On the family of elliptic curves $y^2=x^3-m^{2}x+(pqr{)}^2$

Abstract

In this article, we consider a family of elliptic curves defined by $E_m:y^2=x^3-m^{2}x+(pqr{)}^2$ where $m$ is a positive integer and $p,q,\text{and}r$ are distinct odd primes and study the torsion as well the rank of $E_m(\mathbb{Q})$. More specifically, we proved that if $m\not\equiv 0(\mathrm{mod}3),m\not\equiv 0(\mathrm{mod}4)\text{and}m\equiv 2(\mathrm{mod}{2}^k)$ where $k\ge 5$, then the torsion subgroup of $E_m(\mathbb{Q})$ is trivial and lower bound of the $\mathbb{Q}$ rank of this family of elliptic curves is $2$.

15 pages, some additional references