Invariants
| Level: | $66$ | $\SL_2$-level: | $66$ | Newform level: | $726$ | ||
| Index: | $1320$ | $\PSL_2$-index: | $660$ | ||||
| Genus: | $46 = 1 + \frac{ 660 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 20 }{2}$ | ||||||
| Cusps: | $20$ (none of which are rational) | Cusp widths | $11^{5}\cdot22^{5}\cdot33^{5}\cdot66^{5}$ | Cusp orbits | $5^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $15$ | ||||||
| $\Q$-gonality: | $9 \le \gamma \le 24$ | ||||||
| $\overline{\Q}$-gonality: | $9 \le \gamma \le 24$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 66.1320.46.9 |
Level structure
| $\GL_2(\Z/66\Z)$-generators: | $\begin{bmatrix}35&29\\12&35\end{bmatrix}$, $\begin{bmatrix}37&27\\36&29\end{bmatrix}$, $\begin{bmatrix}49&52\\24&19\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 66.660.46.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: | $2880$ |
Jacobian
| Conductor: | $2^{20}\cdot3^{28}\cdot11^{92}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{10}\cdot2^{14}\cdot4^{2}$ |
| Newforms: | 121.2.a.a$^{4}$, 242.2.a.b$^{2}$, 242.2.a.c$^{2}$, 242.2.a.e$^{2}$, 242.2.a.f$^{2}$, 363.2.a.c$^{2}$, 363.2.a.d$^{2}$, 363.2.a.f$^{2}$, 363.2.a.h$^{2}$, 363.2.a.j$^{2}$, 726.2.a.a, 726.2.a.e, 726.2.a.k, 726.2.a.l |
Rational points
This modular curve has no $\Q_p$ points for $p=7$, and therefore no 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 | Kernel decomposition |
|---|---|---|---|---|---|---|
| 6.24.0-6.a.1.3 | $6$ | $55$ | $55$ | $0$ | $0$ | full Jacobian |
| $X_{S_4}(11)$ | $11$ | $24$ | $12$ | $1$ | $0$ | $1^{9}\cdot2^{14}\cdot4^{2}$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 6.24.0-6.a.1.3 | $6$ | $55$ | $55$ | $0$ | $0$ | full Jacobian |
| 66.440.13-33.a.1.2 | $66$ | $3$ | $3$ | $13$ | $4$ | $1^{7}\cdot2^{11}\cdot4$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 66.2640.96-66.a.1.3 | $66$ | $2$ | $2$ | $96$ | $29$ | $1^{10}\cdot2^{18}\cdot4$ |
| 66.2640.96-66.d.1.4 | $66$ | $2$ | $2$ | $96$ | $40$ | $1^{10}\cdot2^{18}\cdot4$ |
| 66.2640.96-66.i.1.2 | $66$ | $2$ | $2$ | $96$ | $31$ | $1^{10}\cdot2^{18}\cdot4$ |
| 66.2640.96-66.k.1.1 | $66$ | $2$ | $2$ | $96$ | $36$ | $1^{10}\cdot2^{18}\cdot4$ |
| 66.3960.136-66.a.1.3 | $66$ | $3$ | $3$ | $136$ | $42$ | $1^{30}\cdot2^{26}\cdot4^{2}$ |
| 66.3960.146-66.a.1.4 | $66$ | $3$ | $3$ | $146$ | $52$ | $1^{16}\cdot2^{30}\cdot4^{6}$ |
| 66.5280.181-66.a.1.1 | $66$ | $4$ | $4$ | $181$ | $46$ | $1^{63}\cdot2^{32}\cdot4^{2}$ |