Invariants
| Level: | $168$ | $\SL_2$-level: | $8$ | ||||
| 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 $4$ are rational) | Cusp widths | $4^{6}$ | Cusp orbits | $1^{4}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 4G0 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}9&88\\164&65\end{bmatrix}$, $\begin{bmatrix}11&140\\28&141\end{bmatrix}$, $\begin{bmatrix}13&72\\24&157\end{bmatrix}$, $\begin{bmatrix}25&48\\112&127\end{bmatrix}$, $\begin{bmatrix}39&136\\100&1\end{bmatrix}$, $\begin{bmatrix}163&8\\120&13\end{bmatrix}$, $\begin{bmatrix}163&112\\124&47\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 4.24.0.b.1 for the level structure with $-I$) |
| Cyclic 168-isogeny field degree: | $64$ |
| Cyclic 168-torsion field degree: | $3072$ |
| 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 61 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{24}(x^{4}-4x^{3}y+8x^{2}y^{2}+16xy^{3}+16y^{4})^{3}(x^{4}+4x^{3}y+8x^{2}y^{2}-16xy^{3}+16y^{4})^{3}}{y^{4}x^{28}(x-2y)^{4}(x+2y)^{4}(x^{2}+4y^{2})^{4}}$ |
Modular covers
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 168.96.0-8.a.1.7 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-8.a.1.8 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.a.1.16 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.a.1.19 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-56.a.1.16 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-56.a.1.19 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.a.1.32 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.a.1.37 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-8.b.1.4 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-8.b.1.8 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-8.b.2.7 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-8.b.2.8 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.b.1.22 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.b.1.23 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.b.2.18 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.b.2.23 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-56.b.1.22 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-56.b.1.23 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-56.b.2.18 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-56.b.2.23 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.b.1.30 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.b.1.39 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.b.2.29 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.b.2.38 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-8.c.1.4 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-8.c.1.8 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.c.1.16 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.c.1.19 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-56.c.1.16 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-56.c.1.19 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.c.1.32 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.c.1.38 | $168$ | $2$ | $2$ | $0$ |
| 168.96.1-8.g.1.6 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-8.g.1.8 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-8.g.2.5 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-8.g.2.7 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-8.h.1.4 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-8.h.1.8 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-8.h.2.6 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-8.h.2.7 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.n.1.8 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.n.1.12 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.n.2.19 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.n.2.21 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-56.n.1.9 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-56.n.1.11 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-56.n.2.18 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-56.n.2.24 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.n.1.16 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.n.1.23 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.n.2.36 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.n.2.38 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.o.1.8 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.o.1.11 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.o.2.17 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.o.2.24 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-56.o.1.8 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-56.o.1.11 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-56.o.2.18 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-56.o.2.24 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.o.1.14 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.o.1.24 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.o.2.34 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.o.2.48 | $168$ | $2$ | $2$ | $1$ |
| 168.96.2-8.a.1.2 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-8.a.1.6 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-8.a.1.10 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-8.a.1.12 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-24.a.1.2 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-24.a.1.12 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-24.a.1.18 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-24.a.1.21 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-56.a.1.2 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-56.a.1.12 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-56.a.1.18 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-56.a.1.21 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-168.a.1.2 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-168.a.1.30 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-168.a.1.37 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-168.a.1.43 | $168$ | $2$ | $2$ | $2$ |
| 168.144.4-12.b.1.38 | $168$ | $3$ | $3$ | $4$ |
| 168.192.3-12.b.1.41 | $168$ | $4$ | $4$ | $3$ |
| 168.384.11-28.b.1.32 | $168$ | $8$ | $8$ | $11$ |