Invariants
| Level: | $330$ | $\SL_2$-level: | $6$ | ||||
| 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 | $2^{3}\cdot6^{3}$ | 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: | 6I0 |
Level structure
| $\GL_2(\Z/330\Z)$-generators: | $\begin{bmatrix}107&64\\218&165\end{bmatrix}$, $\begin{bmatrix}130&129\\91&134\end{bmatrix}$, $\begin{bmatrix}157&188\\252&215\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 330.24.0.b.1 for the level structure with $-I$) |
| Cyclic 330-isogeny field degree: | $72$ |
| Cyclic 330-torsion field degree: | $5760$ |
| Full 330-torsion field degree: | $38016000$ |
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 |
|---|---|---|---|---|---|
| 6.24.0-6.a.1.3 | $6$ | $2$ | $2$ | $0$ | $0$ |
| 330.16.0-330.b.1.5 | $330$ | $3$ | $3$ | $0$ | $?$ |
| 330.24.0-6.a.1.3 | $330$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 330.144.1-330.e.1.4 | $330$ | $3$ | $3$ | $1$ |
| 330.240.8-330.d.1.4 | $330$ | $5$ | $5$ | $8$ |
| 330.288.7-330.d.1.11 | $330$ | $6$ | $6$ | $7$ |
| 330.480.15-330.bj.1.8 | $330$ | $10$ | $10$ | $15$ |