This note has two main facets, first we determine all modular curves \(X(N)\) (with \(N\geq 7\) which are hyperelliptic or bielliptic. As a consequence, the set \(\Gamma_2(X(N),M):=\{P\in X(N)(L)|[L:M]\leq2\}\) is not finite for some number field \(M\) if and only if \(N=7\) or \(N=8\). Moreover, we obtain that \(\Gamma_2(X(N),\mathbb{Q}(\zeta_N))\) is always finite where \(\zeta_N\) is an \(N\)-th primitive root of unity, hence, in particular, the number of quadratic points is finite for any model \(X_N\) over \(\mathbb{Q}\) of \(X(N)\). Secondly we make available a proof that the automorphism group of \(X(N)\) coincides with the normalizer of \(\Gamma(N)\) in \(\mathrm{PSL}_2(\mathbb{R})\) modulo \(\Gamma(N)\), in particular no exceptional automorphisms appear for \(X(N)\)
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