Invariants
| Level: | $168$ | $\SL_2$-level: | $4$ | ||||
| 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 | $4^{6}$ | 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: | 4G0 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}11&162\\104&49\end{bmatrix}$, $\begin{bmatrix}19&162\\0&79\end{bmatrix}$, $\begin{bmatrix}25&88\\68&51\end{bmatrix}$, $\begin{bmatrix}41&42\\112&11\end{bmatrix}$, $\begin{bmatrix}109&92\\124&163\end{bmatrix}$, $\begin{bmatrix}149&92\\120&163\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 84.24.0.c.1 for the level structure with $-I$) |
| Cyclic 168-isogeny field degree: | $64$ |
| Cyclic 168-torsion field degree: | $3072$ |
| 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 |
|---|---|---|---|---|---|
| 24.24.0-4.b.1.11 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 56.24.0-4.b.1.6 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 168.24.0-84.a.1.3 | $168$ | $2$ | $2$ | $0$ | $?$ |
| 168.24.0-84.a.1.5 | $168$ | $2$ | $2$ | $0$ | $?$ |
| 168.24.0-84.b.1.2 | $168$ | $2$ | $2$ | $0$ | $?$ |
| 168.24.0-84.b.1.6 | $168$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.