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: | 12K3 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}3&106\\118&123\end{bmatrix}$, $\begin{bmatrix}67&94\\162&137\end{bmatrix}$, $\begin{bmatrix}85&24\\54&13\end{bmatrix}$, $\begin{bmatrix}133&36\\114&7\end{bmatrix}$, $\begin{bmatrix}135&22\\32&11\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 168.96.3.cw.1 for the level structure with $-I$) |
| Cyclic 168-isogeny field degree: | $32$ |
| Cyclic 168-torsion field degree: | $768$ |
| Full 168-torsion field degree: | $774144$ |
Rational points
This modular curve has no real points, 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 |
|---|---|---|---|---|---|
| $X_0(3)$ | $3$ | $48$ | $24$ | $0$ | $0$ |
| 56.48.0-56.g.1.5 | $56$ | $4$ | $4$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.96.1-24.bx.1.3 | $24$ | $2$ | $2$ | $1$ | $0$ |
| 56.48.0-56.g.1.5 | $56$ | $4$ | $4$ | $0$ | $0$ |
| 84.96.1-84.b.1.3 | $84$ | $2$ | $2$ | $1$ | $?$ |
| 168.96.1-84.b.1.12 | $168$ | $2$ | $2$ | $1$ | $?$ |
| 168.96.1-24.bx.1.18 | $168$ | $2$ | $2$ | $1$ | $?$ |
| 168.96.1-168.dg.1.2 | $168$ | $2$ | $2$ | $1$ | $?$ |
| 168.96.1-168.dg.1.21 | $168$ | $2$ | $2$ | $1$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 168.384.5-168.ja.1.10 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.ja.2.14 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.ja.3.10 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.ja.4.14 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.jd.1.13 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.jd.2.15 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.jd.3.13 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.jd.4.15 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.om.1.10 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.om.2.14 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.om.3.10 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.om.4.14 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.op.1.13 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.op.2.15 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.op.3.13 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.op.4.15 | $168$ | $2$ | $2$ | $5$ |