ASKSAGE: Sage Q&A Forum - RSS feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Sat, 29 Oct 2016 00:15:35 +0200Elliptic curve defined with parameterhttps://ask.sagemath.org/question/35295/elliptic-curve-defined-with-parameter/For example, I want to create and study the curve y^2 = x^3 + (u)x^2 + (16*u)x over finite fields. This curve parameterizes all elliptic curves with rational 2-torsion subgroups. I get errors when I try to use "u" when defining the curve. Is it possible to define a curve in this way and study it as a family of curves?Fri, 28 Oct 2016 21:58:32 +0200https://ask.sagemath.org/question/35295/elliptic-curve-defined-with-parameter/Answer by castor for <p>For example, I want to create and study the curve y^2 = x^3 + (u)x^2 + (16*u)x over finite fields. This curve parameterizes all elliptic curves with rational 2-torsion subgroups. I get errors when I try to use "u" when defining the curve. Is it possible to define a curve in this way and study it as a family of curves?</p>
https://ask.sagemath.org/question/35295/elliptic-curve-defined-with-parameter/?answer=35298#post-id-35298You may try the following
K.<u> = FunctionField(GF(5,'a'))
E=EllipticCurve([0,u,0,16*u,0])
there are many functions to be applied for your curve E: [SageMath Doc](http://doc.sagemath.org/html/en/reference/curves/sage/schemes/elliptic_curves/ell_generic.html).
Define a point e.g. and compute the double point:
P=E([0,0])
2*P
To determine some more "small" points on your curve:
[E.lift_x(s+t*u, all=True) for s in [0..4] for t in [0..4] if E.is_x_coord(s+t*u)]
which provides:
[[(0 : 0 : 1)], [(4*u : 2*u : 1), (4*u : 3*u : 1)], [(4 : 2 : 1), (4 : 3 : 1)]].Sat, 29 Oct 2016 00:15:35 +0200https://ask.sagemath.org/question/35295/elliptic-curve-defined-with-parameter/?answer=35298#post-id-35298