Invariants
| Level: | $120$ | $\SL_2$-level: | $30$ | Newform level: | $1$ | ||
| Index: | $120$ | $\PSL_2$-index: | $120$ | ||||
| Genus: | $9 = 1 + \frac{ 120 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $30^{4}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 9$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 9$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 30A9 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}7&81\\12&103\end{bmatrix}$, $\begin{bmatrix}39&40\\91&57\end{bmatrix}$, $\begin{bmatrix}96&29\\83&93\end{bmatrix}$, $\begin{bmatrix}105&103\\64&45\end{bmatrix}$, $\begin{bmatrix}108&113\\65&66\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 120.240.9-120.dt.1.1, 120.240.9-120.dt.1.2, 120.240.9-120.dt.1.3, 120.240.9-120.dt.1.4, 120.240.9-120.dt.1.5, 120.240.9-120.dt.1.6, 120.240.9-120.dt.1.7, 120.240.9-120.dt.1.8 |
| Cyclic 120-isogeny field degree: | $72$ |
| Cyclic 120-torsion field degree: | $2304$ |
| Full 120-torsion field degree: | $294912$ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points.
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_{\mathrm{sp}}(3)$ | $3$ | $10$ | $10$ | $0$ | $0$ |
| $X_{S_4}(5)$ | $5$ | $24$ | $24$ | $0$ | $0$ |
| 8.2.0.a.1 | $8$ | $60$ | $60$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 15.60.4.a.1 | $15$ | $2$ | $2$ | $4$ | $1$ |
| 24.24.1.bx.1 | $24$ | $5$ | $5$ | $1$ | $0$ |
| 120.40.2.a.1 | $120$ | $3$ | $3$ | $2$ | $?$ |
| 120.60.4.hn.1 | $120$ | $2$ | $2$ | $4$ | $?$ |
| 120.60.5.n.1 | $120$ | $2$ | $2$ | $5$ | $?$ |