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: | $\begin{bmatrix}5&31\\32&147\end{bmatrix}$, $\begin{bmatrix}37&33\\40&29\end{bmatrix}$, $\begin{bmatrix}61&6\\96&43\end{bmatrix}$, $\begin{bmatrix}63&52\\52&3\end{bmatrix}$, $\begin{bmatrix}109&83\\160&123\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 168.96.3.ml.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
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_0(3)$ | $3$ | $48$ | $24$ | $0$ | $0$ |
| 56.48.0-56.bp.1.3 | $56$ | $4$ | $4$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.96.1-24.ix.1.22 | $24$ | $2$ | $2$ | $1$ | $0$ |
| 56.48.0-56.bp.1.3 | $56$ | $4$ | $4$ | $0$ | $0$ |
| 168.96.1-24.ix.1.3 | $168$ | $2$ | $2$ | $1$ | $?$ |
| 168.96.1-168.za.1.11 | $168$ | $2$ | $2$ | $1$ | $?$ |
| 168.96.1-168.za.1.34 | $168$ | $2$ | $2$ | $1$ | $?$ |
| 168.96.1-168.zt.1.25 | $168$ | $2$ | $2$ | $1$ | $?$ |
| 168.96.1-168.zt.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.bas.1.11 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.bas.2.13 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.bas.3.7 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.bas.4.11 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.baw.1.11 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.baw.2.13 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.baw.3.4 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.baw.4.6 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.bhm.1.11 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.bhm.2.13 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.bhm.3.6 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.bhm.4.10 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.bhq.1.10 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.bhq.2.11 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.bhq.3.7 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.bhq.4.11 | $168$ | $2$ | $2$ | $5$ |