On the Family of Elliptic Curves $y^2=x^3-5pqx$

TL;DR

This paper investigates the ranks and torsion structures of a specific family of elliptic curves defined by $y^2=x^3-5pqx$, providing conditions under which the rank is zero, one, or two over different fields, and characterizing their torsion groups.

Contribution

It establishes explicit conditions on primes p and q that determine the rank and torsion structure of the elliptic curves in this family.

Findings

Rank is zero over $\, \\mathbb{Q}$ and $\, \\mathbb{Q}(i)$ for certain prime congruences.

Rank is one over $\, \\mathbb{Q}$ and two over $\, \\mathbb{Q}(i)$ under specific conditions involving perfect squares.

Torsion subgroup over $\, \\mathbb{Q}$ is isomorphic to $\, \\mathbb{Z}/2\, \\mathbb{Z}$.

Abstract

This article considers the family of elliptic curves given by $E_{pq}: y^2=x^3-5pqx$ and certain conditions on odd primed $p$ and $q$. More specifically, we have proved that if $p \equiv 33 \pmod {40}$ and $ q \equiv 7 \pmod {40}$, then the rank of $E_{pq}$ is zero over both $ \mathbb{Q} $ and $ \mathbb{Q}(i) $. Furthermore, if the primes $ p $ and $q$ are of the form $ 40k + 33 $ and $ 40l + 27$, where $k,l \in \mathbb{Z}$ such that $(25k+ 5l +21)$ is a perfect square, then the given family of elliptic curves has rank one over $\mathbb{Q}$ and rank two over $\mathbb{Q}(i)$. Finally, we have shown that torsion of $E_{pq}$ over $\mathbb{Q}$ is isomorphic to $ \mathbb{Z}/ 2\mathbb{Z}$.