Non-divisible point on a two-parameter family of elliptic curves

Abstract

Let n be a positive integer and t be a non-zero integer. We consider the two-parameter family of elliptic curves over \(\mathbb {Q}\) given by $$\begin{aligned} \mathcal {E}_n(t):y^2=x^3+tx^2-n^2(t+3n^2)x+n^6. \end{aligned}$$We prove a result of non-divisibility of the point \((0,n^3) \in \mathcal {E}_n(t)(\mathbb {Q})\) whenever t is sufficiently large compared to n and \(t^2+3n^2t+9n^4\) is squarefree. Our work extends to this family of elliptic curves a previous study of Duquesne mainly stated for \(n=1\) and \(t>0\).