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: | $\begin{bmatrix}35&90\\132&91\end{bmatrix}$, $\begin{bmatrix}39&44\\112&129\end{bmatrix}$, $\begin{bmatrix}113&144\\4&137\end{bmatrix}$, $\begin{bmatrix}127&46\\32&43\end{bmatrix}$, $\begin{bmatrix}163&22\\44&79\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 168.48.0.g.2 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 |
|---|---|---|---|---|---|
| 12.48.0-12.c.1.2 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 168.48.0-12.c.1.6 | $168$ | $2$ | $2$ | $0$ | $?$ |
| 56.48.0-56.h.1.24 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 168.48.0-56.h.1.1 | $168$ | $2$ | $2$ | $0$ | $?$ |
| 168.48.0-168.t.2.26 | $168$ | $2$ | $2$ | $0$ | $?$ |
| 168.48.0-168.t.2.40 | $168$ | $2$ | $2$ | $0$ | $?$ |
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.q.1.9 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.bc.2.13 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.dc.2.13 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.de.1.14 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.fu.2.1 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.fw.1.15 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.gk.1.15 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.gm.2.10 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.ig.1.15 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.ii.2.10 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.iw.2.2 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.iy.1.16 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.ko.2.13 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.kq.1.14 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.kw.1.10 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.kx.2.15 | $168$ | $2$ | $2$ | $1$ |
| 168.288.8-168.y.2.35 | $168$ | $3$ | $3$ | $8$ |
| 168.384.7-168.u.2.40 | $168$ | $4$ | $4$ | $7$ |