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: | $\begin{bmatrix}13&14\\124&93\end{bmatrix}$, $\begin{bmatrix}27&142\\148&87\end{bmatrix}$, $\begin{bmatrix}54&55\\143&100\end{bmatrix}$, $\begin{bmatrix}97&116\\54&167\end{bmatrix}$, $\begin{bmatrix}107&76\\114&61\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.24.0.ca.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 80 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2\cdot3^3}\cdot\frac{(x+6y)^{24}(11x^{2}-12xy+108y^{2})^{3}(1259x^{6}+1116x^{5}y-61740x^{4}y^{2}-38880x^{3}y^{3}+1077840x^{2}y^{4}-627264xy^{5}+222912y^{6})^{3}}{(x-2y)^{6}(x+6y)^{26}(x^{2}-36xy-60y^{2})^{6}(25x^{2}-132xy+36y^{2})^{2}}$ |
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.c.1.6 | $168$ | $3$ | $3$ | $0$ | $?$ |
| 168.24.0-6.a.1.1 | $168$ | $2$ | $2$ | $0$ | $?$ |
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.ie.1.8 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.ig.1.8 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.ik.1.12 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.im.1.8 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.iz.1.8 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.ja.1.8 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.jf.1.8 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.jg.1.12 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bab.1.8 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bac.1.8 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bae.1.12 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.baf.1.12 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bkv.1.14 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bkw.1.14 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bky.1.12 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bkz.1.14 | $168$ | $2$ | $2$ | $1$ |
| 168.144.1-24.bn.1.3 | $168$ | $3$ | $3$ | $1$ |
| 168.384.11-168.pk.1.16 | $168$ | $8$ | $8$ | $11$ |