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: | $32$ (none of which are rational) | Cusp widths | $4^{8}\cdot8^{8}\cdot12^{8}\cdot24^{8}$ | Cusp orbits | $2^{4}\cdot4^{6}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 32$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 17$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24AO17 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}41&56\\36&67\end{bmatrix}$, $\begin{bmatrix}99&38\\44&135\end{bmatrix}$, $\begin{bmatrix}139&66\\72&115\end{bmatrix}$, $\begin{bmatrix}149&18\\144&53\end{bmatrix}$, $\begin{bmatrix}163&138\\20&95\end{bmatrix}$, $\begin{bmatrix}167&44\\52&45\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 168-isogeny field degree: | $16$ |
| Cyclic 168-torsion field degree: | $384$ |
| Full 168-torsion field degree: | $387072$ |
Rational points
This modular curve has no $\Q_p$ points for $p=5,13$, 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 |
|---|---|---|---|---|---|
| $X_0(3)$ | $3$ | $96$ | $96$ | $0$ | $0$ |
| 56.96.1.bu.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.dg.1 | $24$ | $2$ | $2$ | $7$ | $0$ |
| 56.96.1.bu.2 | $56$ | $4$ | $4$ | $1$ | $1$ |
| 168.192.7.ca.2 | $168$ | $2$ | $2$ | $7$ | $?$ |
| 168.192.7.cf.1 | $168$ | $2$ | $2$ | $7$ | $?$ |
| 168.192.7.jg.1 | $168$ | $2$ | $2$ | $7$ | $?$ |
| 168.192.9.jm.1 | $168$ | $2$ | $2$ | $9$ | $?$ |
| 168.192.9.jr.1 | $168$ | $2$ | $2$ | $9$ | $?$ |
| 168.192.9.og.1 | $168$ | $2$ | $2$ | $9$ | $?$ |