Invariants
| Level: | $168$ | $\SL_2$-level: | $8$ | Newform level: | $1$ | ||
| Index: | $384$ | $\PSL_2$-index: | $192$ | ||||
| Genus: | $5 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$ | ||||||
| Cusps: | $24$ (none of which are rational) | Cusp widths | $8^{24}$ | Cusp orbits | $2^{4}\cdot4^{2}\cdot8$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 5$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8A5 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}1&48\\72&37\end{bmatrix}$, $\begin{bmatrix}9&164\\160&145\end{bmatrix}$, $\begin{bmatrix}53&42\\4&47\end{bmatrix}$, $\begin{bmatrix}75&148\\28&111\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 168.192.5.ht.1 for the level structure with $-I$) |
| Cyclic 168-isogeny field degree: | $64$ |
| Cyclic 168-torsion field degree: | $768$ |
| Full 168-torsion field degree: | $387072$ |
Rational points
This modular curve has no $\Q_p$ points for $p=29$, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.192.1-24.x.2.4 | $24$ | $2$ | $2$ | $1$ | $0$ |
| 56.192.1-56.w.1.12 | $56$ | $2$ | $2$ | $1$ | $1$ |
| 168.192.1-56.w.1.1 | $168$ | $2$ | $2$ | $1$ | $?$ |
| 168.192.1-24.x.2.9 | $168$ | $2$ | $2$ | $1$ | $?$ |
| 168.192.1-168.bq.1.4 | $168$ | $2$ | $2$ | $1$ | $?$ |
| 168.192.1-168.bq.1.22 | $168$ | $2$ | $2$ | $1$ | $?$ |
| 168.192.3-168.bo.1.31 | $168$ | $2$ | $2$ | $3$ | $?$ |
| 168.192.3-168.bo.1.32 | $168$ | $2$ | $2$ | $3$ | $?$ |
| 168.192.3-168.br.1.10 | $168$ | $2$ | $2$ | $3$ | $?$ |
| 168.192.3-168.br.1.20 | $168$ | $2$ | $2$ | $3$ | $?$ |
| 168.192.3-168.bu.1.30 | $168$ | $2$ | $2$ | $3$ | $?$ |
| 168.192.3-168.bu.1.31 | $168$ | $2$ | $2$ | $3$ | $?$ |
| 168.192.3-168.cp.1.14 | $168$ | $2$ | $2$ | $3$ | $?$ |
| 168.192.3-168.cp.1.24 | $168$ | $2$ | $2$ | $3$ | $?$ |