Invariants
| Level: | $168$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
| Index: | $144$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $5 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (none of which are rational) | Cusp widths | $6^{8}\cdot12^{8}$ | Cusp orbits | $2^{4}\cdot4^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 5$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12B5 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}71&78\\165&107\end{bmatrix}$, $\begin{bmatrix}95&82\\111&73\end{bmatrix}$, $\begin{bmatrix}113&22\\54&121\end{bmatrix}$, $\begin{bmatrix}121&96\\141&67\end{bmatrix}$, $\begin{bmatrix}151&38\\15&95\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 168.288.5-168.bpk.1.1, 168.288.5-168.bpk.1.2, 168.288.5-168.bpk.1.3, 168.288.5-168.bpk.1.4, 168.288.5-168.bpk.1.5, 168.288.5-168.bpk.1.6, 168.288.5-168.bpk.1.7, 168.288.5-168.bpk.1.8 |
| Cyclic 168-isogeny field degree: | $32$ |
| Cyclic 168-torsion field degree: | $1536$ |
| Full 168-torsion field degree: | $1032192$ |
Rational points
This modular curve has no real points and no $\Q_p$ points for $p=31$, and therefore no rational points.
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_{\mathrm{sp}}(3)$ | $3$ | $12$ | $12$ | $0$ | $0$ |
| 56.12.0.bi.1 | $56$ | $12$ | $12$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.72.3.ug.1 | $24$ | $2$ | $2$ | $3$ | $0$ |
| 42.72.1.d.1 | $42$ | $2$ | $2$ | $1$ | $1$ |
| 168.48.1.bzm.1 | $168$ | $3$ | $3$ | $1$ | $?$ |
| 168.72.1.bo.1 | $168$ | $2$ | $2$ | $1$ | $?$ |
| 168.72.1.ik.1 | $168$ | $2$ | $2$ | $1$ | $?$ |
| 168.72.3.cvd.1 | $168$ | $2$ | $2$ | $3$ | $?$ |
| 168.72.3.cvq.1 | $168$ | $2$ | $2$ | $3$ | $?$ |
| 168.72.3.ekq.1 | $168$ | $2$ | $2$ | $3$ | $?$ |