Invariants
| Level: | $168$ | $\SL_2$-level: | $24$ | Newform level: | $1$ | ||
| Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $7 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (none of which are rational) | Cusp widths | $6^{8}\cdot24^{4}$ | Cusp orbits | $2^{4}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 12$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 7$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24J7 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}23&114\\48&161\end{bmatrix}$, $\begin{bmatrix}57&40\\104&33\end{bmatrix}$, $\begin{bmatrix}93&16\\40&53\end{bmatrix}$, $\begin{bmatrix}141&106\\40&17\end{bmatrix}$, $\begin{bmatrix}149&46\\88&145\end{bmatrix}$, $\begin{bmatrix}161&40\\88&109\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 168.144.7.bjr.1 for the level structure with $-I$) |
| Cyclic 168-isogeny field degree: | $32$ |
| Cyclic 168-torsion field degree: | $1536$ |
| Full 168-torsion field degree: | $516096$ |
Rational points
This modular curve has no real points and no $\Q_p$ points for $p=5,13,19,61$, and therefore no 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 |
|---|---|---|---|---|---|
| 8.48.0-8.i.1.2 | $8$ | $6$ | $6$ | $0$ | $0$ |
| 21.6.0.b.1 | $21$ | $48$ | $24$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.144.4-24.ch.1.38 | $24$ | $2$ | $2$ | $4$ | $0$ |
| 84.144.3-84.s.1.12 | $84$ | $2$ | $2$ | $3$ | $?$ |
| 168.144.3-84.s.1.35 | $168$ | $2$ | $2$ | $3$ | $?$ |
| 168.144.3-168.caz.1.21 | $168$ | $2$ | $2$ | $3$ | $?$ |
| 168.144.3-168.caz.1.38 | $168$ | $2$ | $2$ | $3$ | $?$ |
| 168.144.3-168.ccq.1.11 | $168$ | $2$ | $2$ | $3$ | $?$ |
| 168.144.3-168.ccq.1.22 | $168$ | $2$ | $2$ | $3$ | $?$ |
| 168.144.4-168.bg.1.5 | $168$ | $2$ | $2$ | $4$ | $?$ |
| 168.144.4-168.bg.1.76 | $168$ | $2$ | $2$ | $4$ | $?$ |
| 168.144.4-24.ch.1.21 | $168$ | $2$ | $2$ | $4$ | $?$ |
| 168.144.4-168.od.1.22 | $168$ | $2$ | $2$ | $4$ | $?$ |
| 168.144.4-168.od.1.38 | $168$ | $2$ | $2$ | $4$ | $?$ |
| 168.144.4-168.qg.1.11 | $168$ | $2$ | $2$ | $4$ | $?$ |
| 168.144.4-168.qg.1.22 | $168$ | $2$ | $2$ | $4$ | $?$ |