# The Mordell-Weil bases for the elliptic curve of the form $\boldsymbol{y^2=x^3-m^2x+n^2}$

## TL;DR

This paper explicitly determines Mordell-Weil bases for specific families of elliptic curves of the form y^2=x^3-m^2x+n^2, extending known rank results and providing explicit basis points under certain conditions.

## Contribution

It explicitly identifies basis points for the Mordell-Weil groups of elliptic curves of the form y^2=x^3-m^2x+n^2, expanding on previous rank results.

## Key findings

- Explicit basis points for E_{1,n} under certain conditions
- Verification of a basis for the rank three part of E_{m,1}
- Extension of known rank results for these elliptic curves

## Abstract

Let $E_{m,n}$ be an elliptic curve over $\mathbb{Q}$ of the form $y^2=x^3-m^2x+n^2$, where $m$ and $n$ are positive integers. Brown and Myers showed that the curve $E_{1,n}$ has rank at least two for all $n$. In the present paper, we specify the two points which can be extended to a basis for $E_{1,n}(\mathbb{Q})$ under certain conditions described explicitly. Moreover, we verify a similar result for the curve $E_{m,1}$, which, however, gives a basis for the rank three part of $E_{m,1}(\mathbb{Q})$.