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arXiv · 2607.16335

Classification of the rank of a certain family of elliptic curves

Arkabrata Ghosh·Bidisha Roy·Richa Sharma
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Abstract

In this article, we study the family of elliptic curves $E_{-2pq}: y^2=x^3-2pqx$, 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 $\legendre{p}{q}$.

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Arkabrata Ghosh, Bidisha Roy, Richa Sharma. 2026-07-16. Classification of the rank of a certain family of elliptic curves. https://arxiv.org/abs/2607.16335

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