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 | $2^{4}\cdot4^{2}\cdot8^{4}$ | Cusp orbits | $2^{5}$ | ||
| 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: | 8O0 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}15&40\\76&161\end{bmatrix}$, $\begin{bmatrix}81&128\\32&113\end{bmatrix}$, $\begin{bmatrix}97&52\\82&5\end{bmatrix}$, $\begin{bmatrix}145&80\\46&121\end{bmatrix}$, $\begin{bmatrix}159&100\\74&119\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 168.48.0.bu.1 for the level structure with $-I$) |
| Cyclic 168-isogeny field degree: | $64$ |
| Cyclic 168-torsion field degree: | $1536$ |
| 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 |
|---|---|---|---|---|---|
| 8.48.0-8.e.1.4 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 168.48.0-8.e.1.9 | $168$ | $2$ | $2$ | $0$ | $?$ |
| 168.48.0-168.u.2.4 | $168$ | $2$ | $2$ | $0$ | $?$ |
| 168.48.0-168.u.2.27 | $168$ | $2$ | $2$ | $0$ | $?$ |
| 168.48.0-168.x.1.14 | $168$ | $2$ | $2$ | $0$ | $?$ |
| 168.48.0-168.x.1.28 | $168$ | $2$ | $2$ | $0$ | $?$ |
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.bz.1.7 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.ce.1.4 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.cy.1.7 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.db.1.11 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.ea.1.8 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.eb.1.8 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.eg.1.2 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.eh.1.2 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.gq.1.6 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.gr.1.6 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.gs.1.6 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.gt.1.4 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.hg.1.5 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.hh.1.9 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.hi.1.8 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.hj.1.8 | $168$ | $2$ | $2$ | $1$ |
| 168.288.8-168.mb.2.34 | $168$ | $3$ | $3$ | $8$ |
| 168.384.7-168.gp.2.23 | $168$ | $4$ | $4$ | $7$ |