Invariants
| Level: | $120$ | $\SL_2$-level: | $12$ | ||||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
| Cusps: | $10$ (of which $2$ are rational) | Cusp widths | $2^{4}\cdot4\cdot6^{4}\cdot12$ | Cusp orbits | $1^{2}\cdot2^{4}$ | ||
| 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: | 12I0 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}9&64\\4&111\end{bmatrix}$, $\begin{bmatrix}9&116\\32&45\end{bmatrix}$, $\begin{bmatrix}47&108\\12&41\end{bmatrix}$, $\begin{bmatrix}67&66\\84&109\end{bmatrix}$, $\begin{bmatrix}73&60\\88&107\end{bmatrix}$, $\begin{bmatrix}79&20\\18&101\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.48.0.o.2 for the level structure with $-I$) |
| Cyclic 120-isogeny field degree: | $24$ |
| Cyclic 120-torsion field degree: | $768$ |
| Full 120-torsion field degree: | $368640$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 11 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^2\cdot3^6}\cdot\frac{x^{48}(27x^{4}+4y^{4})^{3}(2187x^{12}+24300x^{8}y^{4}-2160x^{4}y^{8}+64y^{12})^{3}}{y^{4}x^{60}(3x^{2}-2y^{2})^{6}(3x^{2}+2y^{2})^{6}(9x^{2}-2y^{2})^{2}(9x^{2}+2y^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 60.48.0-6.a.1.7 | $60$ | $2$ | $2$ | $0$ | $0$ |
| 120.48.0-6.a.1.7 | $120$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.