Re: ellgenerators

[John Cremona <john.cremona@gmail.com>]

On 30 May 2012 13:28, Kevin Acres <research@research-systems.com> wrote:
> Hi Max,
>
> If you just want generators, and a lower bound on rank, and in a reasonable
> time, then ratpoints and a few lines of gp script will do that for you.  My
> experience was that sage would run for many days on one curve without giving
> an answer.
>
> John's mwrank program works pretty well, but can (in my experience) bog down
> at heights of 25 or more.  Even so it doesn't take long to make it into a
> command line version, which, once again, you can call from gp.

Note also that mwrank has  a "gp output format" flag for exactly this purpose:

%echo [0,0,0,-35750837071872,63237970697887678464] | mwrank -q -v0 -o
Curve [0,0,0,-35750837071872,63237970697887678464] :	Rank = 1
[[1],[[-2365632,11600547840]]]

>
> My particular problem was that I was looking for generators of MW curves of
> somewhat large conductor and there just wasn't really much out there that I
> could use 'off the shelf'.

Especially when the rank is >1 (so no Heegner points) and there's no
2-torsion (when mwrank can take a long time, agreed!).

John

>
> Best Regards,
>
> Kevin.
>
>
>
> At 10:11 PM 30/05/2012, Max Alekseyev wrote:
>>
>> Kevin, thanks for the idea.
>> I'm currently making system calls to sage in order to get generators
>> when pari fails.
>> Is there any advantage in calling ratpoints instead?
>> Regards,
>> Max
>>
>> On Wed, May 30, 2012 at 3:32 PM, Kevin Acres
>> <research@research-systems.com> wrote:
>> > Hi Max,
>> >
>> > I've recently extracted many 10's of millions of MW curves with
>> > conductors
>> > in the region of 10^32. Â It's fairly easy to issue a system call to
>> > ratpoints from gp in order to get an list of rational points. Â From
>> > there,
>>
>> > it's just a few more lines of gp script to extract the generators and
>> > determine a lower bound for rank. Â The whole process usually takes just
>> > a
>>
>> > few milliseconds per curve.
>> >
>> > Currently I'm processing about 300,000 curves per day in this fashion
>> > but
>> > discarding anything less than rank 8.
>> >
>> > If you want some code samples please let me know.
>> >
>> >
>> > Best Regards,
>> >
>> > Kevin.
>> >
>> >
>> >
>> > At 08:35 PM 30/05/2012, Max Alekseyev wrote:
>> >>
>> >> Why pari cannot compute generators of an elliptic curve, while sage
>> >> can?
>> >> I thought they have roughly the same "power" w.r.t. elliptic curves.
>> >>
>> >> ? ellgenerators(ellinit([-35750837071872,63237970697887678464]))
>> >> Â *** Â  at top-level: ellgenerators(ellini
>> >> Â *** Â  Â  Â  Â  Â  Â  Â  Â  ^--------------------
>> >> Â *** ellgenerators: error opening elldata file:
>> >> `/usr/local/share/pari/elldata/ell1816'.
>> >> Â *** Â  Break loop: type 'break' to go back to GP
>>
>> >>
>> >> sage: EllipticCurve([-35750837071872,63237970697887678464]).gens()
>> >> [(-2365632 : 11600547840 : 1)]
>> >>
>> >> Thanks,
>> >> Max
>> >
>> >
>
>
>