Generators for the elliptic curve $E_{(p,q)} : y^2 = x^3 - p^2x + q^2$

TL;DR

This paper investigates the structure of a family of elliptic curves defined by specific prime parameters, demonstrating the conditions under which two points form a basis, thereby advancing understanding of their group structure.

Contribution

It establishes conditions for two points to form a basis of the elliptic curve group in the family $E_{(p,q)}$, extending previous results on their rank.

Findings

Two points can be extended to a basis under certain conditions.

The rank of the curves is at least two for all primes greater than five.

Full basis recovery is possible under specified conditions.

Abstract

Let $\{E_{(p,q)}\}$ be a family of elliptic curves over a rational field such that we have $E_{(p,q)} : y^2 = x^3 - p^2x + q^2$, where $p$ and $q$ are prime numbers greater than five. Earlier work showed that the elliptic curve $E_{(p,q)}$ had ranked at least two for all $p, q > 5$ and two independent points. This paper shows that two points that can be extended to a basis for $E_{(p,q)}$ under conditions are confident that we will fully recover.