Invariants
| Level: | $168$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $3 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $4$ are rational) | Cusp widths | $6^{4}\cdot12^{4}$ | Cusp orbits | $1^{4}\cdot2^{2}$ | ||
| 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: | 12D3 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}23&87\\108&143\end{bmatrix}$, $\begin{bmatrix}43&3\\36&85\end{bmatrix}$, $\begin{bmatrix}71&78\\138&155\end{bmatrix}$, $\begin{bmatrix}113&116\\96&85\end{bmatrix}$, $\begin{bmatrix}155&44\\126&157\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 168.72.3.cwd.1 for the level structure with $-I$) |
| Cyclic 168-isogeny field degree: | $32$ |
| Cyclic 168-torsion field degree: | $1536$ |
| Full 168-torsion field degree: | $1032192$ |
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 |
|---|---|---|---|---|---|
| 12.72.0-6.a.1.5 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 168.48.1-168.hk.1.4 | $168$ | $3$ | $3$ | $1$ | $?$ |
| 168.48.1-168.hk.1.22 | $168$ | $3$ | $3$ | $1$ | $?$ |
| 168.72.0-6.a.1.2 | $168$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 168.288.5-168.m.1.14 | $168$ | $2$ | $2$ | $5$ |
| 168.288.5-168.ds.1.7 | $168$ | $2$ | $2$ | $5$ |
| 168.288.5-168.ic.1.4 | $168$ | $2$ | $2$ | $5$ |
| 168.288.5-168.ig.1.8 | $168$ | $2$ | $2$ | $5$ |
| 168.288.5-168.qi.1.5 | $168$ | $2$ | $2$ | $5$ |
| 168.288.5-168.qm.1.7 | $168$ | $2$ | $2$ | $5$ |
| 168.288.5-168.rr.1.12 | $168$ | $2$ | $2$ | $5$ |
| 168.288.5-168.rv.1.2 | $168$ | $2$ | $2$ | $5$ |
| 168.288.5-168.bfs.1.5 | $168$ | $2$ | $2$ | $5$ |
| 168.288.5-168.bfv.1.8 | $168$ | $2$ | $2$ | $5$ |
| 168.288.5-168.bgu.1.8 | $168$ | $2$ | $2$ | $5$ |
| 168.288.5-168.bgx.1.6 | $168$ | $2$ | $2$ | $5$ |
| 168.288.5-168.bhw.1.8 | $168$ | $2$ | $2$ | $5$ |
| 168.288.5-168.bhz.1.4 | $168$ | $2$ | $2$ | $5$ |
| 168.288.5-168.bka.1.7 | $168$ | $2$ | $2$ | $5$ |
| 168.288.5-168.bkm.1.7 | $168$ | $2$ | $2$ | $5$ |