I was happy to see that Sage gives you the explicit map between your cubic
to its Weierstrass form. However, rather than having to do so by hand, I
was wondering if Sage is capable of giving the map from the Weierstrass
form to the original cubic, since I'd like a quick way of finding rational
points on the original cubic (the Weierstrass form has positive rank so
it's very quick to generate as many rational points as I want there). If
it's important, [0,0,1] is not a flex point on the original cubic.
R.<x,y,z> = QQ[]
f = 3*y^2*x-y^2*z-2*x*y*z+y*z^2+2*x^3-2*x^2*z
EllipticCurve_from_cubic(f,[0,0,1])
Scheme morphism:
From: Closed subscheme of Projective Space of dimension 2 over
Rational Field defined by:
2*x^3 + 3*x*y^2 - 2*x^2*z - 2*x*y*z - y^2*z + y*z^2
To: Elliptic Curve defined by y^2 + 6*x*y + 256*y = x^3 - 73*x^2
over Rational Field
Defn: Defined on coordinates by sending (x : y : z) to
(1/8*x*y - 1/16*y^2 - 1/8*y*z : -x^2 + 1/8*x*y + 3/16*y^2 + x*z
+ 3/8*y*z : -1/256*y^2)
Thanks
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