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: | $6$ (none of which are rational) | Cusp widths | $2^{4}\cdot8^{2}$ | Cusp orbits | $2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1 \le \gamma \le 2$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8G0 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}27&152\\155&155\end{bmatrix}$, $\begin{bmatrix}33&20\\8&167\end{bmatrix}$, $\begin{bmatrix}139&4\\110&39\end{bmatrix}$, $\begin{bmatrix}151&88\\139&29\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.24.0.bk.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 13 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^6}{3\cdot5^2}\cdot\frac{(4x+y)^{24}(2368x^{4}+2688x^{3}y+1728x^{2}y^{2}+1008xy^{3}+333y^{4})^{3}(4672x^{4}-8448x^{3}y+6912x^{2}y^{2}-3168xy^{3}+657y^{4})^{3}}{(4x+y)^{24}(8x^{2}-3y^{2})^{2}(8x^{2}-36xy+3y^{2})^{8}(24x^{2}-8xy+9y^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 56.24.0-8.m.1.7 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 168.24.0-8.m.1.2 | $168$ | $2$ | $2$ | $0$ | $?$ |
| 168.24.0-12.h.1.1 | $168$ | $2$ | $2$ | $0$ | $?$ |
| 168.24.0-12.h.1.4 | $168$ | $2$ | $2$ | $0$ | $?$ |
| 168.24.0-24.y.1.1 | $168$ | $2$ | $2$ | $0$ | $?$ |
| 168.24.0-24.y.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 |
|---|---|---|---|---|
| 168.144.4-24.fc.1.11 | $168$ | $3$ | $3$ | $4$ |
| 168.192.3-24.fc.1.10 | $168$ | $4$ | $4$ | $3$ |
| 168.384.11-168.hk.1.23 | $168$ | $8$ | $8$ | $11$ |