Invariants
| Level: | $168$ | $\SL_2$-level: | $24$ | Newform level: | $1$ | ||
| Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $9 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $2$ are rational) | Cusp widths | $12^{4}\cdot24^{4}$ | Cusp orbits | $1^{2}\cdot2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 9$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 9$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24C9 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}5&118\\16&25\end{bmatrix}$, $\begin{bmatrix}9&148\\128&69\end{bmatrix}$, $\begin{bmatrix}17&162\\96&77\end{bmatrix}$, $\begin{bmatrix}35&76\\88&61\end{bmatrix}$, $\begin{bmatrix}61&4\\40&77\end{bmatrix}$, $\begin{bmatrix}61&160\\0&5\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 168.144.9.bbf.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 2 rational cusps but no known non-cuspidal 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 |
|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(3)$ | $3$ | $96$ | $48$ | $0$ | $0$ |
| 56.96.1-56.bv.1.1 | $56$ | $3$ | $3$ | $1$ | $1$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.144.4-24.ch.1.38 | $24$ | $2$ | $2$ | $4$ | $0$ |
| 56.96.1-56.bv.1.1 | $56$ | $3$ | $3$ | $1$ | $1$ |
| 168.144.4-168.e.1.21 | $168$ | $2$ | $2$ | $4$ | $?$ |
| 168.144.4-168.e.1.70 | $168$ | $2$ | $2$ | $4$ | $?$ |
| 168.144.4-24.ch.1.32 | $168$ | $2$ | $2$ | $4$ | $?$ |
| 168.144.5-168.d.1.8 | $168$ | $2$ | $2$ | $5$ | $?$ |
| 168.144.5-168.d.1.51 | $168$ | $2$ | $2$ | $5$ | $?$ |