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 primes p and q. More specifically, we have shown 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 \(40\,l + 27\), where \(k,l \in \mathbb {Z}\) such that \((25k+ 5\,l +21)\) 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 the torsion of \(E_{pq}\) over \(\mathbb {Q}\) is isomorphic to \(\mathbb {Z}/ 2\mathbb {Z}\).