On a class of elliptic curves with rank at most two

Rose, H. E.

doi:10.1090/S0025-5718-1995-1297476-3

On a class of elliptic curves with rank at most two
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Math. Comp. 64 (1995), 1251-1265 Request permission

Abstract:

In this note we consider the elliptic curves ${y^2} = {x^3} + px$ defined over $\mathbb {Q}$ for primes p satisfying $p \equiv 1\; \pmod 8$, and review some of their properties. We then compute and list (in the supplement) their ranks, and give, when the rank is positive, the generators of the group of rational points and Mordell-Weil lattice invariant $\tau$ for all primes $p < 50000$ of the form ${m^2} + 64{n^2}$.

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On the equation

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