Invariants
| Level: | $168$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $2^{2}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ | Cusp orbits | $2^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 48$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12P1 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}29&94\\74&15\end{bmatrix}$, $\begin{bmatrix}119&39\\2&121\end{bmatrix}$, $\begin{bmatrix}135&113\\64&149\end{bmatrix}$, $\begin{bmatrix}135&166\\124&39\end{bmatrix}$, $\begin{bmatrix}141&13\\154&87\end{bmatrix}$, $\begin{bmatrix}145&141\\102&7\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 168.48.1.bzo.1 for the level structure with $-I$) |
| Cyclic 168-isogeny field degree: | $32$ |
| Cyclic 168-torsion field degree: | $1536$ |
| Full 168-torsion field degree: | $1548288$ |
Jacobian
| Conductor: | $?$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | not computed |
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
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 3.8.0-3.a.1.1 | $3$ | $12$ | $12$ | $0$ | $0$ | full Jacobian |
| 56.12.0.bk.1 | $56$ | $8$ | $4$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.48.1-12.l.1.10 | $12$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 168.48.0-168.fg.1.9 | $168$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 168.48.0-168.fg.1.22 | $168$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 168.48.0-168.fi.1.2 | $168$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 168.48.0-168.fi.1.13 | $168$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 168.48.1-12.l.1.5 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 168.192.3-168.tc.1.32 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.td.1.12 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.tk.1.32 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.tl.1.28 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.tu.1.29 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.tv.1.25 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.tw.1.31 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.tx.1.29 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.ty.1.26 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.tz.1.18 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.ua.1.30 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.ub.1.26 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.ue.1.28 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.uf.1.32 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.ui.1.28 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.uj.1.16 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.5-168.ct.1.32 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.192.5-168.cv.1.32 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.192.5-168.iz.1.32 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.192.5-168.jb.1.32 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.192.5-168.tj.1.32 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.192.5-168.tl.1.32 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.192.5-168.up.1.32 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.192.5-168.ur.1.32 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-168.bpo.1.1 | $168$ | $3$ | $3$ | $5$ | $?$ | not computed |