# 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?

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You 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. 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)]].

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