Bill Allombert on Tue, 31 Mar 2015 00:47:27 +0200

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new GP function ellfromeqn

Dear PARI developers,

We have added a new GP function ellfromeqn. 
I quote from the documentation:

   Given a genus 1 plane curve, defined by the affine equation f(x,y) = 0, return the
coefficients [a_1,a_2,a_3,a_4,a_6] of a Weierstrass equation for its Jacobian.

   This  allows  to  recover  a Weierstrass model for an elliptic curve given by a general plane
cubic or by a binary quartic or biquadratic model.

   The  function  implements  the f :--->f^* formulae of Artin,  Tate and Villegas  (Advances in
Math. 198 (2005), pp. 366--382).

   In the example below,  the function is used to convert between twisted Edward coordinates and
Weierstrass coordinates.

   ? e = ellfromeqn(a*x^2+y^2-(1+d*x^2*y^2))
   %1 = [0,-a-d,0,-4*d*a,4*d*a^2+4*d^2*a]
   ? E = ellinit(ellfromeqn(y^2-x^2 - 1 +(121665/121666*x^2*y^2)),2^255-19);
   ? ellcard(E)
   %2 = 57896044618658097711785492504343953926856930875039260848015607506283634007912

We thanks Fernando Rodriguez-Villegas for providing the formulae as a GP script.