# elliptic curves in quartic and standard form

y^2 = a*x^4+b*x^3+c*x^2+d*x+e


is birationally equivalent to an elliptic curve in standard Weierstrass form y^2=cubic(x). How to I get sage to exhibit/find the birational transformation that accomplishes that?

[I am specifically interested in knowing all about this class of curves: 8*D*y^2 = (x-2)*(x-1)*(x+1)*(x+2) for integer D. I am pretty new to both SAGE and elliptic curves.]

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Unfortunately I do not know of any online source, but you can take a look into Cassel's book "Lectures on Elliptic Curves". It will tell you how to go from a quartic to a cubic model of an elliptic curve.

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The binary quartic is only birationally equivalent to a Weierstrass equation if it has a rational point. Then, the required transformations are indeed to be found in Cassels. They have not yet been implemented in Sage. If there is no rational point (or you do not know one) then there is a degree 4 map from the quartic to its Jacobian elliptic curve, called a two-covering map. These are used in two-descent, but there is no Sage function which simply inputs a quartic and outputs the cubic. Note that getting the equation of the Jacobian cubic is easy: it is Y^2=X^3-27IX-27*J where I and J are the classical invariants of the quartic. You can read more about this in my paper Classical Invariants and 2-descent on elliptic curves (preprint here), Journal of Symbolic Computation (Proceedings of the Second Magma Conference, Milwaukee, May 1996), Jan/Feb 2001, pages 71-87.

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One can use Jacobian to build elliptic curves:

sage: x, y = polygen(QQ, 'x, y')
sage: Jacobian(y*y-x**4-3*x+5)
Elliptic Curve defined by y^2 = x^3 + 20*x + 9 over Rational Field

( 2014-11-27 18:21:03 +0200 )edit

I do not know if the transformation can be found, but the following works:

sage: D=polygen(QQ,'D')
sage: x,y=polygen(D.parent(),'x,y')
sage: Jacobian(8*D*y^2 - (x-2)*(x-1)*(x+1)*(x+2))
Elliptic Curve defined by y^2 = x^3 - 4672/3*D^2*x + 609280/27*D^3 over Univariate Polynomial Ring in D over Rational Field

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