Tetraelliptic modular curves $X_1(N)$

Abstract

In this paper, we determine all tetraelliptic modular curves $X_1(N)$ over $\mathbb{Q}$, and find some tetraelliptic maps ${\varphi }_N$ from $X_1(N)$ to elliptic curves for those tetraelliptic $X_1(N)$. Also we will construct ${\varphi }_N$ explicitly as rational functions. Moreover, we will show that all ${\varphi }_N$ 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 ${\varphi }_{17}$.

15 pages