On the Mordell–Weil group of the elliptic curve y2=x3+n

We study an infinite family of Mordell curves (i.e. the elliptic curves in the form y2=x3+n, n∈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.