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 $4$ are rational) | Cusp widths | $2^{4}\cdot4\cdot6^{4}\cdot12$ | Cusp orbits | $1^{4}\cdot2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12I0 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}11&12\\68&109\end{bmatrix}$, $\begin{bmatrix}11&82\\98&27\end{bmatrix}$, $\begin{bmatrix}23&60\\102&119\end{bmatrix}$, $\begin{bmatrix}43&116\\28&27\end{bmatrix}$, $\begin{bmatrix}59&10\\62&57\end{bmatrix}$, $\begin{bmatrix}71&46\\0&97\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 12.48.0.a.1 for the level structure with $-I$) |
| Cyclic 120-isogeny field degree: | $24$ |
| Cyclic 120-torsion field degree: | $384$ |
| 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 17 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{(x-y)^{48}(x^{4}-2x^{3}y+6x^{2}y^{2}-2xy^{3}+y^{4})^{3}(x^{12}-6x^{11}y+6x^{10}y^{2}+10x^{9}y^{3}+15x^{8}y^{4}-36x^{7}y^{5}+84x^{6}y^{6}-36x^{5}y^{7}+15x^{4}y^{8}+10x^{3}y^{9}+6x^{2}y^{10}-6xy^{11}+y^{12})^{3}}{y^{6}x^{6}(x-y)^{60}(x+y)^{4}(x^{2}+y^{2})^{6}(x^{2}-4xy+y^{2})^{2}(x^{2}-xy+y^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 120.48.0-6.a.1.7 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 120.48.0-6.a.1.10 | $120$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.