Invariants
| Level: | $168$ | $\SL_2$-level: | $24$ | Newform level: | $1$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $4 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $12^{6}$ | Cusp orbits | $1^{2}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 4$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 4$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12A4 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}89&14\\158&109\end{bmatrix}$, $\begin{bmatrix}106&67\\51&158\end{bmatrix}$, $\begin{bmatrix}107&122\\54&79\end{bmatrix}$, $\begin{bmatrix}120&61\\145&64\end{bmatrix}$, $\begin{bmatrix}151&122\\26&39\end{bmatrix}$, $\begin{bmatrix}156&163\\103&16\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 168.72.4.gi.1 for the level structure with $-I$) |
| Cyclic 168-isogeny field degree: | $64$ |
| Cyclic 168-torsion field degree: | $1536$ |
| Full 168-torsion field degree: | $1032192$ |
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 |
|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(3)$ | $3$ | $48$ | $24$ | $0$ | $0$ |
| 56.48.0-56.x.1.8 | $56$ | $3$ | $3$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.72.2-12.p.1.12 | $24$ | $2$ | $2$ | $2$ | $0$ |
| 56.48.0-56.x.1.8 | $56$ | $3$ | $3$ | $0$ | $0$ |
| 168.72.2-12.p.1.11 | $168$ | $2$ | $2$ | $2$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.