GENERATORS FOR THE ELLIPTIC CURVE E(p,q) : y^2 = x^3 - p^2x + q^2

Abstract

Let $ \lbrace E_{(p,q)} \rbrace $ denote a family of elliptic curves over $ \Q$ as defined by the Weierstrass equation $ E_{(p,q)}\!: y^2=x^3-p^2x+q^2 $ where $p$ and $q$ are both prime numbers greater than 5. As evidence that this has two independent points, we already showed that at least the rank of $ \lbrace E_{(p,q)} \rbrace $ is two. In this study, we show that the two independent points are part of a $\mathbb{Z}$-basis for the quotient of $E_{(p,q)}(\mathbb{Q})$ by its torsion subgroup.