``Congruent number curves'' ny²=x³-x with |n| congruent to 6 mod 8 and less than a million: Positivity of rank, and list of curves of conjectural rank 3

For which numbers n is there a rational Pythagorean triangle of area n? This problem is known to be equivalent to: for which n does the elliptic curve

have a rational point other than the point at infinity and the 2-torsion points with y=0? Such n are said to be ``congruent'' because they are also known to be the numbers that occur as the common difference (``congruum'' in Latin) of a three-term arithmetic progression of squares.

We may, and henceforth do, assume that n is a positive squarefree integer. It is conjectured that if n is congruent mod 8 to 5, 6, or 7, then n is congruent. See this page, concerning the 8k+7 case, for more background on this problem and the relevant conjectures.

Theorem. If n=8k+6 and n<10⁶ then n is a congruent number.

Together with the corresponding results in the 8k+7 and 8k+5 cases, this shows:

E(n) has positive rank for all positive integers n<10⁶ for which the sign of the functional equation for L(E(n),s) is -1.

The proof, combining computation and theory, proceeds as described in the 8k+7 and 8k+5 pages, except that we regard E(n) as a quadratic twist of E(2) rather than of E(1), and split the 8k+6 case into two subcases according to the parity of k, or equivalently to the residue of n modulo 16. [The curve E(2) has Weierstrass equation y²=x³-4x, and is 2-isogenous with the curve y²=x³+x; these curves have conductor 64.] If n is congruent to 14 mod 16, the rational prime 2 is split in the imaginary quadratic field of discriminant -n/2, so Heegner points yield a rational point P(n) on E(n). If n is congruent to 6 mod 16, the prime 2 is inert in that quadratic field, but a ``mock Heegner'' construction applies as in the 8k+5 case, and we again get a rational point P(n). In either case we're done unless P(n) is numerically indistinguishable from a torsion point on E(n) -- which happens for 360 values of n=16m+6, and 295 values of n=16m+14. For each of these 360+295=655 values, we guess that P(n) actually is torsion, and that E(n) has analytic rank at least 3 [by Gross-Zagier in the Heegner subcase, and by the conjectural generalization of Gross-Zagier in the ``mock Heegner'' subcase]. We then expect that E(n) has a rational non-torsion point small enough to find by direct search or by 2-descent with Cremona's MWRANK. In each case we did find at least one such point. In the table at the end of this page we list the 655 n's, and for each one give the numerator and denominator of the x-coordinate of a nontrivial rational solution of ny²=x³-x. This completes the proof.

We believe that all 655 of these curves have rank exactly 3. This is because of an upper bound on the rank, obtained by 2-descent on E(n) and/or on a curve 2-isogenous with E(n). In almost all cases, we obtain an upper bound of 3; in a handful of cases the 2-descent bound is 4 but the parity conjecture reduces this to 3. In all but 30 cases, the 2-part of the Tate-Shafarevich group SHA of E(n) is trivial if E(n) has arithmetic rank 3 as expected. The 30 exceptional cases are the following 22 in the 16k+6 subcase:

42486 = 2*3*73*97	88502 = 2*17*19*137	112326 = 2*3*97*193	202742 = 2*17*67*89
275286 = 2*3*11*43*97	295222 = 2*17*19*457	372742 = 2*17*19*577	392502 = 2*3*11*19*313
487302 = 2*3*241*337	496774 = 2*17*19*769	503206 = 2*11*89*257	563686 = 2*17*59*281
632886 = 2*3*313*337	660822 = 2*3*241*457	667318 = 2*17*19*1033	668166 = 2*3*193*577
767814 = 2*3*73*1753	774598 = 2*11*137*257	820374 = 2*3*73*1873	831878 = 2*17*43*569
875526 = 2*3*337*433	964374 = 2*3*97*1657

and the following 8 in the 16k+14 subcase:

142222 = 2*17*47*89	287246 = 2*31*41*113	564094 = 2*17*47*353	634382 = 2*7*113*401
646366 = 2*7*137*337	679582 = 2*31*97*113	771374 = 2*23*41*409	820046 = 2*17*89*271

In each of these cases, MWRANK finds 3 independent points and confirms arithmetic rank 3 by 2-descent on an isogenous curve, but finds a subgroup of SHA isomorphic with (Z/2Z)².