Invariants
| Level: | $120$ | $\SL_2$-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 $2$ are rational) | Cusp widths | $4^{6}$ | 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: | 4G0 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}3&20\\8&71\end{bmatrix}$, $\begin{bmatrix}5&48\\8&97\end{bmatrix}$, $\begin{bmatrix}23&10\\32&47\end{bmatrix}$, $\begin{bmatrix}73&112\\28&11\end{bmatrix}$, $\begin{bmatrix}81&98\\88&55\end{bmatrix}$, $\begin{bmatrix}93&52\\80&69\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.24.0.e.1 for the level structure with $-I$) |
| Cyclic 120-isogeny field degree: | $48$ |
| Cyclic 120-torsion field degree: | $1536$ |
| Full 120-torsion field degree: | $737280$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 42 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^6\cdot3^2}\cdot\frac{x^{24}(81x^{8}+129024x^{4}y^{4}+1048576y^{8})^{3}}{y^{4}x^{28}(3x^{2}-32y^{2})^{4}(3x^{2}+32y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 40.24.0-4.b.1.7 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 120.24.0-4.b.1.3 | $120$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.