Invariants
| Level: | $168$ | $\SL_2$-level: | $12$ | ||||
| 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 | $1^{2}\cdot3^{2}\cdot4\cdot12$ | 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: | 12E0 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}21&62\\146&147\end{bmatrix}$, $\begin{bmatrix}21&158\\8&99\end{bmatrix}$, $\begin{bmatrix}74&31\\105&40\end{bmatrix}$, $\begin{bmatrix}142&113\\93&2\end{bmatrix}$, $\begin{bmatrix}167&48\\140&157\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.24.0.bw.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 133 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^{18}}\cdot\frac{(x+y)^{24}(x^{2}-24y^{2})^{3}(x^{6}-72x^{4}y^{2}+192x^{2}y^{4}-1536y^{6})^{3}}{y^{12}x^{4}(x+y)^{24}(x^{2}-72y^{2})(x^{2}-8y^{2})^{3}}$ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.6.0.c.1 | $8$ | $8$ | $4$ | $0$ | $0$ |
| 21.8.0-3.a.1.2 | $21$ | $6$ | $6$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 42.24.0-6.a.1.4 | $42$ | $2$ | $2$ | $0$ | $0$ |
| 168.24.0-6.a.1.12 | $168$ | $2$ | $2$ | $0$ | $?$ |
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.bw.1.17 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.cu.1.12 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.dp.1.12 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.dr.1.8 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.if.1.8 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.ig.1.8 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.ii.1.8 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.ij.1.8 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.yv.1.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.yw.1.14 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.yy.1.14 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.yz.1.14 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zh.1.15 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zi.1.12 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zk.1.8 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zl.1.12 | $168$ | $2$ | $2$ | $1$ |
| 168.144.1-24.n.1.1 | $168$ | $3$ | $3$ | $1$ |
| 168.384.11-168.jc.1.32 | $168$ | $8$ | $8$ | $11$ |