On Wed, 29 Nov 2006 09:57:06 -0800, Enrique Acosta
<[EMAIL PROTECTED]> wrote:
> Hi William,
>
>
> I have been trying to compute the ranks of some elliptic curves with
> some complicated but I have ran into trouble. I didn't know where to
> post this so I send it to you in case you are interested.
>
> Here is an example:
>
> E=EllipticCurve([0, -285193/512, 0, 10272094875/131072,
> -115889280609375/268435456])
> E.rank()
>
> I tried it both on my computer and on the online SAGE notebook (the
> worksheet called 0 is unlocked), and it crashes on both.
I think this curve is much too "big" for mwrank to handle.
Possibly this command would work, though I don't know how
long it would take (days? years?):
sage: E=EllipticCurve([0, -285193/512, 0, 10272094875/131072,
-115889280609375/268435456])
sage: F = E.minimal_model()
sage: F.simon_two_descent()
Another possibility would be to try MAGMA, since essentially
all of the world's top people at compute ranks of difficult
elliptic curves have implemented all their code in
MAGMA during the last 3 years, so in some cases it is impressively
good. The commands are similar:
E:=EllipticCurve([0, -285193/512, 0, 10272094875/131072,
-115889280609375/268435456]);
Rank(E);
and the latest version of magma is installed on sage.math (you have
to log in with ssh and type "magma" to start it).
Note that your curve has root number 1:
sage: F.root_number()
1
According to the BSD conjecture (widely believed), this means that
F has even rank. It's also "likely" that the rank is 0, unless
you have a specific reason to believe otherwise.
It would be interesting to know if L(F,1) = 0, i.e., whether the
rank should be positive, but the conductor of your curve is huge,
so I doubt you can compute L(1) in any reasonable amount of time:
sage: E.conductor()
472966246406283960120
William
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