On the Mordell--Weil group of the elliptic curve y^2=x^3+n

TL;DR

This paper investigates the structure of the Mordell--Weil group for a family of elliptic curves y^2=x^3+n, providing explicit points, independence conditions, and methods to compute height bounds to understand the group's basis.

Contribution

It introduces explicit integral points on the family of Mordell curves and demonstrates how to determine their independence and basis within the Mordell--Weil group.

Findings

Explicit integral points are identified for the family y^2=x^3+n.

Conditions under which points are independent are established.

Bounds for canonical heights of points are computed.

Abstract

We study an infinite family of Mordell curves (i.e. the elliptic curves in the form y^2=x^3+n, n \in Z) over Q with three explicit integral points. We show that the points are independent in certain cases. We describe how to compute bounds of the canonical heights of the points. Using the result we show that any pair in the three points can always be a part of a basis of the free part of the Mordell--Weil group.