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: | $\begin{bmatrix}64&11\\67&112\end{bmatrix}$, $\begin{bmatrix}81&4\\52&69\end{bmatrix}$, $\begin{bmatrix}95&106\\92&129\end{bmatrix}$, $\begin{bmatrix}146&47\\121&48\end{bmatrix}$, $\begin{bmatrix}165&50\\40&107\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.12.0.ba.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 1542 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^8\cdot3}\cdot\frac{(3x+y)^{12}(9x^{4}-192x^{2}y^{2}+256y^{4})^{3}}{y^{8}x^{2}(3x+y)^{12}(3x^{2}-64y^{2})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 56.12.0-4.c.1.3 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 84.12.0-4.c.1.2 | $84$ | $2$ | $2$ | $0$ | $?$ |
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.m.1.4 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-24.n.1.4 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-24.bc.1.2 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-24.be.1.6 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-24.bh.1.9 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-24.bi.1.6 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-24.bs.1.8 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-24.bv.1.7 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.co.1.1 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.cq.1.5 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.cs.1.1 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.cu.1.5 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.dm.1.9 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.do.1.6 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.du.1.7 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.dw.1.6 | $168$ | $2$ | $2$ | $0$ |
| 168.72.2-24.cu.1.16 | $168$ | $3$ | $3$ | $2$ |
| 168.96.1-24.iu.1.3 | $168$ | $4$ | $4$ | $1$ |
| 168.192.5-168.gc.1.51 | $168$ | $8$ | $8$ | $5$ |
| 168.504.16-168.di.1.58 | $168$ | $21$ | $21$ | $16$ |