Invariants
| Level: | $168$ | $\SL_2$-level: | $24$ | ||||
| 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 $2$ are rational) | Cusp widths | $1^{2}\cdot3^{2}\cdot4\cdot12$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| 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: | 12E0 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}24&157\\79&126\end{bmatrix}$, $\begin{bmatrix}46&165\\163&56\end{bmatrix}$, $\begin{bmatrix}94&167\\75&92\end{bmatrix}$, $\begin{bmatrix}139&36\\64&29\end{bmatrix}$, $\begin{bmatrix}149&136\\164&69\end{bmatrix}$, $\begin{bmatrix}153&140\\64&101\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 12.24.0.f.1 for the level structure with $-I$) |
| Cyclic 168-isogeny field degree: | $32$ |
| Cyclic 168-torsion field degree: | $1536$ |
| 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 84 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^4}\cdot\frac{x^{24}(3x^{2}+4y^{2})^{3}(3x^{6}+12x^{4}y^{2}+144x^{2}y^{4}+64y^{6})^{3}}{y^{4}x^{36}(x^{2}+4y^{2})^{3}(9x^{2}+4y^{2})}$ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_0(3)$ | $3$ | $12$ | $6$ | $0$ | $0$ |
| 56.12.0-4.b.1.3 | $56$ | $4$ | $4$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 56.12.0-4.b.1.3 | $56$ | $4$ | $4$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.