Invariants
| Level: | $168$ | $\SL_2$-level: | $12$ | 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 | $4^{12}\cdot12^{12}$ | Cusp orbits | $2^{4}\cdot4^{4}$ | ||
| 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: | 12E5 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}33&76\\152&7\end{bmatrix}$, $\begin{bmatrix}51&118\\110&13\end{bmatrix}$, $\begin{bmatrix}51&134\\58&119\end{bmatrix}$, $\begin{bmatrix}107&160\\138&61\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 168.192.5.kc.2 for the level structure with $-I$) |
| Cyclic 168-isogeny field degree: | $32$ |
| Cyclic 168-torsion field degree: | $768$ |
| Full 168-torsion field degree: | $387072$ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.192.1-12.c.2.1 | $12$ | $2$ | $2$ | $1$ | $0$ |
| 168.192.1-12.c.2.6 | $168$ | $2$ | $2$ | $1$ | $?$ |
| 168.192.1-168.lp.1.11 | $168$ | $2$ | $2$ | $1$ | $?$ |
| 168.192.1-168.lp.1.23 | $168$ | $2$ | $2$ | $1$ | $?$ |
| 168.192.1-168.lv.3.6 | $168$ | $2$ | $2$ | $1$ | $?$ |
| 168.192.1-168.lv.3.20 | $168$ | $2$ | $2$ | $1$ | $?$ |
| 168.192.3-168.dh.1.15 | $168$ | $2$ | $2$ | $3$ | $?$ |
| 168.192.3-168.dh.1.20 | $168$ | $2$ | $2$ | $3$ | $?$ |
| 168.192.3-168.eg.2.4 | $168$ | $2$ | $2$ | $3$ | $?$ |
| 168.192.3-168.eg.2.15 | $168$ | $2$ | $2$ | $3$ | $?$ |
| 168.192.3-168.ep.1.1 | $168$ | $2$ | $2$ | $3$ | $?$ |
| 168.192.3-168.ep.1.20 | $168$ | $2$ | $2$ | $3$ | $?$ |
| 168.192.3-168.ev.2.7 | $168$ | $2$ | $2$ | $3$ | $?$ |
| 168.192.3-168.ev.2.30 | $168$ | $2$ | $2$ | $3$ | $?$ |