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: | $\begin{bmatrix}79&144\\78&103\end{bmatrix}$, $\begin{bmatrix}107&150\\106&43\end{bmatrix}$, $\begin{bmatrix}115&66\\30&73\end{bmatrix}$, $\begin{bmatrix}141&14\\44&99\end{bmatrix}$, $\begin{bmatrix}143&56\\124&117\end{bmatrix}$ |
| 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: | $1536$ |
| 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 |
|---|---|---|---|---|---|
| 3.8.0-3.a.1.1 | $3$ | $24$ | $24$ | $0$ | $0$ |
| 56.24.0.g.1 | $56$ | $8$ | $4$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.96.1-24.bx.1.10 | $24$ | $2$ | $2$ | $1$ | $0$ |
| 84.96.1-84.b.1.8 | $84$ | $2$ | $2$ | $1$ | $?$ |
| 168.96.1-84.b.1.6 | $168$ | $2$ | $2$ | $1$ | $?$ |
| 168.96.1-24.bx.1.8 | $168$ | $2$ | $2$ | $1$ | $?$ |
| 168.96.1-168.dg.1.4 | $168$ | $2$ | $2$ | $1$ | $?$ |
| 168.96.1-168.dg.1.35 | $168$ | $2$ | $2$ | $1$ | $?$ |
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.1.1 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.ja.2.1 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.ja.3.11 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.ja.4.11 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.jd.1.6 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.jd.2.6 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.jd.3.13 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.jd.4.13 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.om.1.3 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.om.2.3 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.om.3.9 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.om.4.9 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.op.1.5 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.op.2.5 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.op.3.14 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.op.4.14 | $168$ | $2$ | $2$ | $5$ |