Ask Your Question

points on elliptic curve number field

asked 2015-10-09 05:12:51 -0600

Sha gravatar image

updated 2015-10-09 05:42:44 -0600

kcrisman gravatar image

I have the following elliptic curve : y^2=x^3-3267x+45630 and generator P=[-21,324].

I want to find the general formula to add any point (r,s) to the point ((-15/2)+(27/2)*B,0) where B^2=17.

PARI seems to be pretty straightforward in giving answer. And i think the mistake I did in SAGE is that I should define r and s in number field which I'm not sure how to do it.

The Pari code is as follows :

gp > elladd(E,[r,s],[(-15/2)+(27/2)*B,0])= [(-8*r^3 + (108*B - 60)*r^2 + (1458*B^2 - 1620*B + 450)*r + (8*s^2 + (-19683*B^3 + 32805*B^2 - 18225*B + 3375)))/(8*r^2 + (-216*B + 120)*r + (1458*B^2 - 1620*B + 450)), (-8*s*r^3 + (4374*B^2 - 4860*B + 1350)*s*r + (8*s^3 + (-39366*B^3 + 65610*B^2 - 36450*B + 6750)*s))/(-8*r^3 + (324*B - 180)*r^2 + (-4374*B^2 + 4860*B - 1350)*r + (19683*B^3 - 32805*B^2 + 18225*B - 3375))]
edit retag flag offensive close merge delete

1 answer

Sort by » oldest newest most voted

answered 2016-06-02 15:59:36 -0600

nbruin gravatar image

You haven't told Pari that the point [r,s] actually does lie on E. So pari at this point really just fills in the generic formula with the parameters you've given. The general machinery in Sage is a little more picky (although the computation it does is the same). The good thing is that by the time you're done explaining the situation to sage, it will know that s^2=(r^3-3267*r+45630) and it can use that to simplify the expression more than pari does (note s occurs to higher powers in the answer that pari gives).

sage: E=EllipticCurve([-3267,45630])
sage: k.<B>=NumberField(x^2-17)
sage: kr.<r>=FunctionField(k);
sage: krS.<S>=kr[]
sage: R.<s>=krS.quo(S^2-(r^3-3267*r+45630))
sage: P=E(R)(-15/2+27/2*B,0)
sage: Q=E(R)(r,s)
sage: P,Q
((27/2*B - 15/2 : 0 : 1), (r : s : 1))
sage: P+Q
(((27/2*B - 15/2)*r - 405*B + 3042)/(r - 27/2*B + 15/2) : ((-1215/2*B + 12393/2)/(-r^2 + (27*B - 15)*r + 405/2*B - 6309/2))*s : 1)

As you can see, we need to define 3 fields (plus a polynomial ring to create the relevant function field) before we can properly talk about the points P and Q in the first place, but after that it's smooth sailing. And the formula you get is better simplified than what you'd obtain from pari.

edit flag offensive delete link more


Thank you so much. This has been so useful to me and my calculation. Great answer.!

Sha gravatar imageSha ( 2016-08-29 05:10:36 -0600 )edit

Your Answer


Add Answer

Question Tools


Asked: 2015-10-09 05:12:51 -0600

Seen: 439 times

Last updated: Jun 02 '16