John Cremona on Sun, 26 Mar 2017 19:10:06 +0200

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Re: Selmer

On 26 March 2017 at 17:19, Bill Allombert
<> 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 ?


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).


> (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.