Invariants
| Level: | $168$ | $\SL_2$-level: | $24$ | Newform level: | $1$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $5 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $4^{2}\cdot8^{2}\cdot12^{2}\cdot24^{2}$ | Cusp orbits | $2^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 5$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24I5 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}19&140\\166&117\end{bmatrix}$, $\begin{bmatrix}39&152\\4&95\end{bmatrix}$, $\begin{bmatrix}113&129\\164&115\end{bmatrix}$, $\begin{bmatrix}117&83\\38&99\end{bmatrix}$, $\begin{bmatrix}137&82\\116&105\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 168.96.5.nf.1 for the level structure with $-I$) |
| Cyclic 168-isogeny field degree: | $32$ |
| Cyclic 168-torsion field degree: | $1536$ |
| Full 168-torsion field degree: | $774144$ |
Rational points
This modular curve has no $\Q_p$ points for $p=5,17$, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.96.1-12.o.1.3 | $12$ | $2$ | $2$ | $1$ | $0$ |
| 168.96.1-12.o.1.13 | $168$ | $2$ | $2$ | $1$ | $?$ |
| 168.96.3-168.ce.1.46 | $168$ | $2$ | $2$ | $3$ | $?$ |
| 168.96.3-168.ce.1.47 | $168$ | $2$ | $2$ | $3$ | $?$ |
| 168.96.3-168.cn.1.46 | $168$ | $2$ | $2$ | $3$ | $?$ |
| 168.96.3-168.cn.1.47 | $168$ | $2$ | $2$ | $3$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 168.384.9-168.cki.1.15 | $168$ | $2$ | $2$ | $9$ |
| 168.384.9-168.ckj.1.11 | $168$ | $2$ | $2$ | $9$ |
| 168.384.9-168.ckk.1.13 | $168$ | $2$ | $2$ | $9$ |
| 168.384.9-168.ckl.1.9 | $168$ | $2$ | $2$ | $9$ |
| 168.384.9-168.clo.1.16 | $168$ | $2$ | $2$ | $9$ |
| 168.384.9-168.clp.1.14 | $168$ | $2$ | $2$ | $9$ |
| 168.384.9-168.clq.1.15 | $168$ | $2$ | $2$ | $9$ |
| 168.384.9-168.clr.1.13 | $168$ | $2$ | $2$ | $9$ |
| 168.384.9-168.cme.1.15 | $168$ | $2$ | $2$ | $9$ |
| 168.384.9-168.cmf.1.7 | $168$ | $2$ | $2$ | $9$ |
| 168.384.9-168.cmg.1.16 | $168$ | $2$ | $2$ | $9$ |
| 168.384.9-168.cmh.1.8 | $168$ | $2$ | $2$ | $9$ |
| 168.384.9-168.cmu.1.11 | $168$ | $2$ | $2$ | $9$ |
| 168.384.9-168.cmv.1.5 | $168$ | $2$ | $2$ | $9$ |
| 168.384.9-168.cmw.1.14 | $168$ | $2$ | $2$ | $9$ |
| 168.384.9-168.cmx.1.7 | $168$ | $2$ | $2$ | $9$ |