Optimal curves differing by a 5-isogeny (Q2921809)

For a positive integer \(N\), let \(X_1(N)\), \(X_0(N)\) denote the usual modular curves. Let \(\mathcal{C}\) denote an isogeny class of elliptic curves defined over \(\mathbb{Q}\) of conductor \(N\). For \(i=0, 1\), there is a unique curve \(E_i\in \mathcal{C}\) and a parametrization \(\phi_i:X_i(N)\rightarrow E_i\) such that for any \(E\in \mathcal{C}\) and a parametrization \(\phi'_i:X_i(N)\rightarrow E\), there is an isogeny \(\pi_i:E_i\rightarrow E\) such that \(\pi_i \circ\phi_i=\phi'_i\). The curve \(E_i\) is called the \(X_i(N)\)-optimal curve. Typically, the optimal curves \(E_0\) and \(E_1\) for a given isogeny class are the same, but not always. Based on numerical observations, \textit{W. A. Stein} and \textit{M. Watkins} [Lect. Notes Comput. Sci. 2369, 267--275 (2002; Zbl 1058.11036)] conjecture that \(E_0\) and \(E_1\) differ by a \(5\)-isogeny if and only if \(E_0=X_0(11)\) and \(E_1=X_1(11)\). In this paper, it is proved that this conjecture is true if \(N\) is square-free and not divisible by \(5\). Moreover, in the process of the proof of this result, the authors prove that a conjecture of \textit{T. Hadano} [Nagoya Math. J. 66, 99--108 (1977; Zbl 0343.14013)] on the classification of rational elliptic curves with a rational point of order 5, is not true.