Invariants
| Level: | $120$ | $\SL_2$-level: | $8$ | ||||
| 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 | $2^{4}\cdot4^{2}\cdot8^{4}$ | Cusp orbits | $1^{2}\cdot2^{2}\cdot4$ | ||
| 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: | 8O0 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}67&72\\113&89\end{bmatrix}$, $\begin{bmatrix}67&112\\48&71\end{bmatrix}$, $\begin{bmatrix}71&72\\52&83\end{bmatrix}$, $\begin{bmatrix}79&80\\45&49\end{bmatrix}$, $\begin{bmatrix}113&8\\25&3\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.48.0.be.1 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 7 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^6\cdot3^2}\cdot\frac{x^{48}(81x^{8}-3456x^{6}y^{2}+4608x^{4}y^{4}+98304x^{2}y^{6}+65536y^{8})^{3}(81x^{8}+3456x^{6}y^{2}+4608x^{4}y^{4}-98304x^{2}y^{6}+65536y^{8})^{3}}{y^{4}x^{52}(3x^{2}-16y^{2})^{2}(3x^{2}+16y^{2})^{2}(9x^{4}+256y^{4})^{8}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 40.48.0-8.q.1.4 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 120.48.0-8.q.1.4 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 120.48.0-24.by.1.1 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 120.48.0-24.by.1.14 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 120.48.0-24.bz.2.1 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 120.48.0-24.bz.2.15 | $120$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.