Remarks on $X_{\mathrm{arith}}(N)$ (awaiting review)

The canonical field of definition of $X_{\mathrm{arith}}(N)$ is $\Q$, but this is not true of $X(N)$ except when $N\le 2$, in which case $X(N)=X_{\mathrm{arith}}(N)$.

Hyperelliptic curves: There are no hyperelliptic curves in this family.

Bielliptic curves: There are only 2 in this family: the curve $X_{\textup{arith}}(7)$ is isomorphic to the Klein quartic and the curve $X_{\textup{arith}}(8)$ is isomorphic to the Wiman curve [10.4064/aa161-3-6, MR:3145452].