Invariants
| Level: | $168$ | $\SL_2$-level: | $24$ | Newform level: | $1$ | ||
| Index: | $288$ | $\PSL_2$-index: | $288$ | ||||
| Genus: | $15 = 1 + \frac{ 288 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 20 }{2}$ | ||||||
| Cusps: | $20$ (none of which are rational) | Cusp widths | $6^{8}\cdot12^{4}\cdot24^{8}$ | Cusp orbits | $2^{2}\cdot4^{2}\cdot8$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 28$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 15$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24E15 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}71&54\\4&145\end{bmatrix}$, $\begin{bmatrix}95&90\\24&41\end{bmatrix}$, $\begin{bmatrix}95&112\\40&127\end{bmatrix}$, $\begin{bmatrix}95&148\\12&91\end{bmatrix}$, $\begin{bmatrix}97&142\\160&71\end{bmatrix}$, $\begin{bmatrix}103&34\\76&159\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 168-isogeny field degree: | $64$ |
| Cyclic 168-torsion field degree: | $1536$ |
| Full 168-torsion field degree: | $516096$ |
Rational points
This modular curve has no $\Q_p$ points for $p=13,61$, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.144.8.er.1 | $24$ | $2$ | $2$ | $8$ | $0$ |
| 168.144.7.bar.2 | $168$ | $2$ | $2$ | $7$ | $?$ |
| 168.144.7.bas.1 | $168$ | $2$ | $2$ | $7$ | $?$ |
| 168.144.7.bil.1 | $168$ | $2$ | $2$ | $7$ | $?$ |
| 168.144.8.fu.2 | $168$ | $2$ | $2$ | $8$ | $?$ |
| 168.144.8.fv.1 | $168$ | $2$ | $2$ | $8$ | $?$ |
| 168.144.8.om.2 | $168$ | $2$ | $2$ | $8$ | $?$ |