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: | $\begin{bmatrix}26&111\\3&62\end{bmatrix}$, $\begin{bmatrix}43&12\\10&41\end{bmatrix}$, $\begin{bmatrix}52&75\\45&16\end{bmatrix}$, $\begin{bmatrix}60&103\\97&90\end{bmatrix}$, $\begin{bmatrix}71&106\\56&27\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 120.24.0.fo.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.1.9 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 120.24.0-6.a.1.9 | $120$ | $2$ | $2$ | $0$ | $?$ |
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.dg.1.18 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.gm.1.15 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.jw.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.jy.1.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.bac.1.7 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.bae.1.15 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.bai.1.13 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.bak.1.14 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.byx.1.9 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.byy.1.10 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.bzd.1.6 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.bze.1.15 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.bzl.1.5 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.bzn.1.6 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.bzo.1.6 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.bzq.1.22 | $120$ | $2$ | $2$ | $1$ |
| 120.144.1-120.ci.1.3 | $120$ | $3$ | $3$ | $1$ |
| 120.240.8-120.ja.1.24 | $120$ | $5$ | $5$ | $8$ |
| 120.288.7-120.hmn.1.23 | $120$ | $6$ | $6$ | $7$ |
| 120.480.15-120.biw.1.42 | $120$ | $10$ | $10$ | $15$ |