Invariants
| Level: | $168$ | $\SL_2$-level: | $8$ | ||||
| Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $2^{2}\cdot4^{2}$ | Cusp orbits | $2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1 \le \gamma \le 2$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 4E0 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}37&18\\138&119\end{bmatrix}$, $\begin{bmatrix}85&14\\97&153\end{bmatrix}$, $\begin{bmatrix}125&86\\41&13\end{bmatrix}$, $\begin{bmatrix}143&152\\44&79\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 12.12.0.f.1 for the level structure with $-I$) |
| Cyclic 168-isogeny field degree: | $128$ |
| Cyclic 168-torsion field degree: | $6144$ |
| Full 168-torsion field degree: | $6193152$ |
Models
Smooth plane model Smooth plane model
| $ 0 $ | $=$ | $ 4 x^{2} - 192 y^{2} - 3 z^{2} $ |
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
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 56.12.0-4.b.1.3 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 168.12.0-4.b.1.1 | $168$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 168.72.2-12.r.1.5 | $168$ | $3$ | $3$ | $2$ |
| 168.96.1-12.j.1.7 | $168$ | $4$ | $4$ | $1$ |
| 168.192.5-84.j.1.9 | $168$ | $8$ | $8$ | $5$ |
| 168.504.16-84.r.1.7 | $168$ | $21$ | $21$ | $16$ |