Invariants
| Level: | $168$ | $\SL_2$-level: | $24$ | Newform level: | $1$ | ||
| Index: | $384$ | $\PSL_2$-index: | $384$ | ||||
| Genus: | $17 = 1 + \frac{ 384 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 32 }{2}$ | ||||||
| Cusps: | $32$ (of which $4$ are rational) | Cusp widths | $4^{8}\cdot8^{8}\cdot12^{8}\cdot24^{8}$ | Cusp orbits | $1^{4}\cdot2^{6}\cdot8^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 17$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 17$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24AO17 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}1&108\\150&55\end{bmatrix}$, $\begin{bmatrix}17&156\\66&167\end{bmatrix}$, $\begin{bmatrix}39&136\\52&63\end{bmatrix}$, $\begin{bmatrix}81&16\\154&99\end{bmatrix}$, $\begin{bmatrix}97&112\\54&47\end{bmatrix}$, $\begin{bmatrix}145&64\\158&39\end{bmatrix}$, $\begin{bmatrix}145&96\\108&25\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 168-isogeny field degree: | $16$ |
| Cyclic 168-torsion field degree: | $768$ |
| Full 168-torsion field degree: | $387072$ |
Rational points
This modular curve has 4 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 |
|---|---|---|---|---|---|
| $X_0(3)$ | $3$ | $96$ | $96$ | $0$ | $0$ |
| 56.96.1.n.2 | $56$ | $4$ | $4$ | $1$ | $1$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.192.7.g.2 | $24$ | $2$ | $2$ | $7$ | $0$ |
| 56.96.1.n.2 | $56$ | $4$ | $4$ | $1$ | $1$ |
| 168.192.7.g.2 | $168$ | $2$ | $2$ | $7$ | $?$ |
| 168.192.9.bo.2 | $168$ | $2$ | $2$ | $9$ | $?$ |