We determine all modular curves X(N) (with N ≥ 7) that are hyperelliptic or bielliptic. We also give a proof that the automorphism group of X(N) is PSL₂(ℤ/Nℤ), whence it coincides with the normalizer of Γ(N) in PSL₂(ℝ) modulo ±Γ(N).

For any Eichler order $𝒪(D,N)$ of level $N$ in an indefinite quaternion algebra of discriminant $D$ there is a Fuchsian group $\Gamma (D,N)\subseteq SL(2,ℝ)$ and a Shimura curve $X(D,N)$. We associate to $𝒪(D,N)$ a set $\mathscr{H}\left(𝒪\right(D,N\left)\right)$ of binary quadratic forms which have semi-integer quadratic coefficients, and we develop a classification theory, with respect to $\Gamma (D,N)$, for primitive forms contained in $\mathscr{H}\left(𝒪\right(D,N\left)\right)$. In particular, the classification theory of primitive integral binary quadratic forms by $SL(2,\mathbb{Z})$ is recovered. Explicit fundamental domains for $\Gamma (D,N)$ allow the characterization...