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

For $N=p$ prime modular curve $X_{\mathrm{ns}}(p)$ has the following moduli interpretation. It parameterizes $\bar k$-isomorphism classes of pairs $(E,\mathfrak o)$ where $E$ is an elliptic curve over $k$ and $\mathfrak o$ is a $k$-rational oriented necklace: a $\Gal_k$-stable cyclic permutation of the $p+1$ cyclic subgroups of $E[p](\bar k)$ of order $p$ [arXiv:1402.3498, 10.1017/S0017089517000180].