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 $\mathbb Z/17\mathbb Z$ over cyclic quartic number fields by using the cyclic map $\phi_{17}$.