Re: Selmer

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

On 26 March 2017 at 17:19, Bill Allombert
<Bill.Allombert@math.u-bordeaux.fr> wrote:
> On Sun, Mar 26, 2017 at 07:16:00PM +0430, Benyamin Gholami wrote:
>> Sure Sir !
>> here you are:
>> note that i created a rank 1 surface with torsion Z/2Z*Z/4Z  and i'm
>> searching on this family for curves with rank 6 ( or fine if greater) with
>> codes that you gave me before. my problem is in most of cases it just give
>> "1" as lower bound and the selmer bounds are not indeed accurate. i upload
>> denis simon script and then use sieving. any helps would be appreciated
>> .thanks to you and Prof.Cremona (mr bellabas helped me to write S(E,N) , so
>> thank to him so)
>
> How do you compute the Selmer group with Magma ?

TwoDescent(E);

In this case it returns a list of 15 equations of the form
y^2=quartic, each with a rational map to E, which represent the
nontrivial elements of the 2-Selmer group (which does have order 16).

Magma also has ThreeDescent(E) and FourDescent(E).

John

>
> (The curves you are considering have large coefficients even after the
> minimal model is taken. You are bound to have computational difficulties
> with them.
> You might want to increase the setting
> LIM1 = 5;           \\ Limit for the search of trivial points on quartics
> LIM3 = 50;          \\ Limit for the search of points on ELS quartics
> LIMTRIV = 3;        \\ Limit for the search of trivial points on the elliptic curve
>
> so that ellrank has more chance to find points).
>
> Cheers,
> Bill.
>