Invariants
| Level: | $120$ | $\SL_2$-level: | $12$ | ||||
| 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 | $1^{2}\cdot3^{2}\cdot4\cdot12$ | 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: | 12E0 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}4&51\\55&26\end{bmatrix}$, $\begin{bmatrix}18&77\\23&0\end{bmatrix}$, $\begin{bmatrix}24&47\\23&102\end{bmatrix}$, $\begin{bmatrix}35&58\\84&97\end{bmatrix}$, $\begin{bmatrix}86&111\\9&8\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 120.24.0.ft.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 but none with conductor small enough to be contained within the database of elliptic curves over $\Q$.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.24.0-6.a.1.6 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 120.24.0-6.a.1.5 | $120$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 120.96.1-120.dj.1.34 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.gj.1.10 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.kc.1.13 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.ke.1.8 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.bkw.1.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.bky.1.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.blf.1.3 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.blh.1.7 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.byk.1.3 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.bym.1.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.byt.1.3 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.byv.1.4 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.bzu.1.3 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.bzw.1.3 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.cad.1.11 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.caf.1.24 | $120$ | $2$ | $2$ | $1$ |
| 120.144.1-120.ds.1.10 | $120$ | $3$ | $3$ | $1$ |
| 120.240.8-120.jh.1.17 | $120$ | $5$ | $5$ | $8$ |
| 120.288.7-120.hms.1.42 | $120$ | $6$ | $6$ | $7$ |
| 120.480.15-120.bjb.1.19 | $120$ | $10$ | $10$ | $15$ |