Invariants
| Level: | $168$ | $\SL_2$-level: | $4$ | ||||
| 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 | $2^{2}\cdot4^{2}$ | 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: | 4E0 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}37&100\\102&155\end{bmatrix}$, $\begin{bmatrix}67&86\\76&37\end{bmatrix}$, $\begin{bmatrix}91&30\\162&35\end{bmatrix}$, $\begin{bmatrix}121&6\\120&1\end{bmatrix}$, $\begin{bmatrix}155&166\\122&77\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 28.12.0.b.1 for the level structure with $-I$) |
| Cyclic 168-isogeny field degree: | $128$ |
| Cyclic 168-torsion field degree: | $6144$ |
| 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 631 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^4\cdot3^4\cdot7^2}\cdot\frac{x^{12}(49x^{4}+4032x^{2}y^{2}+331776y^{4})^{3}}{y^{4}x^{16}(7x^{2}+576y^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.12.0-2.a.1.1 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 168.12.0-2.a.1.2 | $168$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 168.48.0-28.b.1.2 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-28.c.1.2 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-28.c.1.8 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-56.d.1.1 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-56.d.1.3 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-84.e.1.5 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-84.e.1.8 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-56.g.1.1 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-56.g.1.2 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-84.g.1.5 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-84.g.1.8 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.l.1.2 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.l.1.7 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.r.1.1 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.r.1.8 | $168$ | $2$ | $2$ | $0$ |
| 168.72.2-84.b.1.9 | $168$ | $3$ | $3$ | $2$ |
| 168.96.1-84.b.1.8 | $168$ | $4$ | $4$ | $1$ |
| 168.192.5-28.d.1.10 | $168$ | $8$ | $8$ | $5$ |
| 168.504.16-28.d.1.3 | $168$ | $21$ | $21$ | $16$ |