# Elliptic curve over binary field in Sage

I searched in tutorial, but I haven't found any information about that. Can I write in Sage Elliptic curve over binary field (for example y^2+xy=x^3+g^3x^2+(g^3+1) over F(2^4))

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I guess g stands for "the" generator of the Field F(2^4).

You can get the documentation about the ways to construct elliptic curves by typing:

sage: EllipticCurve?


Then you can try something along the lines:

sage: F = GF(2^4, 'g') ; F
Finite Field in g of size 2^4
sage: F.inject_variables()
Defining g
sage: R.<x,y> = F[] ; R
Multivariate Polynomial Ring in x, y over Finite Field in g of size 2^4
sage: C = EllipticCurve(y^2+x*y-x^3-g^3*x^2-(g^3+1)) ; C
Elliptic Curve defined by y^2 + x*y = x^3 + g^3*x^2 + (g^3+1) over Finite Field in g of size 2^4

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Thanks for help. One more question. I define two points and try do operation like this

M=C((g^8,g^8))

N=C((0,g^7))

O=M-N

print O

But I haven't got cordinates of O

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When you type sage: O (g^3 + 1 : g^2 + g : 1) You got projective coordinates. You can do: sage: O.dehomogenize(2) (g^3 + 1, g^2 + g) and check: sage: C(g^3 + 1, g^2 + g) == O True