Trielliptic modular curves $X_1(N)$
Volume 206 / 2022
Acta Arithmetica 206 (2022), 171-188
MSC: Prmiary 11G18; Secondary 11G30.
DOI: 10.4064/aa220527-7-10
Published online: 8 November 2022
Abstract
We determine all trielliptic modular curves $X_1(N)$ over $\mathbb Q$, and construct explicit trielliptic maps from trielliptic $X_1(N)$ to elliptic curves. By using the trielliptic map constructed for $X_1(21)$, we find six non-cuspidal points of $X_1(21)$ defined over the cubic number field $\mathbb Q(\zeta _9)^+$. So the trielliptic map explains the sporadic elliptic curves over $\mathbb Q(\zeta _9)^+$ with 21-torsion found by Najman [Math. Res. Lett. 23 (2016)].
