On the family of elliptic curves \(Y^{2}=X^{3}-T^{2}X+1\)
Let \(E\) be the elliptic curve over \(\mathbb{Q}(T)\) given by the equation \(Y^2 = X^3 -T^2X + 1\). In this paper the author shows that the Mordell-Weil group \(E(\mathbb{Q}(T))\) is isomorphic to \(\mathbb{Z}^3\) and is generated by \((0,1), (-1,T), (-T,1)\). Furthermore he shows that \(E(\mathbb{C}(T))\) is isomorphic to \(\mathbb{Z}^4\) and is generated by \((0,1), (-1,T), (-T,1)\), and \((\zeta_6,\zeta_6^{-1}T)\), where \(\zeta_6\) denotes the primitive sixth root of unity \((1+\sqrt{-3})/2\). A subfamily of higher rank is investigated, too.
scientific article; zbMATH DE number 6062711