Cryptography Reference
In-Depth Information
-3404372409, 2800656, -3416385600, 5958184124547072,
10091699281/2737152, [195.1547871847901607239497645,
75.00000000000000000000000000, -270.1547871847901607239497645],
0.1986024692687475355260042188, 0.1567132675477145982613047883*I,
-6.855899811988574944063544705, -21.22835194662770142565252843*I,
0.03112364190214999895971387115]
The output contains several parameters for the curve (type
?ellinit
to
see an explanation). For example, the periods
ω
1
=
i
0
.
156713
...
and
ω
2
=
0
.
198602
...
are entries. The
j
-invariant is the 13th entry:
? e1[13]
%2 = 10091699281/2737152
Here is the curve
E
2
:
y
2
=
x
3
+ 73:
? e2=ellinit([0,0,0,0,73])
%3 = [0, 0, 0, 0, 73, 0, 0, 292, 0, 0, -63072, -2302128, 0,
[-4.179339196381231892056376349, 2.089669598190615946028188174
+ 3.619413915098187674530455654*I, 2.089669598190615946028188174
-3.619413915098187674530455654*I], 2.057651708004923756251055780,
-1.028825854002461878125527890+0.5939928837575679811100134634*I,
-2.644469941892436553395125300, 1.322234970946218276697562650
-2.290178149223208371431388983*I, 1.222230471806529890431614914]
We can add the p oints (2
,
9) and (3
,
10), which lie on the curve:
? elladd(e2,[2,9],[3,10])
%4 = [-4, -3]
We can compute the 3rd multiple of (2
,
9):
? ellpow(e2,[2,9],3)
%5 = [5111/625, -389016/15625]
The torsion subgroup of the Mordell-Weil group can be computed:
? elltors(e1)
%6 = [10, [10], [[3, 1944]]]
? elltors(e2)
%7 = [1, [], []]
The first output says that the torsion subgroup of
E
1
(
Q
) has order 10, it
is cyclic of order 10, and it is generated by the point (3
,
1944). The second
output says that the torsion subgroup of
E
2
(
Q
) is trivial.
The number of points on an elliptic curve mod a prime
p
has the form
p
+1
a
p
. The value of
a
13
for
E
1
is computed as follows:
? ellap(e1,13)
%8=4
−
Therefore, there are 13 + 1
−
4 = 10 points on
E
1
mod 13.
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