Invariants
| Level: | $330$ | $\SL_2$-level: | $30$ | Newform level: | $1$ | ||
| Index: | $144$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $9 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $6^{4}\cdot30^{4}$ | Cusp orbits | $2^{2}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 16$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 9$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 30K9 |
Level structure
| $\GL_2(\Z/330\Z)$-generators: | $\begin{bmatrix}106&265\\133&263\end{bmatrix}$, $\begin{bmatrix}187&241\\160&53\end{bmatrix}$, $\begin{bmatrix}319&290\\31&103\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 330-isogeny field degree: | $144$ |
| Cyclic 330-torsion field degree: | $11520$ |
| Full 330-torsion field degree: | $12672000$ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 30.72.1.o.1 | $30$ | $2$ | $2$ | $1$ | $1$ |
| 165.72.3.w.2 | $165$ | $2$ | $2$ | $3$ | $?$ |
| 330.72.3.w.1 | $330$ | $2$ | $2$ | $3$ | $?$ |
| 330.72.5.df.1 | $330$ | $2$ | $2$ | $5$ | $?$ |
| 330.72.5.dp.1 | $330$ | $2$ | $2$ | $5$ | $?$ |
| 330.72.5.eg.1 | $330$ | $2$ | $2$ | $5$ | $?$ |
| 330.72.5.el.1 | $330$ | $2$ | $2$ | $5$ | $?$ |