Invariants
| Level: | $168$ | $\SL_2$-level: | $8$ | ||||
| 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 | $2^{4}\cdot8^{2}$ | 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: | 8G0 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}11&72\\70&115\end{bmatrix}$, $\begin{bmatrix}27&128\\37&81\end{bmatrix}$, $\begin{bmatrix}29&112\\26&141\end{bmatrix}$, $\begin{bmatrix}53&8\\42&137\end{bmatrix}$, $\begin{bmatrix}57&16\\44&47\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 168.24.0.cx.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 |
|---|---|---|---|---|---|
| 24.24.0-8.n.1.5 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 56.24.0-8.n.1.6 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 84.24.0-84.g.1.3 | $84$ | $2$ | $2$ | $0$ | $?$ |
| 168.24.0-84.g.1.10 | $168$ | $2$ | $2$ | $0$ | $?$ |
| 168.24.0-168.ba.1.18 | $168$ | $2$ | $2$ | $0$ | $?$ |
| 168.24.0-168.ba.1.21 | $168$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 168.96.0-168.dz.1.4 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.dz.2.8 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.ea.1.4 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.ea.2.8 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.eb.1.4 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.eb.2.8 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.ec.1.4 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.ec.2.8 | $168$ | $2$ | $2$ | $0$ |
| 168.144.4-168.ld.1.27 | $168$ | $3$ | $3$ | $4$ |
| 168.192.3-168.mv.1.27 | $168$ | $4$ | $4$ | $3$ |
| 168.384.11-168.lb.1.4 | $168$ | $8$ | $8$ | $11$ |