Invariants
| Level: | $168$ | $\SL_2$-level: | $8$ | ||||
| Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $1^{2}\cdot2\cdot8$ | Cusp orbits | $1^{2}\cdot2$ | ||
| 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: | 8C0 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}74&143\\3&50\end{bmatrix}$, $\begin{bmatrix}137&30\\90&41\end{bmatrix}$, $\begin{bmatrix}143&66\\32&37\end{bmatrix}$, $\begin{bmatrix}152&93\\89&64\end{bmatrix}$, $\begin{bmatrix}167&46\\148&9\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.12.0.y.1 for the level structure with $-I$) |
| Cyclic 168-isogeny field degree: | $64$ |
| Cyclic 168-torsion field degree: | $3072$ |
| Full 168-torsion field degree: | $6193152$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 1284 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^{12}\cdot3}\cdot\frac{x^{12}(3x^{2}-24xy+32y^{2})^{3}(3x^{2}+24xy+32y^{2})^{3}}{y^{8}x^{14}(3x^{2}-128y^{2})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 28.12.0-4.c.1.2 | $28$ | $2$ | $2$ | $0$ | $0$ |
| 168.12.0-4.c.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 |
|---|---|---|---|---|
| 168.48.0-24.l.1.9 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-24.n.1.3 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-24.u.1.1 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-24.w.1.3 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-24.bk.1.3 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-24.bn.1.3 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-24.bp.1.5 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-24.bq.1.3 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.cg.1.11 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.ci.1.9 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.ck.1.1 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.cm.1.9 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.de.1.16 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.dg.1.13 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.di.1.12 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.dk.1.3 | $168$ | $2$ | $2$ | $0$ |
| 168.72.2-24.ck.1.26 | $168$ | $3$ | $3$ | $2$ |
| 168.96.1-24.is.1.31 | $168$ | $4$ | $4$ | $1$ |
| 168.192.5-168.fw.1.60 | $168$ | $8$ | $8$ | $5$ |
| 168.504.16-168.cu.1.59 | $168$ | $21$ | $21$ | $16$ |