Invariants
| Level: | $330$ | $\SL_2$-level: | $30$ | Newform level: | $1$ | ||
| Index: | $432$ | $\PSL_2$-index: | $216$ | ||||
| Genus: | $13 = 1 + \frac{ 216 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (of which $2$ are rational) | Cusp widths | $6^{6}\cdot30^{6}$ | Cusp orbits | $1^{2}\cdot2^{3}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 13$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 13$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 30H13 |
Level structure
| $\GL_2(\Z/330\Z)$-generators: | $\begin{bmatrix}31&288\\110&203\end{bmatrix}$, $\begin{bmatrix}121&300\\90&217\end{bmatrix}$, $\begin{bmatrix}141&242\\10&93\end{bmatrix}$, $\begin{bmatrix}194&265\\165&196\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 330.216.13.bg.2 for the level structure with $-I$) |
| Cyclic 330-isogeny field degree: | $48$ |
| Cyclic 330-torsion field degree: | $1920$ |
| Full 330-torsion field degree: | $4224000$ |
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 |
|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(3)$ | $3$ | $144$ | $72$ | $0$ | $0$ |
| 110.144.1-110.c.1.2 | $110$ | $3$ | $3$ | $1$ | $?$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 30.216.6-30.a.2.16 | $30$ | $2$ | $2$ | $6$ | $0$ |
| 110.144.1-110.c.1.2 | $110$ | $3$ | $3$ | $1$ | $?$ |
| 330.144.5-330.bi.2.5 | $330$ | $3$ | $3$ | $5$ | $?$ |
| 330.216.6-30.a.2.1 | $330$ | $2$ | $2$ | $6$ | $?$ |