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: | $8$ (none of which are rational) | Cusp widths | $6^{4}\cdot12^{4}$ | Cusp orbits | $2^{2}\cdot4$ | ||
| 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: | 12G3 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}1&70\\136&53\end{bmatrix}$, $\begin{bmatrix}13&56\\56&65\end{bmatrix}$, $\begin{bmatrix}61&144\\158&131\end{bmatrix}$, $\begin{bmatrix}107&128\\26&51\end{bmatrix}$, $\begin{bmatrix}127&56\\8&89\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 168.72.3.bq.1 for the level structure with $-I$) |
| Cyclic 168-isogeny field degree: | $128$ |
| Cyclic 168-torsion field degree: | $3072$ |
| Full 168-torsion field degree: | $1032192$ |
Rational points
This modular curve has no real points and no $\Q_p$ points for $p=19$, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.72.2-12.d.1.3 | $12$ | $2$ | $2$ | $2$ | $0$ |
| 168.72.1-168.a.1.1 | $168$ | $2$ | $2$ | $1$ | $?$ |
| 168.72.1-168.a.1.7 | $168$ | $2$ | $2$ | $1$ | $?$ |
| 168.72.2-12.d.1.5 | $168$ | $2$ | $2$ | $2$ | $?$ |
| 168.72.2-168.e.1.6 | $168$ | $2$ | $2$ | $2$ | $?$ |
| 168.72.2-168.e.1.21 | $168$ | $2$ | $2$ | $2$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 168.288.7-168.bq.1.1 | $168$ | $2$ | $2$ | $7$ |
| 168.288.7-168.bq.1.8 | $168$ | $2$ | $2$ | $7$ |
| 168.288.7-168.bu.1.7 | $168$ | $2$ | $2$ | $7$ |
| 168.288.7-168.bu.1.10 | $168$ | $2$ | $2$ | $7$ |
| 168.288.7-168.bv.1.1 | $168$ | $2$ | $2$ | $7$ |
| 168.288.7-168.bv.1.24 | $168$ | $2$ | $2$ | $7$ |
| 168.288.7-168.cb.1.2 | $168$ | $2$ | $2$ | $7$ |
| 168.288.7-168.cb.1.15 | $168$ | $2$ | $2$ | $7$ |
| 168.288.7-168.sm.1.2 | $168$ | $2$ | $2$ | $7$ |
| 168.288.7-168.sm.1.15 | $168$ | $2$ | $2$ | $7$ |
| 168.288.7-168.sq.1.4 | $168$ | $2$ | $2$ | $7$ |
| 168.288.7-168.sq.1.13 | $168$ | $2$ | $2$ | $7$ |
| 168.288.7-168.st.1.8 | $168$ | $2$ | $2$ | $7$ |
| 168.288.7-168.st.1.9 | $168$ | $2$ | $2$ | $7$ |
| 168.288.7-168.sx.1.11 | $168$ | $2$ | $2$ | $7$ |
| 168.288.7-168.sx.1.14 | $168$ | $2$ | $2$ | $7$ |