Invariants
| Level: | $168$ | $\SL_2$-level: | $4$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (none of which are rational) | Cusp widths | $4^{6}$ | Cusp orbits | $2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1 \le \gamma \le 2$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 4G0 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}1&52\\26&91\end{bmatrix}$, $\begin{bmatrix}31&102\\84&143\end{bmatrix}$, $\begin{bmatrix}129&110\\58&65\end{bmatrix}$, $\begin{bmatrix}149&90\\74&163\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 168.24.0.d.1 for the level structure with $-I$) |
| Cyclic 168-isogeny field degree: | $128$ |
| Cyclic 168-torsion field degree: | $6144$ |
| Full 168-torsion field degree: | $3096576$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.24.0-8.a.1.1 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 56.24.0-8.a.1.3 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 84.24.0-84.b.1.3 | $84$ | $2$ | $2$ | $0$ | $?$ |
| 168.24.0-84.b.1.6 | $168$ | $2$ | $2$ | $0$ | $?$ |
| 168.24.0-168.b.1.3 | $168$ | $2$ | $2$ | $0$ | $?$ |
| 168.24.0-168.b.1.13 | $168$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 168.144.4-168.k.1.9 | $168$ | $3$ | $3$ | $4$ |
| 168.192.3-168.da.1.27 | $168$ | $4$ | $4$ | $3$ |
| 168.384.11-168.n.1.29 | $168$ | $8$ | $8$ | $11$ |