Computer Packages - Elliptic Curves: Number Theory and Cryptography - page 491

(11 : 4 : 1), (11 : 9 : 1) @ }
The @ 's in the last output specifies that the entries are a set indexed by
positive integers.
Let's compute the Weil pairing e 3 ((0 , 3) , (5 , 1)) on the curve E 5 : y 2 = x 2 +2
over F 7 , as in Example 11.5.
> E5:= EllipticCurve([0, GF(7)!2]);
> WeilPairing( E5![0,3], E5![5,1], 3);
4

The answer is 4, which is a cube root of unity in F 7 . Note that this is the
inverse of the Weil pairing used elsewhere in this topic (cf. Remark 11.11).
We can compute the Mordell-Weil group E 2 ( Q ):
> MordellWeilGroup(E2);
Abelian Group isomorphic to Z + Z
Defined on 2 generators (free)
> Generators(E2);
[(2: 9: 1),(-4: 3: 1)]
To find a command that computes, for example, Mordell-Weil groups, type
?MordellWeil or ?Mordell to get an example or a list of examples.
To quit Magma, type
<Ctrl>D

For much more on Magma, go to
http://magma.maths.usyd.edu.au/magma/htmlhelp/MAGMA.htm
For elliptic curves, click on the Arithmetic Geometry link.

D.3 SAGE

Sage is an open source computer algebra package that can be downloaded
for free from www.sagemath.org/ . For general information, see the web site,
which also contains a tutorial and documentation.
The following is the transcript of a session, with commentary.
The session starts:
Linux sage 2.6.17-12-386 #2 Sun Sep 23 22:54:19 UTC 2007 i686
The programs included with the Ubuntu system are free software;
the exact distribution terms for each program are described in
the individual files in /usr/share/doc/*/copyright.
Ubuntu comes with ABSOLUTELY NO WARRANTY, to the extent
permitted by applicable law.

| SAGE Version 2.8.8.1, Release Date: 2007-10-21 |