|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. Cheers, Bill.