On the Family of Elliptic Curves $y^2=x^3-5pqx$

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 $p\equiv 33(\mathrm{m}\mathrm{o}\mathrm{d}40)$ and $q\equiv 7(\mathrm{m}\mathrm{o}\mathrm{d}40)$, 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 $\mathbb{Z}\mathrm{/}2\mathbb{Z}$.