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: | 24V3 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}15&109\\124&33\end{bmatrix}$, $\begin{bmatrix}25&141\\120&31\end{bmatrix}$, $\begin{bmatrix}95&105\\12&47\end{bmatrix}$, $\begin{bmatrix}115&146\\156&161\end{bmatrix}$, $\begin{bmatrix}135&110\\68&99\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 168.96.3.nx.1 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 | $24$ | $2$ | $2$ | $1$ | $0$ |
| 168.48.0-168.dz.1.9 | $168$ | $4$ | $4$ | $0$ | $?$ |
| 168.96.1-84.p.1.3 | $168$ | $2$ | $2$ | $1$ | $?$ |
| 168.96.1-84.p.1.25 | $168$ | $2$ | $2$ | $1$ | $?$ |
| 168.96.1-24.ix.1.6 | $168$ | $2$ | $2$ | $1$ | $?$ |
| 168.96.1-168.zx.1.25 | $168$ | $2$ | $2$ | $1$ | $?$ |
| 168.96.1-168.zx.1.44 | $168$ | $2$ | $2$ | $1$ | $?$ |
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.bds.1.10 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.bds.2.9 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.bds.3.3 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.bds.4.6 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.bdw.1.9 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.bdw.2.10 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.bdw.3.4 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.bdw.4.2 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.bkm.1.10 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.bkm.2.9 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.bkm.3.2 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.bkm.4.4 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.bkq.1.9 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.bkq.2.10 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.bkq.3.6 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.bkq.4.3 | $168$ | $2$ | $2$ | $5$ |