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

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 $m\not\equiv 0(\mathrm{mod}4)$ and $m\equiv 2(\mathrm{mod}64)$ with neither $p$ nor $q$ divides $m$.

15