Invariants
| Level: | $168$ | $\SL_2$-level: | $12$ | 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: | $12$ (none of which are rational) | Cusp widths | $4^{6}\cdot12^{6}$ | Cusp orbits | $2^{6}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 4$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 3$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12L3 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}57&34\\94&129\end{bmatrix}$, $\begin{bmatrix}57&38\\76&29\end{bmatrix}$, $\begin{bmatrix}87&22\\58&81\end{bmatrix}$, $\begin{bmatrix}137&6\\110&115\end{bmatrix}$, $\begin{bmatrix}153&8\\82&143\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 168.96.3.fj.2 for the level structure with $-I$) |
| Cyclic 168-isogeny field degree: | $32$ |
| Cyclic 168-torsion field degree: | $1536$ |
| Full 168-torsion field degree: | $774144$ |
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 |
|---|---|---|---|---|---|
| 24.96.2-24.b.2.12 | $24$ | $2$ | $2$ | $2$ | $0$ |
| 84.96.1-84.b.1.8 | $84$ | $2$ | $2$ | $1$ | $?$ |
| 168.96.0-168.o.2.7 | $168$ | $2$ | $2$ | $0$ | $?$ |
| 168.96.0-168.o.2.48 | $168$ | $2$ | $2$ | $0$ | $?$ |
| 168.96.1-84.b.1.11 | $168$ | $2$ | $2$ | $1$ | $?$ |
| 168.96.2-24.b.2.5 | $168$ | $2$ | $2$ | $2$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 168.384.5-168.oe.2.10 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.og.3.9 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.og.4.9 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.oj.1.18 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.oj.2.19 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.om.3.9 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.om.4.9 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.po.3.5 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.po.4.5 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.pq.1.1 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.pq.2.1 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.qc.3.6 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.qc.4.7 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.qe.1.1 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.qe.2.1 | $168$ | $2$ | $2$ | $5$ |