Re: question on inconsistent results for a curve and ellrank() command

[Aurel Page <aurel.page@normalesup.org>]

When fixing the random seed and varying the precision, you get the same 
results:

? setrand(1);
? ellrank(E,1)
% = [2, 6, 0, [[-68349492379463773635627069600/12769, 
21338619687688795682928514252421827970088000/1442897], 
[24769306495896066304571392, 150515871036796575853516933113202757632]]]
? \p 100
   realprecision = 115 significant digits (100 digits displayed)
? setrand(1);
? ellrank(E,1)
% = [2, 6, 0, [[-68349492379463773635627069600/12769, 
21338619687688795682928514252421827970088000/1442897], 
[24769306495896066304571392, 150515871036796575853516933113202757632]]]
? \p 200
   realprecision = 211 significant digits (200 digits displayed)
? setrand(1);
? ellrank(E,1)
% = [2, 6, 0, [[-68349492379463773635627069600/12769, 
21338619687688795682928514252421827970088000/1442897], 
[24769306495896066304571392, 150515871036796575853516933113202757632]]]
? \p 500
   realprecision = 500 significant digits
? setrand(1);
? ellrank(E,1)
% = [2, 6, 0, [[-68349492379463773635627069600/12769, 
21338619687688795682928514252421827970088000/1442897], 
[24769306495896066304571392, 150515871036796575853516933113202757632]]]

Aurel

On 07/03/2024 08:34, Aurel Page wrote:
> Dear Randall,
>
> Bill will confirm, but I think ellrank uses a random search, so the 
> points found may not be the same on several runs. I don't think this 
> has anything to do with the precision.
> For instance I get:
> ? ellrank(E,1)
> % = [2, 6, 0, [[-821148234320121552464836608/121, 
> 22227642712180677905417915898820314918912/1331]]]
> ? ellrank(E,1)
> % = [2, 6, 0, [[-371764877985768391740000, 
> 1733635091788436792108632377956904000], [482298396787552594734284800, 
> 10726073559724236251459173261972663500800]]]
> ? ellrank(E,1)
> % = [2, 6, 0, [[1910041765787625231041280, 
> 6699967211953306552674675136155240960]]]
> ? ellrank(data,1)
> % = [4, 6, 0, [[-2604689033065838647568640, 
> 8678086907094504656562057134370808320], [-1406975789099557808651520, 
> 5167717070222070579741742214157519360], 
> [1005530876255752038297064960/81, 
> 44725001024807552828328543783945972536320/729]]]
>
> Note that if you want to start again with higher effort because you 
> did not find all the points, you can provide the previously found 
> points to ellrank (optional argument points).
>
> Cheers,
> Aurel
>
> On 07/03/2024 02:12, American Citizen wrote:
> > Hello:
> >
> > See attached gp-pari file.
> >
> > While running my programs I stumbled across an inconsistent results 
> > which caused me to take a careful look.
> >
> > Here are those results involving a certain elliptic curve:
> >
> > E = [0, 12304376939822994932659524, 0, 
> > -3648259547053109398533591982691229700073064038400, 0]
> >
> > 3 points of the 4 points Mordell-Weil basis (set) [ I had to do some 
> > work to finally settle down on these 3 points ]
> >
> > p = [[-999779060917633308399360, 
> > 3866142351194747967608412122674782720], [-293883589543938115962240, 
> > 1452405131779215974552917691531685120], [-1406975789099557808651520, 
> > 5167717070222070579741742214157519360]]
> >
> > I am trying to recover the 4th point right now.
> >
> > However when I run the ellrank(E,{effort}) command, differing results 
> > come up.
> >
> > > [0, 12304376939822994932659524, 0, 
> > > -3648259547053109398533591982691229700073064038400, 0]
> > > [[-999779060917633308399360, 3866142351194747967608412122674782720], 
> > > [-293883589543938115962240, 1452405131779215974552917691531685120], 
> > > [-1406975789099557808651520, 5167717070222070579741742214157519360]]
> > > e is a rank 4 curve with SHA > 1
> > > setting precision = 19
> > > ellrank(E)
> > > [0, 6, 0, []]
> > > ellrank(E,2)
> > > [2, 6, 0, [[-293883589543938115962240, 
> > > 1452405131779215974552917691531685120]]]
> > > ellrank(E,3)
> > > setting precision = 38
> > > ellrank(E)
> > > [0, 6, 0, []]
> > > ellrank(E,2)
> > > [2, 6, 0, [[-6105797019069920466053760, 
> > > 15917389406358134595823780451205131520], 
> > > [-1910041765787625231041280, 6699967211953306552674675136155240960]]]
> > > ellrank(E,3)
> > > setting precision = 57
> > > ellrank(E)
> > > [0, 6, 0, []]
> > > ellrank(E,2)
> > > [2, 6, 0, [[-2604689033065838647568640, 
> > > 8678086907094504656562057134370808320]]]
> > > ellrank(E,3)
> > > setting precision = 84
> > > ellrank(E)
> > > [0, 6, 0, []]
> > > ellrank(E,2)
> > > [2, 6, 0, [[482298396787552594734284800, 
> > > 10726073559724236251459173261972663500800]]]
> > > ellrank(E,3)
> > > setting precision = 96
> > > ellrank(E)
> > > [0, 6, 0, []]
> > > ellrank(E,2)
> > > [0, 6, 0, []]
> > > ellrank(E,3)
> > > setting precision = 115
> > > ellrank(E)
> > > [0, 6, 0, []]
> > > ellrank(E,2)
> > > [2, 6, 0, [[-1406975789099557808651520, 
> > > 5167717070222070579741742214157519360], [-999779060917633308399360, 
> > > 3866142351194747967608412122674782720]]]
> > > ellrank(E,3)
> > How would one know which exact precision and which effort setting to 
> > use, to recover at least 2 if not 3 points for the Mordell Weil 
> > basis??? Right now it appears that this is just hit-or-miss. So how 
> > do we aim to get the maximum hit? Some of the attempts come up 
> > empty-handed. This doesn't seem right.
> >
> > Randall