Invariants
| Level: | $168$ | $\SL_2$-level: | $24$ | 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 $2$ are rational) | Cusp widths | $2^{4}\cdot6^{4}\cdot8^{2}\cdot24^{2}$ | Cusp orbits | $1^{2}\cdot2^{5}$ | ||
| 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: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24W3 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}12&157\\139&30\end{bmatrix}$, $\begin{bmatrix}28&153\\153&4\end{bmatrix}$, $\begin{bmatrix}53&88\\68&9\end{bmatrix}$, $\begin{bmatrix}67&80\\156&143\end{bmatrix}$, $\begin{bmatrix}82&35\\81&104\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 168.96.3.qc.2 for the level structure with $-I$) |
| Cyclic 168-isogeny field degree: | $16$ |
| Cyclic 168-torsion field degree: | $768$ |
| Full 168-torsion field degree: | $774144$ |
Rational points
This modular curve has 2 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 |
|---|---|---|---|---|---|
| 12.96.0-12.c.3.3 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 168.96.0-12.c.3.23 | $168$ | $2$ | $2$ | $0$ | $?$ |
| 168.96.1-168.zw.1.6 | $168$ | $2$ | $2$ | $1$ | $?$ |
| 168.96.1-168.zw.1.21 | $168$ | $2$ | $2$ | $1$ | $?$ |
| 168.96.2-168.f.1.7 | $168$ | $2$ | $2$ | $2$ | $?$ |
| 168.96.2-168.f.1.17 | $168$ | $2$ | $2$ | $2$ | $?$ |
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.ly.4.23 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.rg.1.6 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.ua.1.4 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.ui.1.1 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.we.4.13 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.wj.4.7 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.ye.4.10 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.yo.4.6 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.zt.1.3 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.zw.2.3 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.bba.1.1 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.bbb.1.4 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.bbp.4.8 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.bbs.4.13 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.bdu.4.8 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.bdv.4.14 | $168$ | $2$ | $2$ | $5$ |