Invariants
| Level: | $120$ | $\SL_2$-level: | $12$ | ||||
| Index: | $32$ | $\PSL_2$-index: | $16$ | ||||
| Genus: | $0 = 1 + \frac{ 16 }{12} - \frac{ 0 }{4} - \frac{ 4 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $4\cdot12$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $4$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12B0 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}21&100\\46&63\end{bmatrix}$, $\begin{bmatrix}43&119\\75&8\end{bmatrix}$, $\begin{bmatrix}75&67\\119&70\end{bmatrix}$, $\begin{bmatrix}80&51\\87&95\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 120.16.0.a.2 for the level structure with $-I$) |
| Cyclic 120-isogeny field degree: | $72$ |
| Cyclic 120-torsion field degree: | $2304$ |
| Full 120-torsion field degree: | $1105920$ |
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 |
|---|---|---|---|---|---|
| 24.16.0-6.a.1.5 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 60.16.0-6.a.1.4 | $60$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 120.96.2-120.b.2.22 | $120$ | $3$ | $3$ | $2$ |
| 120.96.3-120.a.1.11 | $120$ | $3$ | $3$ | $3$ |
| 120.128.1-120.a.1.14 | $120$ | $4$ | $4$ | $1$ |
| 120.160.4-120.a.1.3 | $120$ | $5$ | $5$ | $4$ |
| 120.192.7-120.c.2.14 | $120$ | $6$ | $6$ | $7$ |
| 120.320.11-120.g.2.9 | $120$ | $10$ | $10$ | $11$ |