Invariants
| Level: | $168$ | $\SL_2$-level: | $8$ | ||||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
| Cusps: | $10$ (none of which are rational) | Cusp widths | $4^{8}\cdot8^{2}$ | Cusp orbits | $2^{3}\cdot4$ | ||
| 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: | 8N0 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}33&70\\76&3\end{bmatrix}$, $\begin{bmatrix}59&126\\0&43\end{bmatrix}$, $\begin{bmatrix}115&96\\40&7\end{bmatrix}$, $\begin{bmatrix}125&142\\96&139\end{bmatrix}$, $\begin{bmatrix}157&138\\144&143\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 168.48.0.j.1 for the level structure with $-I$) |
| Cyclic 168-isogeny field degree: | $64$ |
| Cyclic 168-torsion field degree: | $3072$ |
| Full 168-torsion field degree: | $1548288$ |
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 |
|---|---|---|---|---|---|
| 24.48.0-12.c.1.16 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 56.48.0-56.i.1.6 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 168.48.0-12.c.1.11 | $168$ | $2$ | $2$ | $0$ | $?$ |
| 168.48.0-56.i.1.19 | $168$ | $2$ | $2$ | $0$ | $?$ |
| 168.48.0-168.u.2.11 | $168$ | $2$ | $2$ | $0$ | $?$ |
| 168.48.0-168.u.2.55 | $168$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 168.192.1-168.cb.2.11 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.cm.1.11 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.dg.1.11 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.di.2.11 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.fy.1.13 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.ga.2.6 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.go.2.6 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.gq.1.13 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.ik.2.11 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.im.1.11 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.ja.1.11 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.jc.2.7 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.ks.1.13 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.ku.2.10 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.ky.2.10 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.kz.1.11 | $168$ | $2$ | $2$ | $1$ |
| 168.288.8-168.bf.1.48 | $168$ | $3$ | $3$ | $8$ |
| 168.384.7-168.y.2.8 | $168$ | $4$ | $4$ | $7$ |