Invariants
| Level: | $168$ | $\SL_2$-level: | $8$ | Newform level: | $1$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $3 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (of which $4$ are rational) | Cusp widths | $8^{12}$ | Cusp orbits | $1^{4}\cdot2^{2}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 3$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 3$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8B3 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}23&20\\72&37\end{bmatrix}$, $\begin{bmatrix}113&100\\116&27\end{bmatrix}$, $\begin{bmatrix}137&164\\60&109\end{bmatrix}$, $\begin{bmatrix}161&116\\48&107\end{bmatrix}$, $\begin{bmatrix}167&100\\16&15\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 168.96.3.cp.1 for the level structure with $-I$) |
| Cyclic 168-isogeny field degree: | $64$ |
| Cyclic 168-torsion field degree: | $1536$ |
| Full 168-torsion field degree: | $774144$ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.96.0-8.c.1.9 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 56.96.0-8.c.1.4 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 168.96.1-168.o.1.10 | $168$ | $2$ | $2$ | $1$ | $?$ |
| 168.96.1-168.o.1.22 | $168$ | $2$ | $2$ | $1$ | $?$ |
| 168.96.2-168.a.1.22 | $168$ | $2$ | $2$ | $2$ | $?$ |
| 168.96.2-168.a.1.24 | $168$ | $2$ | $2$ | $2$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 168.384.5-168.hr.1.8 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.hr.2.16 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.hs.1.6 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.hs.2.12 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.ht.1.8 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.ht.2.16 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.hv.1.6 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.hv.2.12 | $168$ | $2$ | $2$ | $5$ |