Re: what's going on here?

My guess is, but I haven't looked at the code behind it: The effort relates to searching on an integral model. It takes some effort to find the generator in the minimal model. If it did search on the initial model then it might find it earlier.

e2 = ellminimalmodel(e);
? ellrank(e2)
%4 = [1, 1, 2, []]
? ellrank(e2,1)
%5 = [1, 1, 2, [[-4009511/4, 876467493/8]]]

This is a rare instance I think of a purposely built curve. In general, there is a good chance that the height of points is smallest in the minimal model.

Chris

On Wed, 4 Feb 2026 at 08:17, Aurel Page <aurel.page@normalesup.org> wrote:

Dear Randall,

You simply need to increase the search effort a bit:

? ellrank(E,1)
% = [1, 1, 2, [[1/4, 1/8]]]

Maybe we can tune the default better, I am not sure.

Best,
Aurel

On 04/02/2026 00:28, American Citizen wrote:
> This is a question on the ellrank() command.
>
> I have e = [0, -3860/957, 0, 965/957, 0]
>
> ? E=ellinit(e);
> ? ellrank(E)
> %7 = [1, 1, 2, []]
>
> Huh??? what?
>
> ? [x,y]=[1/4,-1/8]
> %8 = [1/4, -1/8]
> ? ellordinate(E,1/4)
> %9 = [1/8, -1/8]
>
> this is a rather simple elliptic curve of rank=1,
>
> mwrank confirms the Mordell-Weil basis of [1/4,-1/8]
>
> I am using
>
>> GP/PARI CALCULATOR Version 2.18.1 (development
>> git-5d3c5ac62a)
>> amd64 running linux (x86-64/GMP-6.3.0 kernel) 64-bit version
>> compiled: Aug 23 2025, gcc version 13.3.0 (SUSE Linux)
>> threading engine: pthread, nbthreads = 6
>> (readline v7.0 enabled, extended help enabled)
> if I use ellrank(E,1) then I get the MW basis.
>
> I am just curious as to why this is missed, with no search effort
> specified. Should the ellrank command be rewritten with search effort
> set to 1 automatically?
>
> Randall
>
>