Re: Selmer

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

On the curve [0, -438879732513040081/428593506250000, 0,
-112353213812869130368/26787094140625,
3081846839375070007684953094049946244/717548412498706207275390625]
mwrank gives a lower rank bound of 1 (with a point of infinite order)
and an upper bound of 2.

You must distinguish between "2-Selmer rank" which is the rank of the
2-Selmer group (which in turn is an elementary abelian 2-group), and
"the upper bound on the rank of E obtained by doing a 2-descent".
This curve has torsion Z/2 x Z/4 so the torsion already contributes 2
to the 2-Selmer rank, as that contains E/2E.  Secondly, both mwrank
and Magma do not actually do a 2-descent on such a curve but use
descent via 2-isogeny, and also higher descents.  This gives better
bounds on the rank of E, which is hard to interpret as the rank of a
more complicated Selmer group.  mwrank's output tries to explain this
quite carefully.  (Note that Sage will use mwrank if you ask for the
rank of this curve, but it is not easy to get non-default parameters
passed to mwrank from Sage, so I was running mwrank independently
using the command-line flag -p500.  Also, in the Sage to mwrank
interface it does not allow rational coefficients which is a pity
since mwrank itself does allow these.)

Sage also has Simon's script which by default gives nothing very
helpful (E.simon_two_descent() just returns (0,10,[]) which means that
the lower bound is 0, the upper bound is 10, and no ponts of infinite
order are found -- with the default parameters).

These results are consistent.  Since E has 3 2-isogenies, there are 3
different descents by 2-isogeny, and these do not give the same rank
bounds.  mwrank tries all 3 and picks the best, while Simon's ell.gp
just uses one.  The first descent via 2-isogeny finds a rank bound of
8 from a Selmer bound of 10, the difference coming from the torsion.
The other two give rank bound 2.

The use of descent to find rank bounds on elliptic curves does require
some knowledge of what you are doing since there are many variations
and the underlying theory is not simple.

On 26 March 2017 at 14:07, Bill Allombert
<Bill.Allombert@math.u-bordeaux.fr> wrote:
> On Sun, Mar 26, 2017 at 02:12:39AM +0430, Benyamin Gholami wrote:
>> i think Denis Simon Script has a bug:
>> for the curve below if didn't have a mistake , his script compute 2 selmer
>> rank 10 but magma compute 4 . whats wrong?
>>
>> EllipticCurve([0, -438879732513040081/428593506250000, 0,
>> -112353213812869130368/26787094140625,
>> 3081846839375070007684953094049946244/717548412498706207275390625]);
>
> Please details the full computation you did with PARI and MAGMA that
> leads to this conclusion, this would help finding the discrepancy.
>
> Cheers,
> Bill
>