Invariants
| Level: | $330$ | $\SL_2$-level: | $6$ | ||||
| Index: | $16$ | $\PSL_2$-index: | $8$ | ||||
| Genus: | $0 = 1 + \frac{ 8 }{12} - \frac{ 0 }{4} - \frac{ 2 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $2\cdot6$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $2$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 6C0 |
Level structure
| $\GL_2(\Z/330\Z)$-generators: | $\begin{bmatrix}59&145\\300&97\end{bmatrix}$, $\begin{bmatrix}109&6\\16&173\end{bmatrix}$, $\begin{bmatrix}315&94\\37&51\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 30.8.0.a.1 for the level structure with $-I$) |
| Cyclic 330-isogeny field degree: | $216$ |
| Cyclic 330-torsion field degree: | $17280$ |
| Full 330-torsion field degree: | $114048000$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 196 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 8 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^4\cdot5^3}\cdot\frac{x^{8}(15x^{2}+16y^{2})^{3}(135x^{2}+16y^{2})}{y^{2}x^{14}}$ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 10.2.0.a.1 | $10$ | $8$ | $4$ | $0$ | $0$ |
| 33.8.0-3.a.1.1 | $33$ | $2$ | $2$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 33.8.0-3.a.1.1 | $33$ | $2$ | $2$ | $0$ | $0$ |
| 330.8.0-3.a.1.1 | $330$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 330.48.0-30.a.1.3 | $330$ | $3$ | $3$ | $0$ |
| 330.48.1-30.c.1.1 | $330$ | $3$ | $3$ | $1$ |
| 330.80.2-30.c.1.3 | $330$ | $5$ | $5$ | $2$ |
| 330.96.3-30.c.1.5 | $330$ | $6$ | $6$ | $3$ |
| 330.160.5-30.c.1.6 | $330$ | $10$ | $10$ | $5$ |
| 330.192.7-330.a.1.16 | $330$ | $12$ | $12$ | $7$ |