-504a_{0}a_{4}a_{2}a_{6}-180a_{0}a_{4}a_{3}a_{5}-180a_{1}a_{3}a_{2}a_{6}+4a_{1}a_{4}a_{2}a_{5}\\\
&-540a_{0}a_{5}a_{1}a_{6}-80a_{1}^{2}a_{5}^{2}\\\
J_{6}=&\;176a_{1}^{2}a_{5}^{2}a_{3}^{2}+64a_{1}^{2}a_{5}^{2}a_{4}a_{2}+1600a_{1}^{3}a_{5}a_{4}a_{6}+1600a_{1}a_{5}^{3}a_{0}a_{2}-160a_{0}a_{4}^{4}a_{2}-96a_{0}^{2}a_{4}^{3}a_{6}\\\
&+60a_{0}a_{4}^{3}a_{3}^{2}+72a_{1}a_{3}^{4}a_{5}-24a_{1}a_{3}^{3}a_{4}^{2}-119880a_{0}^{3}a_{6}^{3}-320a_{1}^{3}a_{5}^{3}-160a_{2}^{4}a_{4}a_{6}-96a_{2}^{3}a_{0}a_{6}^{2}\\\
&+60a_{2}^{3}a_{3}^{2}a_{6}-24a_{2}^{2}a_{3}^{3}a_{5}+8a_{2}^{2}a_{3}^{2}a_{4}^{2}-900a_{2}^{2}a_{1}^{2}a_{6}^{2}-24a_{2}^{3}a_{4}^{3}-36a_{2}^{4}a_{5}^{2}-36a_{1}^{2}a_{4}^{4}\\\
&+424a_{0}a_{4}^{2}a_{2}^{2}a_{6}-2240a_{1}^{2}a_{5}^{2}a_{0}a_{6}+2250a_{1}^{3}a_{3}a_{6}^{2}+492a_{0}a_{4}^{2}a_{2}a_{3}a_{5}+20664a_{0}^{2}a_{4}a_{6}^{2}a_{2}\\\
&+3060a_{0}^{2}a_{4}a_{6}a_{3}a_{5}-468a_{0}a_{4}a_{3}^{2}a_{2}a_{6}-198a_{0}a_{4}a_{3}^{3}a_{5}-640a_{0}a_{4}a_{2}^{2}a_{5}^{2}+3472a_{0}a_{4}a_{2}a_{5}a_{1}a_{6}\\\
&-18600a_{0}a_{4}a_{1}^{2}a_{6}^{2}-876a_{0}a_{4}^{2}a_{1}a_{6}a_{3}+492a_{1}a_{3}a_{2}^{2}a_{4}a_{6}-238a_{1}a_{3}^{2}a_{2}a_{4}a_{5}+76a_{1}a_{3}a_{2}a_{4}^{3}\\\
&+3060a_{1}a_{3}a_{0}a_{6}^{2}a_{2}+1818a_{1}a_{3}^{2}a_{0}a_{6}a_{5}+26a_{1}a_{3}a_{2}^{2}a_{5}^{2}-1860a_{1}^{2}a_{3}a_{2}a_{5}a_{6}+330a_{1}^{2}a_{3}^{2}a_{6}a_{4}\\\
&+76a_{2}^{3}a_{4}a_{3}a_{5}-876a_{2}^{2}a_{0}a_{6}a_{3}a_{5}+616a_{2}^{3}a_{5}a_{1}a_{6}+2250a_{0}^{2}a_{5}^{3}a_{3}-10044a_{0}^{2}a_{6}^{2}a_{3}^{2}\\\
&+28a_{1}a_{4}^{2}a_{2}^{2}a_{5}-640a_{1}^{2}a_{4}^{2}a_{2}a_{6}+26a_{1}^{2}a_{4}^{2}a_{3}a_{5}-1860a_{1}a_{4}a_{0}a_{5}^{2}a_{3}+616a_{1}a_{4}^{3}a_{0}a_{5}\\\
&-18600a_{0}^{2}a_{5}^{2}a_{6}a_{2}+59940a_{0}^{2}a_{5}a_{6}^{2}a_{1}+330a_{0}a_{5}^{2}a_{3}^{2}a_{2}+162a_{0}a_{6}a_{3}^{4}-900a_{0}^{2}a_{5}^{2}a_{4}^{2}\\\
&-198a_{1}a_{3}^{3}a_{2}a_{6}\\\
J_{10}=&\;a_{6}^{-1}Res_{X}\left(f,\frac{\partial f}{\partial X}\right)\\\
\end{split}$ (34)
$\begin{split}&t_{2}^{6}t_{3}-15265260t_{2}^{5}t_{3}-27949860t_{2}^{4}t_{3}^{2}-118098t_{2}^{3}t_{3}^{3}+14693280768t_{2}^{5}+93437786558880t_{2}^{4}t_{3}\\\
&-878290475269680t_{2}^{3}t_{3}^{2}+85811055510240t_{2}^{2}t_{3}^{3}-1139016237660t_{2}t_{3}^{4}+3486784401t_{3}^{5}\\\
&-223154201664000000t_{2}^{4}-287728673929542000000t_{2}^{3}t_{3}-2469658010168691000000t_{2}^{2}t_{3}^{2}\\\
&-109818018101695500000t_{2}t_{3}^{3}-70607384120250000t_{3}^{4}+1355661775108800000000000t_{2}^{3}\\\
&+433843541357670112500000000t_{2}^{2}t_{3}-662569101476807962500000000t_{2}t_{3}^{2}\\\
&+571919811374025000000000t_{3}^{3}-4117822641892980000000000000000t_{2}^{2}\\\
&-327077365625983809843750000000000t_{2}t_{3}-2316275236064801250000000000000t_{3}^{2}\\\
&+6253943137374963375000000000000000000t_{2}+4690457353031222531250000000000000000t_{3}\\\
&-3799270455955290250312500000000000000000000=0\\\ \end{split}$ $\displaystyle
a_{111}$ $\displaystyle=\frac{2}{9}\;(A^{2}C-6BC+9D),$ $\displaystyle a_{112}$
$\displaystyle=\frac{1}{9}\;(2B^{3}+4ABC+12C^{2}+3AD),$ $\displaystyle
a_{113}$
$\displaystyle=\,a_{122}\,=\frac{1}{9}\,\left(AB^{3}+\frac{4}{3}\;A^{2}BC+4B^{2}C+6AC^{2}+3BD\right),$
$\displaystyle a_{123}$
$\displaystyle=\frac{1}{18}\,\left(2B^{4}+4AB^{2}C+\frac{4}{3}\,A^{2}C^{2}+4BC^{2}+3ABD+12CD\right),$
$\displaystyle a_{133}$
$\displaystyle=\frac{1}{18}\left(AB^{4}+\frac{4}{3}\,A^{2}B^{2}C+\frac{16}{3}\,B^{3}C+\frac{26}{3}\,ABC^{2}+8C^{3}+3B^{2}D+2ACD\right),$
$\displaystyle a_{222}$
$\displaystyle=\frac{1}{9}\left(3B^{4}+6AB^{2}C+\frac{8}{3}\,A^{2}C^{2}+2BC^{2}-3CD\right),$
$\displaystyle a_{223}$
$\displaystyle=\frac{1}{18}\left(-\frac{2}{3}\,B^{3}C-\frac{4}{3}\,ABC^{2}-4C^{3}+9B^{2}D+8ACD\right),$
$\displaystyle a_{233}$
$\displaystyle=\frac{1}{18}\left(B^{5}+2AB^{3}C+\frac{8}{9}\,A^{2}BC^{2}+\frac{2}{3}\,B^{2}C^{2}-BCD+9D^{2}\right),$
$\displaystyle a_{333}$
$\displaystyle=\frac{1}{36}\left(-2B^{4}C-4AB^{2}C^{2}-\frac{16}{9}\,A^{2}C^{3}-\frac{4}{3}\,BC^{3}+9B^{3}D+12ABCD+20C^{2}D\right)$

## References