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For nonzero rational D, which may be taken to be a squarefree integer, letE_D be the elliptic curve Dy^2=x^3-x over Q arising in the "congruent number"problem. It is known that the L-function of E_D has sign -1, and thus oddanalytic 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 arithmeticrank of each of these curves E_D is odd, and therefore positive. We prove thatE_D has positive rank for each D such that |D| is in one of the abovecongruence classes mod 8 and also satisfies |D|<10^6. Our proof iscomputational: we use the modular parametrization of E_1 or E_2 to construct arational point P_D on each E_D from CM points on modular curves, and computeP_D to enough accuracy to usually distinguish it from any of the rationaltorsion points on E_D. In the 1375 cases in which we cannot numericallydistinguish P_D from a torsion point of E_D, we surmise that P_D is in fact atorsion point but that E_D has rank 3, and prove that the rank is positive bysearching for and finding a non-torsion rational point. We also report on theconjectural extension to |D|<10^7 of the list of curves E_D whose analytic rankis odd and greater than 1, which raises several new questions.

Curves Dy 2 = x 3 — x of Odd Analytic Rank