Elliptic Curves of Type y2 = x3 - 3pqx Having Ranks Zero and One

The group of rational points on an elliptic curve over Q is always a finitely generated Abelian group, hence isomorphic to Z(r) x G with G a finite Abelian group. Here, r is the rank of the elliptic curve. In this paper, we determine sufficient conditions that need to be set on the prime numbers p and q so that the elliptic curve E : y(2) = x(3) - 3pqx over Q would possess a rank zero or one. Specifically, we verify that if distinct primes p and q satisfy the congruence p = q = 5 (mod 24), then E has rank zero. Furthermore, if p = 5 (mod 12) is considered instead of a modulus of 24, then E has rank zero or one. Lastly, for primes of the form p = 24k + 17 and q = 24l + 5, where 9k + 3l + 7 is a perfect square, we show that E has rank one.