Invariants
| Level: | $66$ | $\SL_2$-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 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 66.48.0.13 |
Level structure
| $\GL_2(\Z/66\Z)$-generators: | $\begin{bmatrix}5&56\\42&49\end{bmatrix}$, $\begin{bmatrix}20&31\\63&62\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 66.24.0.a.1 for the level structure with $-I$) |
| Cyclic 66-isogeny field degree: | $12$ |
| Cyclic 66-torsion field degree: | $240$ |
| Full 66-torsion field degree: | $79200$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 38 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^{20}}{3^6\cdot11^3}\cdot\frac{(x+4y)^{24}(9x^{2}-30xy-8y^{2})^{3}(29889x^{6}+903960x^{5}y-291600x^{4}y^{2}-11232000x^{3}y^{3}+38465280x^{2}y^{4}-50227200xy^{5}+30543872y^{6})^{3}}{(x+4y)^{24}(3x-4y)^{6}(3x+28y)^{2}(9x^{2}-8xy+80y^{2})^{6}(225x^{2}-552xy+592y^{2})^{2}}$ |
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 | $6$ | $2$ | $2$ | $0$ | $0$ |
| 66.16.0-66.a.1.2 | $66$ | $3$ | $3$ | $0$ | $0$ |
| 66.24.0-6.a.1.1 | $66$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.