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I have an elliptic curve say y^2=x^3 8. I pick a point P=(1,3) and an arbitrary point Q=(x,y). I want to add the two points on Sage to leave my results in x and y. How do I do that

asked 1 year ago

Poncho gravatar image

I want a general formula for addition of two points on an elliptic curve that can be generated by sage.

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answered 1 year ago

dan_fulea gravatar image

We construct the function field F of the quotient ring of Z[x,y] (well, Q also works, but will write fractions not as i like them) w.r.t. the ideal generated by y2+x3+8. We arrange that in F the two elements have these names. Then consider the elliptic curve with a4=0, a6=8, either directly over F, or use a base change from Q to F. There are two points, (1,3) and (x,y), and we can add them (generically, something has to be arranged in case of the specialization x=1 in practice).

R.<X,Y> = PolynomialRing(QQ)
S.<x,y> = R.quotient(-Y^2 + X^3 + 8)
F = FractionField(S)
E = EllipticCurve(F, [0, 8])
P1, P2 = E.point((1, 3)), E.point((x, y))

print(f"P1 + P2 has the components:\n{(P1 + P2).xy()}")

And we get:

P1 + P2 has the components:
((x^2 + x - 6*y + 16)/(x^2 - 2*x + 1), (-9*x^2 + 3*x*y - 3*y^2 + 33*y - 72)/(3*x^2 - y^2 - 3*x + 9))
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Asked: 1 year ago

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Last updated: Nov 20 '23