On the Elliptic Curves of the Form y2 = x3 − pqx

By the Mordell- Weil theorem, the group of rational points on an elliptic curve over a number field is a finitely generated abelian group. This paper studies the rank of the family Epq : y2 = x3 ? pqx of elliptic curves, where p and q are distinct primes. We give infinite families of elliptic curves of the form y2 = x3 ? pqx with rank two, three and four, assuming a conjecture of Schinzel and Sierpinski is true.