Images of isogeny classes on modular elliptic curves (Q1769978)

Let \(E\) be a modular elliptic curve with modular parametrization \(\pi: X_0 (N) \to E\), both \(E\) and \(\pi\) being defined over a number field \(K\) with algebraic closure \(\overline K\), and let \(S \subset X_0 (N) (\overline K)\) be an infinite set of points corresponding to elliptic curves which are isogenous to each other, but not isogenous to \(E\). It is shown that \(\pi(S)\) generates in \(E(\overline K)\) a subgroup of infinite rank and of finite torsion.