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

Abstract

The group of rational points on an elliptic curve over Q is always a finitely generated Abelian group, hence isomorphic to Zr×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:y2=x3−3pqx 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(mod24), then E has rank zero. Furthermore, if p≡5(mod12) 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.