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

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 $m \not \equiv 0 \pmod{3}, m \not \equiv 0 \pmod{4} ~\text{and}~ m \equiv 2 \pmod {2^{k}}$ where $k \geq 5$, 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 $2$.