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: | $\begin{bmatrix}85&100\\5&97\end{bmatrix}$, $\begin{bmatrix}97&96\\126&149\end{bmatrix}$, $\begin{bmatrix}115&60\\24&125\end{bmatrix}$, $\begin{bmatrix}153&136\\74&41\end{bmatrix}$, $\begin{bmatrix}159&140\\149&3\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.24.0.bj.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, including 42 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^3\cdot3}\cdot\frac{x^{24}(81x^{8}+51840x^{6}y^{2}+1234944x^{4}y^{4}+5898240x^{2}y^{6}+1048576y^{8})^{3}}{y^{2}x^{26}(3x^{2}-32y^{2})^{8}(3x^{2}+32y^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 56.24.0-8.n.1.3 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 168.24.0-8.n.1.4 | $168$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 168.96.0-24.bk.1.4 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bk.1.5 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bk.2.1 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bk.2.8 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bl.1.2 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bl.1.7 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bl.2.3 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bl.2.6 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.dg.1.5 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.dg.1.12 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.dg.2.6 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.dg.2.11 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.dh.1.6 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.dh.1.11 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.dh.2.3 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.dh.2.14 | $168$ | $2$ | $2$ | $0$ |
| 168.144.4-24.fb.1.11 | $168$ | $3$ | $3$ | $4$ |
| 168.192.3-24.fb.1.16 | $168$ | $4$ | $4$ | $3$ |
| 168.384.11-168.hj.1.42 | $168$ | $8$ | $8$ | $11$ |