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 | $1^{2}\cdot3^{2}\cdot4\cdot12$ | 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: | 12E0 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}72&5\\7&44\end{bmatrix}$, $\begin{bmatrix}104&33\\119&4\end{bmatrix}$, $\begin{bmatrix}136&71\\55&66\end{bmatrix}$, $\begin{bmatrix}145&86\\60&89\end{bmatrix}$, $\begin{bmatrix}156&37\\115&54\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 168.24.0.fi.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.1.9 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 168.24.0-6.a.1.13 | $168$ | $2$ | $2$ | $0$ | $?$ |
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.dg.1.33 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.gm.1.8 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.jw.1.2 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.jy.1.9 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.baa.1.15 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bac.1.14 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bag.1.15 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bai.1.7 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.byv.1.13 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.byw.1.5 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bzb.1.14 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bzc.1.14 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bzj.1.5 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bzl.1.13 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bzm.1.11 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bzo.1.16 | $168$ | $2$ | $2$ | $1$ |
| 168.144.1-168.bo.1.8 | $168$ | $3$ | $3$ | $1$ |
| 168.384.11-168.ry.1.60 | $168$ | $8$ | $8$ | $11$ |