Re: New GP function ellrank (2-descent)

[Bill Allombert <Bill.Allombert@math.u-bordeaux.fr>]

On Tue, Feb 16, 2021 at 02:47:51PM +0000, John Cremona wrote:
> There is something wrong with this example.  The first one is a
> 2-cover of 2170b1 but the second is not.  So rational points on the
> second cannot help you find points on that elliptic curve.

I am looking again at this example.
Denis program find the following set of quartics
(reduced using Cremona-Stoll reduction):

1277*x^4+202*x^3+11877*x^2+6680*x+46800
1364*x^4+292*x^3+12837*x^2-6916*x+41748
1973*x^4+2874*x^3+18681*x^2+7892*x+24212
468*x^4+668*x^3+11877*x^2+2020*x+127700
4745552*x^4+7271768*x^3+19077981*x^2+12368046*x+8955533
5108*x^4+9812*x^3+18933*x^2+10788*x+13268
6053*x^4+3946*x^3+18681*x^2+5748*x+7892
7488*x^4+12304*x^3+19101*x^2+13112*x+10832
8000*x^4+14800*x^3+23757*x^2+11802*x+8477
8525*x^4+14370*x^3+21897*x^2+10036*x+7892

The smallest rational points are:

1:[[-69957/407849,35677573217664/166340806801]]->407849
2:[[339591/68258,1274199043488/1164788641]]->  ->339591
3:[[-612623/271333,25483980869760/73621596889]]->612623
4:[[-4078490/69957,118925244058880/1631327283]]->4078490
5:???
6:[[-168946/407849,17838786608832/166340806801]]->407849
7:[[-542666/612623,50967961739520/375306940129]]->612623
8:[[984644/69957,29731311014720/1631327283]]    ->984644
9:[[41321/7192,5688388587/1616402]]             ->41321
10:[[271333/170645,2548398086976/5823943205]]   ->271333

The quartic #9 has a much smaller rational points than the others.
Given that ratpoints is quadratic in the height of x, and
that generating new quartics is fast, it makes sense to try several
of them.
In my example, I did a search of bound 70000 and find #9.
To find the second best one, I would need a bound of 280000
which would take 16 more times, which is more than the time
to run hyperellratpoints(,70000) on the 10 quartics.

Does that make more sense now ?
Do you have a better reduction procedure ?

Cheers,
Bill.