Invariants
| Level: | $168$ | $\SL_2$-level: | $12$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $2^{3}\cdot6^{3}$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| 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: | 6I0 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}56&97\\167&156\end{bmatrix}$, $\begin{bmatrix}73&158\\126&59\end{bmatrix}$, $\begin{bmatrix}75&20\\134&33\end{bmatrix}$, $\begin{bmatrix}139&132\\166&29\end{bmatrix}$, $\begin{bmatrix}148&147\\61&50\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 168.24.0.fk.1 for the level structure with $-I$) |
| Cyclic 168-isogeny field degree: | $32$ |
| Cyclic 168-torsion field degree: | $1536$ |
| Full 168-torsion field degree: | $3096576$ |
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 |
|---|---|---|---|---|---|
| 12.24.0-6.a.1.6 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 168.24.0-6.a.1.11 | $168$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 168.96.1-168.zg.1.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zi.1.6 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zm.1.17 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zo.1.11 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.blj.1.13 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bll.1.12 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.blm.1.7 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.blo.1.3 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.byx.1.11 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.byz.1.9 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bza.1.5 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bzc.1.2 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bzt.1.3 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bzu.1.7 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bzz.1.5 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.caa.1.24 | $168$ | $2$ | $2$ | $1$ |
| 168.144.1-168.cp.1.12 | $168$ | $3$ | $3$ | $1$ |
| 168.384.11-168.sa.1.33 | $168$ | $8$ | $8$ | $11$ |