Tetraelliptic modular curves $X_1(N)$

In this paper, we determine all tetraelliptic modular curves $X_1(N)$ over $\mathbb{Q}$, and find some tetraelliptic maps ${\phi }_N$ from $X_1(N)$ to elliptic curves for those tetraelliptic $X_1(N)$. Also we will construct ${\phi }_N$ explicitly as rational functions. Moreover, we will show that all ${\phi }_N$ we found are Galois and find elliptic curves with torsion subgroup $\mathbb{Z}\mathrm{/}17\mathbb{Z}$ over cyclic quartic number fields by using the cyclic map ${\phi }_{17}$.