## ``Congruent number curves'' ny2=x3-x with |n| congruent to 7 mod 8 and less than ten million with rank at least 3, assuming the conjecture of Birch and Swinnerton-Dyer

In this page we reported and used the list of |n|=8k+7<106 for which the ``congruent number'' curve

E(n): n y2 = x3 - x

has analytic rank at least 3.

Extending this computation to 107 would take about two weeks on a 500MHz machine running the same gp program. But most of this time would be spent on values of |n| for which E(n) can be proved to have rank at most 1 by a 2-descent between E(n) and the 2-isogenous curve ny2=x3+4x. [As n grows, the bound from this 2-descent on the rank of E(n) exceeds any given integer r with probability approaching 100%; but this approach is very slow, and of the integers |n|=8k+7<107, only 16.2% pass this test even for r=1.]

For the 106 computation, it was worthwhile to try all n, including those (86% in that range) for which the arithmetic rank cannot exceed 1, because this made it possible to prove that every E(n) in that range with |n|=8k+7 has positive rank. But we do not expect to be able in reasonable time to find a non-torsion point on each of the 2392 curves listed below. Thus we carried out the computation only for those |n|=8k+7<107 that also passed the 2-descent test for rank at least 3. Hence we cannot claim unconditionally to have listed all cases of analytic rank at least 3 with |n|=8k+7<107; but we do know that the list is complete assuming the ``analytic rank = arithmetic rank'' conjecture of Birch and Swinnerton-Dyer.

Actually one also needs the assumption that none of these curves has a nonzero Heegner point that is numerically indistinguishable from a torsion point. I set the threshold to 10-8, working in 19-digit precision, but in fact all points below that threshold were within 10-15 of a torsion point.

A graph (PostScript, landscape mode) of the function f(N)={number of |n|<N in the list} can be found here.

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