Invariants
| Level: | $66$ | $\SL_2$-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 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 66.16.0.3 |
Level structure
| $\GL_2(\Z/66\Z)$-generators: | $\begin{bmatrix}1&24\\31&47\end{bmatrix}$, $\begin{bmatrix}22&45\\59&28\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 66.8.0.a.1 for the level structure with $-I$) |
| Cyclic 66-isogeny field degree: | $36$ |
| Cyclic 66-torsion field degree: | $720$ |
| Full 66-torsion field degree: | $237600$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 207 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 8 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^2\cdot5^2\cdot11^3}\cdot\frac{x^{8}(33x^{2}-100y^{2})^{3}(297x^{2}-100y^{2})}{y^{2}x^{14}}$ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 3.8.0-3.a.1.1 | $3$ | $2$ | $2$ | $0$ | $0$ |
| 22.2.0.a.1 | $22$ | $8$ | $4$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 3.8.0-3.a.1.1 | $3$ | $2$ | $2$ | $0$ | $0$ |
| 66.8.0-3.a.1.2 | $66$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 66.48.0-66.a.1.2 | $66$ | $3$ | $3$ | $0$ |
| 66.48.1-66.b.1.1 | $66$ | $3$ | $3$ | $1$ |
| 66.192.7-66.c.1.8 | $66$ | $12$ | $12$ | $7$ |
| 66.880.30-66.c.1.4 | $66$ | $55$ | $55$ | $30$ |
| 66.880.32-66.c.1.8 | $66$ | $55$ | $55$ | $32$ |
| 66.1056.39-66.c.1.4 | $66$ | $66$ | $66$ | $39$ |
| 132.64.1-132.a.1.2 | $132$ | $4$ | $4$ | $1$ |
| 198.48.0-198.a.1.1 | $198$ | $3$ | $3$ | $0$ |
| 198.48.1-198.a.1.1 | $198$ | $3$ | $3$ | $1$ |
| 198.48.2-198.a.1.3 | $198$ | $3$ | $3$ | $2$ |
| 330.80.2-330.a.1.5 | $330$ | $5$ | $5$ | $2$ |
| 330.96.3-330.a.1.1 | $330$ | $6$ | $6$ | $3$ |
| 330.160.5-330.a.1.7 | $330$ | $10$ | $10$ | $5$ |