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 | $2^{2}\cdot4^{3}\cdot8$ | 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: | 8J0 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}45&4\\136&117\end{bmatrix}$, $\begin{bmatrix}81&128\\124&21\end{bmatrix}$, $\begin{bmatrix}113&4\\44&81\end{bmatrix}$, $\begin{bmatrix}129&100\\98&81\end{bmatrix}$, $\begin{bmatrix}161&16\\162&71\end{bmatrix}$, $\begin{bmatrix}163&92\\68&63\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.24.0.i.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, including 90 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^4}{3^2}\cdot\frac{x^{24}(1296x^{8}+864x^{6}y^{2}+180x^{4}y^{4}+12x^{2}y^{6}+y^{8})^{3}}{y^{8}x^{28}(3x^{2}+y^{2})^{2}(6x^{2}+y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 56.24.0-4.b.1.5 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 168.24.0-4.b.1.7 | $168$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.