Tetraelliptic modular curves $X_1(N)$

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 construct $\phi _N$ explicitly as rational functions. Moreover, we show that all $\phi _N$ we have 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 $\phi _{17}$.