Invariants
| Level: | $168$ | $\SL_2$-level: | $4$ | ||||
| Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $2^{2}\cdot4^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 4E0 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}15&32\\85&139\end{bmatrix}$, $\begin{bmatrix}131&4\\112&159\end{bmatrix}$, $\begin{bmatrix}145&28\\139&15\end{bmatrix}$, $\begin{bmatrix}149&40\\105&103\end{bmatrix}$, $\begin{bmatrix}157&132\\70&19\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 168.12.0.v.1 for the level structure with $-I$) |
| Cyclic 168-isogeny field degree: | $64$ |
| Cyclic 168-torsion field degree: | $3072$ |
| Full 168-torsion field degree: | $6193152$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points but none with conductor small enough to be contained within the database of elliptic curves over $\Q$.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.12.0-4.c.1.1 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 56.12.0-4.c.1.2 | $56$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 168.48.0-168.dc.1.5 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.dc.1.12 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.dd.1.2 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.dd.1.24 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.dg.1.1 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.dg.1.16 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.dh.1.3 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.dh.1.14 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.dw.1.8 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.dw.1.9 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.dx.1.1 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.dx.1.16 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.ea.1.1 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.ea.1.16 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.eb.1.5 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.eb.1.12 | $168$ | $2$ | $2$ | $0$ |
| 168.72.2-168.cr.1.15 | $168$ | $3$ | $3$ | $2$ |
| 168.96.1-168.zp.1.9 | $168$ | $4$ | $4$ | $1$ |
| 168.192.5-168.ft.1.5 | $168$ | $8$ | $8$ | $5$ |
| 168.504.16-168.cr.1.10 | $168$ | $21$ | $21$ | $16$ |