Re: New elldata package minor update (updated generators)

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

If anyone is interested, I have more elliptic curves which it would be
easy to provide in the same format as the ones in the existing elldata
package (label, coefficients, generators), with some issues around the
labelling:

- for prime conductors up to 10^10, combining the Stein-Watkins
database (which is incomplete) with Bennett-Gherga-Rechnitser
(incomplete), I have generators up to 3*10^8 (these curves are now in
the LMFDB) and for all curves of rank>=2.  One can use ellheegner to
find the rank 1 generators, but there are a *lot* of them.

- all curves whose conductors are divisible only by 2,3,5,7, by
Matschke, van Bommel and others.  (In the LMFDB).   They also have all
conductors supported on {2,3,5,7,11} (another 433022 curves) and
{...,13} (another 3889408) but I have not computed their generators.

For the primes up to 3*10^8 there are 19 of rank 5 (see
http://www.lmfdb.org/EllipticCurve/Q/?rank=5), which ellrank() handles
beautifully, for example

? E = ellinit([0, 0, 1, -79, 342])
%1 = [0, 0, 1, -79, 342, 0, -158, 1369, -6241, 3792, -295704,
-19047851, -54526169088/19047851, Vecsmall([1]), [Vecsmall([128,
-1])], [0, 0, 0, 0, 0,
0, 0, 0]]
? ellglobalred(E)
%2 = [19047851, [1, 0, 0, 0], 1, Mat([19047851, 1]), [[1, 5, 0, 1]]]
? ellrank(E)
%3 = [5, 5, [[5, 8], [0, 18], [10, 23], [12, 33], [4, 9]]]
? ##
  ***   last result computed in 32 ms.

This compares rather well with mwrank (3.5s, but only 396ms without
saturation, which is a fairer comparison, though Bill has a good
saturation implementation on the way).

John

On Mon, 1 Mar 2021 at 18:10, Bill Allombert
<Bill.Allombert@math.u-bordeaux.fr> wrote:
>
> Dear PARI users,
>
> I have made a minor update to the elldata package.
> The list of curves are strictly identical to the previous version, only the
> generating set given by ellgenerators() is different for some curves.
>
> In particular, for the three curves below, the previous elldata package
> reported incorrect results for ellgenerators():
> 69206d1, 284193m5, 284193m6
> (the group generated by ellgenerators() wa a subgroup of finite index
> of the Mordell-Weil group).
>
> John Cremona has updated his Elliptic Curve Data to list smaller
> generators, see <https://github.com/JohnCremona/ecdata/>
> I have updated the PARI package elldata to match.
>
> For detail, see
> http://johncremona.github.io/ecdata/release_notes.md
>
> This package contains elliptic curves defined over the rationals representing
> all isogeny classes for all conductor at most 500000. Theses data are needed by
> ellsearch, ellidentify and ellgenerators and require PARI/GP 2.2.11 and up.
>
> The package can be downloaded from
> <http://pari.math.u-bordeaux.fr/packages.html>
>
> Thanks to John for providing this very useful database!
>
> If diskspace is an issue, you can compress all the elldata/ell*
> files using gzip, GP is able to read gziped files.
>
> Cheers,
> Bill.
>