This article considers the family of elliptic curves given by $E_{ pq}: y^2=x^3-5pqx$ and certain conditions on odd primed $p$ and $q$. More specifically, we have proved that if $p \equiv 33 \pmod { 40}$ and $ q \equiv 7 \pmod { 40}$, then the rank of $E_{ pq}$ is zero over both $ \mathbb{ Q} $ and $ \mathbb{ Q}(i) $. Furthermore, if the primes $ p $ and $q$ are of the form $ 40k + 33 $ and $ 40l + 27$, where $k,l \in \mathbb{ Z}$ such that $(25k+ 5l +21)$ is a perfect square, then the given family of elliptic curves has rank one over $\mathbb{ Q}$ and rank two over $\mathbb{ Q}(i)$. Finally, we have shown that torsion of $E_{ pq}$ over $\mathbb{ Q}$ is isomorphic to $ \mathbb{ Z}/ 2\mathbb{ Z}$.
Categories: math.NT