Classification of the rank of a certain family of elliptic curves

2026/07/16 by Arkabrata Ghosh, Bidisha Roy, Richa Sharma
#math.NT

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Abstract

In this article, we study the family of elliptic curves E_-2pq: y²=x³-2pqx, where p and q are distinct odd primes. Using a 2-isogeny methods and some elementary techniques, we obtain explicit possibilities for the Mordell--Weil ranks, conditional on the Parity Conjecture. Moreover, in the rank-one case, we are also able to derive explicit conditions that are independent of the parity conjecture. Moreover, the main results depend only on the residue classes of (p,q) modulo 8 and the Legendre symbols \legendrepq.

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