Invariants
| Level: | $168$ | $\SL_2$-level: | $84$ | Newform level: | $1$ | ||
| Index: | $384$ | $\PSL_2$-index: | $192$ | ||||
| Genus: | $13 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (all of which are rational) | Cusp widths | $2\cdot4\cdot6\cdot12\cdot14\cdot28\cdot42\cdot84$ | Cusp orbits | $1^{8}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 13$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 13$ | ||||||
| Rational cusps: | $8$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 84G13 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}22&17\\165&14\end{bmatrix}$, $\begin{bmatrix}23&144\\164&73\end{bmatrix}$, $\begin{bmatrix}47&64\\30&109\end{bmatrix}$, $\begin{bmatrix}123&118\\152&103\end{bmatrix}$, $\begin{bmatrix}159&160\\146&19\end{bmatrix}$, $\begin{bmatrix}165&100\\98&139\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 168.192.13.pd.1 for the level structure with $-I$) |
| Cyclic 168-isogeny field degree: | $4$ |
| Cyclic 168-torsion field degree: | $192$ |
| Full 168-torsion field degree: | $387072$ |
Rational points
This modular curve has 8 rational cusps but no known non-cuspidal 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 |
|---|---|---|---|---|---|
| $X_0(7)$ | $7$ | $48$ | $24$ | $0$ | $0$ |
| 24.48.1-24.er.1.10 | $24$ | $8$ | $8$ | $1$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.48.1-24.er.1.10 | $24$ | $8$ | $8$ | $1$ | $0$ |
| 84.192.5-42.a.1.47 | $84$ | $2$ | $2$ | $5$ | $?$ |
| 168.192.5-42.a.1.27 | $168$ | $2$ | $2$ | $5$ | $?$ |