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: | $\begin{bmatrix}25&128\\42&65\end{bmatrix}$, $\begin{bmatrix}41&148\\58&105\end{bmatrix}$, $\begin{bmatrix}89&32\\82&35\end{bmatrix}$, $\begin{bmatrix}95&124\\72&49\end{bmatrix}$, $\begin{bmatrix}155&8\\8&111\end{bmatrix}$, $\begin{bmatrix}157&140\\60&97\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.24.0.l.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 25 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^4\cdot3^8\cdot7^4}\cdot\frac{(7x+6y)^{24}(2401x^{8}-254016x^{4}y^{4}+26873856y^{8})^{3}}{y^{8}x^{8}(7x+6y)^{24}(7x^{2}-72y^{2})^{2}(7x^{2}+72y^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.24.0-4.b.1.2 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 168.24.0-4.b.1.9 | $168$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.