Re: question on possibly extending the ell2cover(E) command

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

To all:

I used 3 AI engines to help me out, building from scratch and allowing 
each AI engine to correct each other (Claude Opus / OpenAI ChatGPT / 
Google Gemini) I had to keep all 3 AI engines on a very tight leash

I used John's approach.

There is yet more work to do, I have to minimize the two-covers and this 
is NOT a trivial task.

But at least I have something working now, which is my request. 
100.testdata is a rank 6 congruent number elliptic curve, 5 independent 
points are known. ( I am trying to find the 6th and last generator). 
ell2coverP.gp is the code I am using, but this is a preliminary draft 
version.

It took 1.224 secs to locate the last two-cover for the 6th point. I 
attach both programs for your information. If you spot something which 
needs improvement in ell2coverP.gp, please let me know.

Thank you.

- Randall


Randall

Randall

On 10/21/26 6:09 AM, John Cremona wrote:
> Mathematically the answer is certainly yes.  The way 2-descent is implemented in pari (Bill will confirm or correct me) is to form the cubic number field (or algebra) K=Q(theta)=Q[x]/(f(x)) where the curve is y^2=f(x), determine a certain finite subgroup of K*/(K*)^2 and convert each element a in there into a quartic q(x), whose solubility gives a point on E(Q)/2E(Q).  (Explicitly, a solution to the quartic lets you fins a representative for the coset "a mod squares" of the form x0-theta with x0 rational, and this x0 is the x-coordinate of the point.
>
> If you have a point (x0,y0) then you know that x0-theta is in the image of E(Q)/2E(Q) inside K*/(K*)^2, so you can cut down the finite subgroup you are looking at to one of half the size.
>
> My mwrank does such a sieving, though it does not use the cubic field K, instead doing something different (but leading to an equivalent conclusion), as decribed in my paper "On the computation of Mordell-Weil and 2-Selmer Groups of Elliptic Curves" Rocky Mountain Journal of Mathematics (2002) vol. 32, no. 3, pp.953--967. There's a pdf here: https://johncremona.github.io/papers/filter.pdf .  In the same paper there's a description of how to test that points on a curve are independent without using heights, only arithmetic modulo p for a few primes and linear algebra over F_2.  (I used that to check the independence of the published points on a rank 30 curve recently, but have not yet done the rank 31 or 32 curves announced even more recently by Elkies and Klagsbrun.)
>
> mwrank uses this "Selmer sieve" as follows:  it *first* searches for small points on the curve and maps these to quartics as above, giving a 'known' subgroup of the Selmer group which is known to be in the image of E(Q)/2E(Q), and only then searches for quartics which -- if there are any and they are soluble -- give points which are guaranteed to be independent of the ones found in the search.  For very many curves, the points found in the initial search already generate E(Q), in which case all the descent step does is to give a proof that they do, with no more quartics found (and hence no point-searching needed on any quartics).  But when there is a large generator (as I suspect is the case for the curves Randall cares about) the descent step does have work to do.
>
> John
>
>
>
> On Tuesday, October 20th, 2026 at 9:19 PM, American Citizen <website.reader3@gmail.com> wrote:
>
> > We read:
> >
> > ? ?ell2cover
> > ell2cover(E): if E is an elliptic curve over Q, returns a basis of the
> > set of everywhere locally soluble 2-covers of the curve E. For each
> > cover a pair [R,P] is returned where y^2-R(x) is a quartic curve and P
> > belongs to E(k), where k = Q(x)[y] / (y^2-R(x)).
> >
> > Is there a possible way to extend this command so that it could be
> > modified by known independent points on the curve, ie. only provide two
> > covers which might contain the other unknown points, i.e.
> >
> > ell2cover(E,P) returns a set of everywhere locally soluble 2-covers
> > which are not included in the coverage of independent points P
> >
> > Randall
> >
> > p.s. This probably involved the generators of the 2-Selmer group
d = 20705509974170914386;
e = [0,0,0,-428718143290491219724075383655341756996,0];
p = [[40821495094439589586,224774993632097101574585869280],[583973001734845932179373/3481,442849163016799695584747045881237473/205379],[-257947107601149552144/3721,1237395404522267842932163821970440/226981],[5044545810935808592071876940479525/118158400064761,313331479898803659884334596656817605911379122622365/1284389961633556538509],[3746109238344814059266937663846/179451362689,922992683158670394153672656947730660840190360/76018647908226113]];

Attachment: ell2coverP.gp
Description: application/gnuplot