This page is semi-redundant with http://wiki.l-functions.org/LfunctionsAndModularFormsII/CentralValues/32A_EvenRankTwists
Gonzalo found those d's for which the coefficient of the weight 3/2 lift vanished. Noam then post-processed these using some GP/PARI script, that considered descent by 2-isogeny. The results were:
1 mod 8, up to 10^10:
70778066 d's with zero coefficient, of which
25570630 are squarefree, and of these
4992744 have selmer bound 4,
181264 have selmer bound 6, and
582 have selmer bound 8.
3 mod 8, up to 10^10:
62320480 d's with zero coefficient, of which
19940705 are squarefree, and of these
4418976 have selmer bound 4,
139863 have selmer bound 6, and
286 have selmer bound 8.
10 mod 16, up to 10^10:
34789316 d's with zero coefficients, of which
12684954 are squarefree (gp in 5 minutes...), and of these
4431668 have selmer bound 4,
111590 have selmer bound 6, and only
26 have selmer bound 8.
2 mod 16, up to 10^10:
37312299 d's with zero coefficients, of which
14405271 are squarefree (gp in 5 minutes...), and of these
5158646 have selmer bound 4,
163713 have selmer bound 6, and
246 have selmer bound 8.Also "mwrank -s" (Selmer only) was used for a full 2-descent on all isogenous curves for those twists with Selmer bound 6 or more - this ran through about 4000 an hour. From this, we were left with 1033 possible twists of rank 6 or more up to 10^{10}. Further descents were used to show that the only rank 6 twist up to 10^{10} is that which was already found by Rogers, namely d=6611719866.
Mark has adapted Denis Simon's ellQ.gp GP/PARI script to the current case, and can find the Selmer bound at a much higher rate (about 50 per second). The use of isogenous curves can also be useful in obtaining the true rank. From the 19599604 million curves with Selmer bound 4 or more, there are 969744+862818+1205091+1196003 or 4233656 possible rank 4+ twists after considering all isogenous curves. The previous factoid forgets that Noam only used a 2-isogeny-descent scan on the curve itself, and not a full 2-descent. This leaves only 3059567 candidates for rank 4.
For these remaining 3.06 million curves, we used a CasselsTatePairing (due to Steve Donnelly in MAGMA) on all curves in the isogeny class. This eliminated all but 143105 (plus the Rogers twist) of the rank 4 candidates. This took about 40 cpu-days (5 days on an 8-cluster). We then searched for points up to x-height 10^4 on all 2-covers (of all isogenous curves) -- the results seem to confirm our initial guess that many of the twists have rank 2 and 4-Sha for all curves in the isogeny class, as the counts of 0-4 independent points were 12532+52251+58686+8318+11318 (so maybe only about 15% with rank 4). The current machinery of a CasselsTatePairing for a 2-cover with a 4-cover is still not fully robust, though a similar method has been developed by T. A. Fisher in the specific case where the curve has a rational 2-torsion point (as here). However, it still looks unlikely (at present) that we will actually be able to prove which twists have rank 2 versus rank 4 for the totality of our remaining dataset. One idea for project completion is to change the curve to 36A, where the existence of a 3-isogeny could help.
None of the steps currently carried out is overly time-consuming, and there is hope to extend the data at least to 10^{11} in the rank 4 case. In particular, doing 2-descents with the GP/PARI code takes less than a cpu-day (times four, for each isogenous curve), while the CasselsTate machinery took about 6 cpu-weeks. Multiplying these by 10 still leaves us with reasonable time estimates.
We don't expect that many rank 4 curves - for comparison, in an experiment with odd parity twists (see this PDF), Watkins found about 90000 of rank 3 (or more) up to 10^9.