Re: New GP function ellrank (2-descent)

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

Bill,

Can you add some documentation for the parameter 'effort'?  (That name
sounds very Magma-like, by the way).   I assume that it is related to
a bound on a search for rational points on 2-covers (quartics), but is
it linear or exponential?

Also what does the output of ellrankinint() contain?   (If you say
that it is very technical and I should not worry about it, I will feel
like a Mathematica user!)   In particular, does it contain the
quartics which represent elements of (or generators of) the 2-Selmer
group?  If so, then I think that users might like to see these.

John

On Sat, 13 Feb 2021 at 18:03, Bill Allombert
<Bill.Allombert@math.u-bordeaux.fr> wrote:
>
> On Fri, Feb 05, 2021 at 04:53:27PM +0100, Bill Allombert wrote:
> > Dear PARI developers,
> >
> > I have added a new GP function 'ellrank' to the master branch.
> > This is a port of Denis Simon GP script ellQ.gp.
> > However the interface is different, it returns [r,R,V] where r is a
> > lower bound for the rank, R is an upper bound and V is a list of point.
> >
> > It should compare favorably to John Cremona mwrank (which use a
> > different algorithm), except it does not do saturation yet, but this
> > will be added soon.
>
> I have added a GP function ellsaturation that can be used to saturate
> the points.
>
> { E=ellinit([0,1,0,-19357973048906456166239827272707359553313344,
>     21187731957757821187375878909257489490487412099497964528006317056]);
> }
> R = ellrank(E,100);
> R1=matdet(ellheightmatrix(E,R[3]))
> \\%3 = 29871087490016654.513984426820550500851
> S = ellsaturation(E, R[3],200);
> R2=matdet(ellheightmatrix(E,S))
> \\%5 = 312848498549.62405624139280925577340965
> R1/R2
> \\%6 = 95480.999999999999999999999999999999760
>
> Cheers,
> Bill
>