This article considers the family of elliptic curves given by Ep,q:y2=x3-pqx, where p and q are distinct odd primes. More specifically, we show that p≡7(mod8), q≡5(mod8), and pq=-1, then the ranks of both Ep,q(Q) and Ep,q(Q(i)) are 0. In the second theorem, we prove that if (i) p≡3(mod8), (ii) q≡5(mod8), (iii) pq=-1, and (iv) 2(p+q) is a perfect square, then the rank of Ep,q(Q) is 1 and the rank of Ep,q(Q(i)) is 2. Finally, by using the BSD and parity conjectures, we prove that if (i) p≡5(mod8), (ii) q≡3(mod8), and (iii) pq=-1, then the rank of Ep,q(Q) is 1 over Q and the rank of Ep,q(Q(i)) is 2.