Invariants
| Level: | $168$ | $\SL_2$-level: | $6$ | ||||
| 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 | $2^{3}\cdot6^{3}$ | 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: | 6I0 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}18&19\\29&34\end{bmatrix}$, $\begin{bmatrix}63&32\\94&47\end{bmatrix}$, $\begin{bmatrix}87&148\\82&51\end{bmatrix}$, $\begin{bmatrix}100&51\\129&124\end{bmatrix}$, $\begin{bmatrix}157&60\\114&115\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.24.0.cb.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 63 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2\cdot3^3}\cdot\frac{(x-6y)^{24}(7x^{2}+60xy-36y^{2})^{3}(1457x^{6}-10260x^{5}y-26820x^{4}y^{2}-21600x^{3}y^{3}+572400x^{2}y^{4}+1218240xy^{5}+2078784y^{6})^{3}}{(x-6y)^{26}(x+2y)^{6}(5x^{2}-12xy+84y^{2})^{6}(29x^{2}+84xy+180y^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 42.24.0-6.a.1.4 | $42$ | $2$ | $2$ | $0$ | $0$ |
| 168.16.0-24.d.1.7 | $168$ | $3$ | $3$ | $0$ | $?$ |
| 168.24.0-6.a.1.8 | $168$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 168.96.1-24.ih.1.8 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.ij.1.8 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.in.1.12 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.ip.1.8 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.jc.1.8 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.jd.1.8 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.ji.1.8 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.jj.1.12 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bah.1.8 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bai.1.12 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bak.1.12 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bal.1.8 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.blb.1.12 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.blc.1.14 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.ble.1.14 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.blf.1.14 | $168$ | $2$ | $2$ | $1$ |
| 168.144.1-24.bk.1.3 | $168$ | $3$ | $3$ | $1$ |
| 168.384.11-168.pl.1.16 | $168$ | $8$ | $8$ | $11$ |