Curves D y² = x³ - x of odd analytic rank

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For nonzero rational D, which may be taken to be a squarefree integer, let ED be the elliptic curve Dy²=x³-x over Q arising in the "congruent number" problem. It is known that the L-function of ED has sign -1, and thus odd analytic rank, if and only if |D| is congruent to 5, 6, or 7 mod 8. For such D, we expect by the conjecture of Birch and Swinnerton-Dyer that the arithmetic rank of each of these curves ED is odd, and therefore positive. We prove that ED has positive rank for each D such that |D| is in one of the above congruence classes mod 8 and also satisfies |D|<10⁶. Our proof is computational: we use the modular parametrization of E₁ or E₂ to construct a rational point PD on each ED from CM points on modular curves, and compute PD to enough accuracy to usually distinguish it from any of the rational torsion points on ED. In the 1375 cases in which we cannot numerically distinguish PD from a torsion point of ED, we surmise that PD is in fact a torsion point but that ED has rank 3, and prove that the rank is positive by searching for and finding a non-torsion rational point. We also report on the conjectural extension to |D|<10⁷ of the list of curves ED whose analytic rank is odd and greater than 1, which raises several new questions.