Generators for the elliptic curve \(y^2=x^3-nx\)
Let \(E_n\) be the elliptic curve \(y^2=x^3-nx\) with \(n\in \mathbb{N}-\{0\}\). Let \(E_n(\mathbb{Q})\) be the set of rational points, then its torsion group \(E(\mathbb{Q})_{\text{tor}}\) is isomorphic to \(\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}\) (resp. \(\mathbb{Z}/2\mathbb{Z}\)) when \(n\) is (resp. is not) a square. Imposing some conditions on \(n\) (namely: \(n\) not a square, 4th-power free and \(n=st\) for some non square integers \(s\) and \(t\) such that \(t-s=\alpha^2\) and \(m^4s-t=\beta^2\) for some \(\alpha\), \(\beta\), \(m\in \mathbb{N}-\{0\}\,\)) the authors find explicit points \(G_1=(-s,s\alpha)\) and \(G_2=(m^2s,ms\beta)\) which are among the generators of \(E(\mathbb{Q})\) if \(m=\) 2 or 3 (the same should be true for \(m\geq 4\) but a proof is provided only for \(n\geq m^{26}\)). The proof is by direct computation: first a 2-descent procedure shows that the points are independent modulo \(E(\mathbb{Q})_{\text{tor}}\,\), then the computation of the heights and a theorem of \textit{S. Siksek} [Rocky Mt. J. Math. 25, No.~4, 1501--1538 (1995; Zbl 0852.11028)] yield the result. In the final section the authors provide infinitely many examples of families of curves which verify the assumptions of the main theorem and a way of constructing them via quadratic forms.
scientific article; zbMATH DE number 5955273
scientific article