On a family of elliptic curves of rank at least $2$

TL;DR

This paper investigates a specific family of elliptic curves defined over rational numbers, proving they have trivial torsion subgroup and a rank of at least two under certain conditions on parameters and primes.

Contribution

It establishes the triviality of the torsion subgroup and provides conditions ensuring the rank is at least two for this family of elliptic curves.

Findings

Torsion subgroup of the family is trivial.

Rank is at least two under specified conditions.

Conditions on m, p, q determine rank and torsion properties.

Abstract

Let $C_{m} : y^{2} = x^{3} - m^{2}x +p^{2}q^{2}$ be a family of elliptic curves over $\mathbb{Q}$, where $m$ is a positive integer and $p, q$ are distinct odd primes. We study the torsion part and the rank of $C_m(\mathbb{Q})$. More specifically, we prove that the torsion subgroup of $C_{m}(\mathbb{Q})$ is trivial and the $\mathbb{Q}$-rank of this family is at least $2$, whenever $m \not \equiv 0 \pmod 4$ and $m \equiv 2 \pmod {64}$ with neither $p$ nor $q$ divides $m$.