People
Delaunay, (Elkies), Hart, Hiary, Huynh, Rubinstein, Watkins
Sources
Here is a paper of Rogers from 2000. http://www.expmath.org/restricted/9/9.4/rogers.ps
The idea is from Gouvêa and Mazur, and is that a (u,v) pair gives a point on a unique twist Dy^2=f(x), where D is the squarefree part of f(u/v). In our case of the congruent number curve, this polynomial is uv(u-v)(u+v), and so the squarefree computation is much easier. We loop over (u,v) pairs, and compute the twist these points lie on. In other words, we loop over points rather than looping over twists. We also don't bother with hashing as Rogers did, and just compute the 2-Selmer group of every D candidate via fast F_2-linear-algebra.
Rogers later found a rank 7 twist (797507543735). This is the only one known. It takes us less than a cpu-hour to find it (indeed, about 3 minutes if we optimise).
- The smallest rank 5 twist is 48272239.
- The smallest rank 6 twist is 6611719866.
- The smallest rank 7 twist is 797507543735.
Each goes up by a factor of a bit more than 100x. We might then expect a rank 8 twist before 10^{15}, and even more so by 10^{16}. However, we don't even find any more rank 7 twists when searching much further.
A paper of Dujella, Janfada, and Salami gives some more rank 6 curves are found (maybe 35 in total), but nothing more of rank 7. http://www.cs.uwaterloo.ca/journals/JIS/VOL12/Janfada/janfada3.html
Searching
The Rogers curve can be found from (u,v) with (u,v)=(79873,235280). We can search up to 1 million in an hour, so finding this point is essentially trivial. We searched (using sage.math.washington.edu under "National Science Foundation Grant No. DMS-0821725") up to 100 million, with more stringent squarefree bounds, taking about 1000 cpu-hours (maybe 3 days on 17 cpus). The scaling is as H^2 when we search to H, but we throw away D values that are larger than 2^{55}, as we should already find much of interest before that point. However, we found nothing new.
Granville's heuristic
Watkins will talk at 2:30pm on Thursday about this, and explain the data and the new Elkies search method.
A new idea of Elkies
This is from an email of Noam D. Elkies.
[...] let's try this: for pairwise relatively prime d1,d2,d3,d4 with d1*d2*d3*d4 in some range [H1,H2], and find short vectors in the lattice in the (u,v) plane consisting of integer vectors defined by
d_1^2 | u and d_2^2 | v and d_3^2 | (u+v) and d_4^2 | (u-v).
In gp I get this lattice as the columns of L where
r = lift( (d_2/d_1)^2 * chinese(Mod(-1,d_3^2), Mod(1,d_4^2)) );
L = [r*d_1^2, (d_1*d_3*d_4)^2; d_2^2, 0];
By symmetry you can assume d1<d2, d3<d4, and I think d1<d3. This does about H^4 work to find twists of size about H^8, which should be at least comparable with trying all (u,v) up to H^4 even if we require that two of u,v,u+v,u-v have small representatives mod squares.
I tried this for d1,d2,d3,d4 in [5,300] and (H1,H2) = (10^6, 3*10^8), using the 2-descent bound and checking for large Mestre sum (up to p=10^5 as you did) for any survivors with rank bound >4. (Earlier runs quickly found 29274 and other rank 4 examples.) I used only the minimal nonzero vector of the lattice, "R = qflll(L); L = L*R; uv = L[,1];" -- probably qfminim would have been a better choice.
It took 10.5 hours on a single processor, and found several rank-5 twists (though not the smallest one 48272239, which did turn up once in an earlier search) and also the minimal rank-6 twist 6611719866, three times (twice for [u,v]=[45602, 19675825] and once for [11439744, 272679169]). I haven't yet seen 797507543735, let alone a new example of rank 7 or higher. I did find one case, 1726644858239, with an even larger Mestre sum and bound 7, though mwrank quickly reports that one of the isogenous curves reduces the Selmer bound to 5 and soon afterwards finds 5 independent generators to get rank 5 and |Sha|=4.
New work with the Elkies method
We have implemented a faster version of this new search method. The old PARI-based code already finds the rank 7 twist with the parameters (d1,d2,d3,d4)=(40,167,9,389) leading to (u,v)=(18822400,13526165), where it doesn't matter that these latter values are not coprime. We are currently experimenting with different parameters for H1 and H2, and can do the computation up to 500 in around an hour. UPDATE: We ran it up to 1000 in 15 hours. As before, we run the fast 2-Selmer test on every D we run across, and then compute Mestre sums, followed by descents on isogenous curves, then finally we actually try to find more points to certify the rank. We still have no new rank 7 twist.