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: | $\begin{bmatrix}19&56\\42&97\end{bmatrix}$, $\begin{bmatrix}27&56\\86&143\end{bmatrix}$, $\begin{bmatrix}59&88\\10&37\end{bmatrix}$, $\begin{bmatrix}103&96\\68&155\end{bmatrix}$, $\begin{bmatrix}137&56\\1&87\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.24.0.bf.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 55 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^2\cdot3^2\cdot7}\cdot\frac{x^{24}(2401x^{8}-2963520x^{6}y^{2}+136152576x^{4}y^{4}-1254113280x^{2}y^{6}+429981696y^{8})^{3}}{y^{2}x^{26}(7x^{2}-144y^{2})^{2}(7x^{2}+144y^{2})^{8}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.24.0-8.n.1.1 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 84.24.0-28.g.1.2 | $84$ | $2$ | $2$ | $0$ | $?$ |
| 168.24.0-28.g.1.4 | $168$ | $2$ | $2$ | $0$ | $?$ |
| 168.24.0-8.n.1.1 | $168$ | $2$ | $2$ | $0$ | $?$ |
| 168.24.0-56.ba.1.3 | $168$ | $2$ | $2$ | $0$ | $?$ |
| 168.24.0-56.ba.1.13 | $168$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 168.96.0-56.bg.1.5 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-56.bg.2.5 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-56.bh.1.5 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-56.bh.2.5 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.dr.1.2 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.dr.2.4 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.ds.1.4 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.ds.2.8 | $168$ | $2$ | $2$ | $0$ |
| 168.144.4-168.ir.1.29 | $168$ | $3$ | $3$ | $4$ |
| 168.192.3-168.lp.1.37 | $168$ | $4$ | $4$ | $3$ |
| 168.384.11-56.dv.1.3 | $168$ | $8$ | $8$ | $11$ |