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

[American Citizen <website.reader3@gmail.com>]

Aurel:

Thank you for the reply. It turns out that the curve does have SHA > 1, 
my best estimate is SHA=9

For those who like to know, this elliptic curve is a bit unusual, as the 
4th generator is significantly higher than the other 3.

> e = 
> [0,12304376939822994932659524,0,-3648259547053109398533591982691229700073064038400,0]
> MW_pts = 
> [[1400650711366853768290560,4666571880720187680622289494327841280],[597507505680042881763840,1557660631597565910427943559692743680],[1910041765787625231041280,6699967211953306552674675136155240960],[2283738942442691504242557493982452515296885368573255561307043758442240927185853664/139918493230305356926881857257071496605320691716223246289,143967496803053198802470727018467319173904232793442308182221768412335727985088750240137534120161342906625450912826244360000/1655055948180891329722008983187988514269081118289967648775651182429065015843146253337]]
> regulator = 44936.543916...
> est_SHA = 9 (from approx. L-series calculations)
> pt_heights = [6.4951..., 6.9582..., 7.0529..., 144.5035...]
Will I have to just do multiple runs, and select the run which gives the 
most found points?

For nearly all of the 328 elliptic curves I am using, ellrank() did find 
a representative set of the Mordell-Weil basis, only that my program 
flagged the unusual results coming from this curve and that is how I 
found out what was happening. I will accept that setting the precision 
is not the controlling factor here.

Randall

On 3/6/24 23:34, 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