This article considers the family of elliptic curves given by \(E_{p,q}: y^{2}=x^{3}-pqx\) , where p and q are distinct odd primes. More specifically, we show that \(p \equiv 7 \pmod {8}\) , \(q \equiv 5 \pmod {8}\) , and \(\left( \dfrac{p}{q} \right) =-1\) , then the ranks of both \(E_{p,q}(\mathbb {Q})\) and \(E_{p,q}(\mathbb {Q}(i))\) are 0. In the second theorem, we prove that if (i) \(p \equiv 3 \pmod {8}\) , (ii) \(q \equiv 5 \pmod {8}\) , (iii) \(\left( \dfrac{p}{q} \right) =-1\) , and (iv) \(2(p+q)\) is a perfect square, then the rank of \(E_{p,q}(\mathbb {Q})\) is 1 and the rank of \(E_{p,q}(\mathbb {Q}(i))\) is 2. Finally, by using the BSD and parity conjectures, we prove that if (i) \(p \equiv 5 \pmod {8}\) , (ii) \(q \equiv 3 \pmod {8}\) , and (iii) \(\left( \dfrac{p}{q} \right) =-1\) , then the rank of \(E_{p,q}(\mathbb {Q})\) is 1 over \(\mathbb {Q}\) and the rank of \(E_{p,q}(\mathbb {Q}(i))\) is 2.
