Elliptic curve over 16-bit binary field

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To see how to create an elliptic curve, see:

sage: EllipticCurve?

To create a 16-bit binary field, you can do:

sage: K = GF(2^16)
sage: K
Finite Field in z16 of size 2^16

If you want to use the generator of the field:

sage: K.inject_variables()
Defining z16

Then, you can define the elleptic curve you want :

sage:  E = EllipticCurve(K, [2, 3*z16^12,1,1,z16]) ; E
Elliptic Curve defined by y^2 + y = x^3 + z16^12*x^2 + x + z16 over Finite Field in z16 of size 2^16

This is of course an arbitrary example, you have to construct the curve that satisfies your needs.

To see quickly what you can do with the curve, you can use [TAB] completion:

sage: E.[TAB]