Invariants
| Level: | $168$ | $\SL_2$-level: | $56$ | Newform level: | $1$ | ||
| Index: | $504$ | $\PSL_2$-index: | $252$ | ||||
| Genus: | $16 = 1 + \frac{ 252 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (none of which are rational) | Cusp widths | $7^{6}\cdot14^{3}\cdot56^{3}$ | Cusp orbits | $3^{2}\cdot6$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 30$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 16$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 56B16 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}2&119\\147&142\end{bmatrix}$, $\begin{bmatrix}38&17\\31&116\end{bmatrix}$, $\begin{bmatrix}100&161\\49&124\end{bmatrix}$, $\begin{bmatrix}106&51\\135&62\end{bmatrix}$, $\begin{bmatrix}122&67\\17&24\end{bmatrix}$, $\begin{bmatrix}123&26\\2&95\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 168.252.16.cu.1 for the level structure with $-I$) |
| Cyclic 168-isogeny field degree: | $64$ |
| Cyclic 168-torsion field degree: | $3072$ |
| Full 168-torsion field degree: | $294912$ |
Rational points
This modular curve has no $\Q_p$ points for $p=67$, and therefore no rational points.
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(7)$ | $7$ | $24$ | $12$ | $0$ | $0$ |
| 24.24.0-24.y.1.9 | $24$ | $21$ | $21$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.24.0-24.y.1.9 | $24$ | $21$ | $21$ | $0$ | $0$ |
| 56.252.7-28.c.1.12 | $56$ | $2$ | $2$ | $7$ | $0$ |
| 168.252.7-28.c.1.23 | $168$ | $2$ | $2$ | $7$ | $?$ |