Invariants
| Level: | $168$ | $\SL_2$-level: | $24$ | Newform level: | $1$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $3 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (of which $4$ are rational) | Cusp widths | $2^{2}\cdot4^{3}\cdot6^{2}\cdot8\cdot12^{3}\cdot24$ | Cusp orbits | $1^{4}\cdot2^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 3$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 3$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24AA3 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}37&116\\12&65\end{bmatrix}$, $\begin{bmatrix}49&36\\12&67\end{bmatrix}$, $\begin{bmatrix}83&84\\108&29\end{bmatrix}$, $\begin{bmatrix}89&52\\108&67\end{bmatrix}$, $\begin{bmatrix}101&10\\96&163\end{bmatrix}$, $\begin{bmatrix}115&38\\52&165\end{bmatrix}$, $\begin{bmatrix}135&94\\128&49\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 168.96.3.dy.1 for the level structure with $-I$) |
| Cyclic 168-isogeny field degree: | $16$ |
| Cyclic 168-torsion field degree: | $768$ |
| Full 168-torsion field degree: | $774144$ |
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 |
|---|---|---|---|---|---|
| $X_0(3)$ | $3$ | $48$ | $24$ | $0$ | $0$ |
| 56.48.0-56.i.1.6 | $56$ | $4$ | $4$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.96.1-12.b.1.24 | $24$ | $2$ | $2$ | $1$ | $0$ |
| 56.48.0-56.i.1.6 | $56$ | $4$ | $4$ | $0$ | $0$ |
| 168.96.1-12.b.1.37 | $168$ | $2$ | $2$ | $1$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.