Computer Packages - Elliptic Curves: Number Theory and Cryptography - page 486

Appendix D

Computer Packages

There are several computer algebra packages available that do calculations
on elliptic curves. In this appendix, we give a brief introduction to three of
the major packages. Rather than give explanations of the structure of these
packages, we simply include some examples of some computations that can
be performed with them. The reader should consult the documentation that
is available online or with the packages to see numerous other possibilities.

D.1 Pari

Pari/GP is a free computer algebra system for number theory calculations.
It can be downloaded from http://pari.math.u-bordeaux.fr/ .
Here is a transcript of a session, with commentary.

GP/PARI CALCULATOR Version 2.3.0 (released)
i686 running linux (ix86 kernel) 32-bit version
compiled: Aug 16 2007, gcc-3.4.4 20050721 (Red Hat 3.4.4-2)
(readline v4.3 enabled [was v5.0 in Configure], extended help
available)
Copyright (C) 2000-2006 The PARI Group

PARI/GP is free software, covered by the GNU General Public
License, and comes WITHOUT ANY WARRANTY WHATSOEVER.
Type ? for help, \ q to quit. Type ?12 for how to get moral
(and possibly technical) support.
parisize = 4000000, primelimit = 500000

First, we need to enter and initialize an elliptic curve. Let [ a 1 ,a 2 ,a 3 ,a 4 ,a 6 ]
be the coecients for the curve in generalized Weierstrass form. Start with
the curve of Example 9.3: E 1 : y 2 = x 3 − 58347 x + 3954150.

? e1=ellinit([0,0,0,-58347,3954150])
%1 = [0, 0, 0, -58347, 3954150, 0, -116694, 15816600,