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: | $\begin{bmatrix}25&82\\102&157\end{bmatrix}$, $\begin{bmatrix}27&14\\104&37\end{bmatrix}$, $\begin{bmatrix}82&17\\165&122\end{bmatrix}$, $\begin{bmatrix}144&107\\103&148\end{bmatrix}$, $\begin{bmatrix}149&112\\30&107\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.12.0.m.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 1248 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^4\,\frac{x^{12}(4x^{4}+8x^{2}y^{2}+y^{4})^{3}}{y^{2}x^{20}(8x^{2}+y^{2})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 84.12.0-4.c.1.2 | $84$ | $2$ | $2$ | $0$ | $?$ |
| 168.12.0-4.c.1.6 | $168$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 168.48.0-8.h.1.9 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-8.j.1.3 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-8.p.1.1 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-8.r.1.2 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-56.be.1.3 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-24.bg.1.3 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-56.bg.1.5 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-24.bi.1.3 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-56.bi.1.1 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-24.bk.1.1 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-56.bk.1.5 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-24.bm.1.3 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.cw.1.3 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.cy.1.14 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.da.1.1 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.dc.1.13 | $168$ | $2$ | $2$ | $0$ |
| 168.72.2-24.ci.1.16 | $168$ | $3$ | $3$ | $2$ |
| 168.96.1-24.iq.1.2 | $168$ | $4$ | $4$ | $1$ |
| 168.192.5-56.bk.1.7 | $168$ | $8$ | $8$ | $5$ |
| 168.504.16-56.ci.1.18 | $168$ | $21$ | $21$ | $16$ |