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 | $2^{4}\cdot8^{2}$ | 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: | 8G0 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}1&40\\5&13\end{bmatrix}$, $\begin{bmatrix}19&88\\111&1\end{bmatrix}$, $\begin{bmatrix}23&96\\103&95\end{bmatrix}$, $\begin{bmatrix}31&72\\17&67\end{bmatrix}$, $\begin{bmatrix}97&112\\45&107\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.24.0.bn.1 for the level structure with $-I$) |
| Cyclic 120-isogeny field degree: | $24$ |
| Cyclic 120-torsion field degree: | $768$ |
| 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 44 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{2^2}{5}\cdot\frac{(5x+2y)^{24}(2560000x^{8}+5120000x^{7}y+2560000x^{6}y^{2}-640000x^{5}y^{3}-1046400x^{4}y^{4}-406400x^{3}y^{5}-75200x^{2}y^{6}-6800xy^{7}-239y^{8})^{3}}{y^{2}(4x+y)^{2}(5x+2y)^{24}(20x^{2}+10xy+y^{2})^{2}(40x^{2}+20xy+3y^{2})^{8}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.24.0-8.n.1.5 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 120.24.0-8.n.1.5 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 60.24.0-20.h.1.1 | $60$ | $2$ | $2$ | $0$ | $0$ |
| 120.24.0-20.h.1.5 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 120.24.0-40.z.1.8 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 120.24.0-40.z.1.13 | $120$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.