Invariants
| Level: | $168$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $3 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $4$ are rational) | Cusp widths | $6^{4}\cdot12^{4}$ | Cusp orbits | $1^{4}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 3$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 3$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12D3 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}43&150\\126&43\end{bmatrix}$, $\begin{bmatrix}53&122\\66&49\end{bmatrix}$, $\begin{bmatrix}89&90\\156&149\end{bmatrix}$, $\begin{bmatrix}107&107\\72&145\end{bmatrix}$, $\begin{bmatrix}149&137\\102&55\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 168.72.3.diq.1 for the level structure with $-I$) |
| Cyclic 168-isogeny field degree: | $32$ |
| Cyclic 168-torsion field degree: | $1536$ |
| Full 168-torsion field degree: | $1032192$ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.72.0-6.a.1.8 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 42.72.0-6.a.1.1 | $42$ | $2$ | $2$ | $0$ | $0$ |
| 168.48.1-168.hm.1.8 | $168$ | $3$ | $3$ | $1$ | $?$ |
| 168.48.1-168.hm.1.30 | $168$ | $3$ | $3$ | $1$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 168.288.5-168.d.1.6 | $168$ | $2$ | $2$ | $5$ |
| 168.288.5-168.dd.1.3 | $168$ | $2$ | $2$ | $5$ |
| 168.288.5-168.ha.1.3 | $168$ | $2$ | $2$ | $5$ |
| 168.288.5-168.hd.1.3 | $168$ | $2$ | $2$ | $5$ |
| 168.288.5-168.nf.1.5 | $168$ | $2$ | $2$ | $5$ |
| 168.288.5-168.ni.1.3 | $168$ | $2$ | $2$ | $5$ |
| 168.288.5-168.oa.1.4 | $168$ | $2$ | $2$ | $5$ |
| 168.288.5-168.od.1.3 | $168$ | $2$ | $2$ | $5$ |
| 168.288.5-168.bqm.1.3 | $168$ | $2$ | $2$ | $5$ |
| 168.288.5-168.bqo.1.7 | $168$ | $2$ | $2$ | $5$ |
| 168.288.5-168.bra.1.5 | $168$ | $2$ | $2$ | $5$ |
| 168.288.5-168.brc.1.2 | $168$ | $2$ | $2$ | $5$ |
| 168.288.5-168.bsq.1.3 | $168$ | $2$ | $2$ | $5$ |
| 168.288.5-168.bss.1.2 | $168$ | $2$ | $2$ | $5$ |
| 168.288.5-168.bts.1.3 | $168$ | $2$ | $2$ | $5$ |
| 168.288.5-168.bua.1.3 | $168$ | $2$ | $2$ | $5$ |