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: | $12$ (none of which are rational) | Cusp widths | $2^{4}\cdot6^{4}\cdot8^{2}\cdot24^{2}$ | Cusp orbits | $2^{6}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 4$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 3$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24W3 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}36&119\\115&68\end{bmatrix}$, $\begin{bmatrix}41&82\\140&27\end{bmatrix}$, $\begin{bmatrix}76&165\\37&116\end{bmatrix}$, $\begin{bmatrix}78&85\\79&120\end{bmatrix}$, $\begin{bmatrix}108&65\\73&92\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 168.96.3.qr.2 for the level structure with $-I$) |
| Cyclic 168-isogeny field degree: | $16$ |
| Cyclic 168-torsion field degree: | $384$ |
| Full 168-torsion field degree: | $774144$ |
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 |
|---|---|---|---|---|---|
| 24.96.1-24.ix.1.22 | $24$ | $2$ | $2$ | $1$ | $0$ |
| 168.96.0-84.c.1.15 | $168$ | $2$ | $2$ | $0$ | $?$ |
| 168.96.0-84.c.1.31 | $168$ | $2$ | $2$ | $0$ | $?$ |
| 168.96.1-24.ix.1.24 | $168$ | $2$ | $2$ | $1$ | $?$ |
| 168.96.2-168.g.1.13 | $168$ | $2$ | $2$ | $2$ | $?$ |
| 168.96.2-168.g.1.34 | $168$ | $2$ | $2$ | $2$ | $?$ |
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.qw.1.1 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.rs.2.16 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.vg.1.1 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.vi.1.7 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.xi.1.5 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.xm.1.3 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.yw.1.9 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.za.1.3 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.baq.4.1 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.baw.2.13 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.bbg.2.1 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.bbm.1.7 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.bdc.1.5 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.bdi.2.4 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.bds.2.9 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.bdy.3.3 | $168$ | $2$ | $2$ | $5$ |