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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 $C_m\left(\mathbb{Q}\right)$. More specifically, we prove that the torsion subgroup of $C_m\left(\mathbb{Q}\right)$ is trivial and the $\mathbb{Q}$-rank of this family is at least 2, whenever $m\neg \equiv 0(mod3)$, $m\neg \equiv 0(mod4)$ and $m\equiv 2(mod64)$ with neither $p$ nor $q$ dividing $m$.