# Elliptic curves - morphism

Consider the example from the documentation:

sage: R.<u,v,t> = QQ[]
sage: Jacobian(u^3+v^3+t, variables=[u,v])
Elliptic Curve defined by y^2 = x^3 + (-27/4*t^2) over
Multivariate Polynomial Ring in u, v, t over Rational Field


how to obtain the morphism in this case?

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You can do this:

sage: T.<t> = FunctionField(QQ)
sage: R.<u,v> = T[]
sage: f = u^3+v^3+t
sage: h = Jacobian(f, curve=Curve(f)); h
Scheme morphism:
From: Affine Plane Curve over Rational function field in t over Rational Field defined by u^3 + v^3 + t
To:   Elliptic Curve defined by y^2 = x^3 + (-27/4*t^2) over Rational function field in t over Rational Field
Defn: Defined on coordinates by sending (u, v) to
((-t^3)*u^4*v^4 + (-t^4)*u^4*v + (-t^4)*u*v^4 : 1/2*t^3*u^6*v^3 + (-1/2*t^3)*u^3*v^6 + (-1/2*t^4)*u^6 + 1/2*t^4*v^6 + 1/2*t^5*u^3 + (-1/2*t^5)*v^3 : t^3*u^3*v^3)

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Thank you very much, it work well.

( 2019-10-22 11:42:14 +0100 )edit