Cryptography Reference
In-Depth Information
sage: EllipticCurve(GF(229),[0,-1]).abelian group()
(Multiplicative Abelian Group isomorphic to C42 x C6,
((62 :
25 :
1), (113 :
14 :
1)))
Z
42
⊕
Z
6
, and it has the listed generators.
Let's compute the rank and generators of the Mordell-Weil group
E
2
(
Q
):
sage:
Therefore,
E
5
(
F
229
)
E2.rank()
2
sage: E2.gens()
[(-4 : 3 : 1),(2: 9 : 1)]
The generators are the generators of the nontorsion part. The command
does not yield generators of the torsion subgroup. For these, use the command
E.torsion subgroup().gen()
used previously.
To find the periods
ω
1
and
ω
2
of
E
1
:
sage: E1.period lattice.0
0.1986024692687475355260042188...
sage: E1.period lattice.1
0.1567132675477145982613047883...*I
The
j
-invariant of
E
1
is
sage: E1.j
invariant()
10091699281/2737152
To find a list of commands that start with a given string of letters, type
those letters and then press the “Tab” key:
sage: Ell ('Tab')
Ellipsis EllipticCurve from c4c6
EllipticCurve EllipticCurve from cubic
To find out about the command
EllipticCurve
,type
sage: EllipticCurve?
The output is a description with some examples.
For more on SAGE, go to
http://www.sagemath.org
.
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