This article considers the family of elliptic curves given by and certain conditions on odd primes p and q. More specifically, we have shown that if and, then the rank of is zero over both and. Furthermore, if the primes p and q are of the form and, where such that is a perfect square, then the given family of elliptic curves has rank one over and rank two over. Finally, we have shown that the torsion of over is isomorphic to.