Bielliptic and Hyperelliptic modular curves X(N) and the group Aut(X(N))

We determine all modular curves X(N) (with $N\geq 7$) which are hyperelliptic or bielliptic. We make available a proof that the automorphism group of X(N) coincides with the normalizer of $Γ(N)$ in $\operatorname{PSL}_2(\mathbb{R})$.

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