Invariants
| Level: | $120$ | $\SL_2$-level: | $4$ | ||||
| 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: | $\begin{bmatrix}15&14\\76&83\end{bmatrix}$, $\begin{bmatrix}35&64\\96&37\end{bmatrix}$, $\begin{bmatrix}61&28\\36&65\end{bmatrix}$, $\begin{bmatrix}79&84\\12&113\end{bmatrix}$, $\begin{bmatrix}99&110\\100&111\end{bmatrix}$, $\begin{bmatrix}113&46\\36&101\end{bmatrix}$ |
| 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.1.11 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 60.24.0-4.b.1.2 | $60$ | $2$ | $2$ | $0$ | $0$ |
| 120.24.0-24.a.1.3 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 120.24.0-24.a.1.8 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 120.24.0-24.b.1.2 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 120.24.0-24.b.1.7 | $120$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.