Invariants
| Level: | $168$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
| Index: | $72$ | $\PSL_2$-index: | $36$ | ||||
| Genus: | $2 = 1 + \frac{ 36 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $6^{2}\cdot12^{2}$ | Cusp orbits | $2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12B2 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}19&146\\44&23\end{bmatrix}$, $\begin{bmatrix}61&54\\78&73\end{bmatrix}$, $\begin{bmatrix}77&109\\58&33\end{bmatrix}$, $\begin{bmatrix}91&93\\142&59\end{bmatrix}$, $\begin{bmatrix}145&165\\36&43\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 168.36.2.bb.1 for the level structure with $-I$) |
| Cyclic 168-isogeny field degree: | $128$ |
| Cyclic 168-torsion field degree: | $6144$ |
| Full 168-torsion field degree: | $2064384$ |
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 |
|---|---|---|---|---|---|
| 12.36.1-12.a.1.5 | $12$ | $2$ | $2$ | $1$ | $0$ |
| 168.24.0-168.g.1.3 | $168$ | $3$ | $3$ | $0$ | $?$ |
| 168.36.1-12.a.1.1 | $168$ | $2$ | $2$ | $1$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 168.144.3-168.px.1.7 | $168$ | $2$ | $2$ | $3$ |
| 168.144.3-168.py.1.5 | $168$ | $2$ | $2$ | $3$ |
| 168.144.3-168.qe.1.1 | $168$ | $2$ | $2$ | $3$ |
| 168.144.3-168.qf.1.5 | $168$ | $2$ | $2$ | $3$ |
| 168.144.3-168.rc.1.12 | $168$ | $2$ | $2$ | $3$ |
| 168.144.3-168.rd.1.4 | $168$ | $2$ | $2$ | $3$ |
| 168.144.3-168.rj.1.5 | $168$ | $2$ | $2$ | $3$ |
| 168.144.3-168.rk.1.3 | $168$ | $2$ | $2$ | $3$ |
| 168.144.3-168.se.1.5 | $168$ | $2$ | $2$ | $3$ |
| 168.144.3-168.sf.1.1 | $168$ | $2$ | $2$ | $3$ |
| 168.144.3-168.sl.1.3 | $168$ | $2$ | $2$ | $3$ |
| 168.144.3-168.sm.1.6 | $168$ | $2$ | $2$ | $3$ |
| 168.144.3-168.tg.1.4 | $168$ | $2$ | $2$ | $3$ |
| 168.144.3-168.th.1.2 | $168$ | $2$ | $2$ | $3$ |
| 168.144.3-168.tn.1.7 | $168$ | $2$ | $2$ | $3$ |
| 168.144.3-168.to.1.1 | $168$ | $2$ | $2$ | $3$ |