Invariants
| Level: | $66$ | $\SL_2$-level: | $66$ | Newform level: | $726$ | ||
| Index: | $1584$ | $\PSL_2$-index: | $792$ | ||||
| Genus: | $55 = 1 + \frac{ 792 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$ | ||||||
| Cusps: | $24$ (of which $4$ are rational) | Cusp widths | $11^{6}\cdot22^{6}\cdot33^{6}\cdot66^{6}$ | Cusp orbits | $1^{4}\cdot5^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $17$ | ||||||
| $\Q$-gonality: | $11 \le \gamma \le 24$ | ||||||
| $\overline{\Q}$-gonality: | $11 \le \gamma \le 24$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 66.1584.55.5 |
Level structure
| $\GL_2(\Z/66\Z)$-generators: | $\begin{bmatrix}7&60\\6&37\end{bmatrix}$, $\begin{bmatrix}19&12\\40&47\end{bmatrix}$, $\begin{bmatrix}25&54\\36&41\end{bmatrix}$, $\begin{bmatrix}53&33\\48&37\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 66.792.55.a.1 for the level structure with $-I$) |
| Cyclic 66-isogeny field degree: | $2$ |
| Cyclic 66-torsion field degree: | $40$ |
| Full 66-torsion field degree: | $2400$ |
Jacobian
| Conductor: | $2^{21}\cdot3^{35}\cdot11^{101}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{15}\cdot2^{16}\cdot4^{2}$ |
| Newforms: | 11.2.a.a$^{4}$, 33.2.a.a$^{2}$, 66.2.a.a, 66.2.a.b, 66.2.a.c, 121.2.a.b$^{4}$, 242.2.a.c$^{2}$, 242.2.a.e$^{2}$, 242.2.a.f$^{2}$, 363.2.a.d$^{2}$, 363.2.a.f$^{2}$, 363.2.a.g$^{2}$, 363.2.a.h$^{2}$, 363.2.a.j$^{2}$, 726.2.a.b, 726.2.a.g, 726.2.a.k, 726.2.a.l |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal 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 | $6$ | $66$ | $66$ | $0$ | $0$ | full Jacobian |
| $X_{\mathrm{sp}}^+(11)$ | $11$ | $24$ | $12$ | $2$ | $1$ | $1^{13}\cdot2^{16}\cdot4^{2}$ |
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 | $6$ | $66$ | $66$ | $0$ | $0$ | full Jacobian |
| 66.528.17-33.a.1.7 | $66$ | $3$ | $3$ | $17$ | $6$ | $1^{10}\cdot2^{12}\cdot4$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 66.3168.109-66.a.1.8 | $66$ | $2$ | $2$ | $109$ | $26$ | $1^{38}\cdot2^{8}$ |
| 66.3168.109-66.b.1.1 | $66$ | $2$ | $2$ | $109$ | $51$ | $1^{38}\cdot2^{8}$ |
| 66.3168.109-66.c.1.8 | $66$ | $2$ | $2$ | $109$ | $26$ | $1^{38}\cdot2^{8}$ |
| 66.3168.109-66.d.1.1 | $66$ | $2$ | $2$ | $109$ | $43$ | $1^{38}\cdot2^{8}$ |
| 66.3168.115-66.a.1.7 | $66$ | $2$ | $2$ | $115$ | $30$ | $1^{14}\cdot2^{21}\cdot4$ |
| 66.3168.115-66.b.1.8 | $66$ | $2$ | $2$ | $115$ | $44$ | $1^{14}\cdot2^{21}\cdot4$ |
| 66.3168.115-66.c.1.5 | $66$ | $2$ | $2$ | $115$ | $38$ | $1^{14}\cdot2^{21}\cdot4$ |
| 66.3168.115-66.d.1.6 | $66$ | $2$ | $2$ | $115$ | $42$ | $1^{14}\cdot2^{21}\cdot4$ |
| 66.3168.115-66.e.1.1 | $66$ | $2$ | $2$ | $115$ | $43$ | $1^{40}\cdot2^{10}$ |
| 66.3168.115-66.f.1.5 | $66$ | $2$ | $2$ | $115$ | $31$ | $1^{40}\cdot2^{10}$ |
| 66.3168.115-66.g.1.1 | $66$ | $2$ | $2$ | $115$ | $39$ | $1^{40}\cdot2^{10}$ |
| 66.3168.115-66.h.1.5 | $66$ | $2$ | $2$ | $115$ | $35$ | $1^{40}\cdot2^{10}$ |
| 66.4752.175-66.a.1.7 | $66$ | $3$ | $3$ | $175$ | $56$ | $1^{28}\cdot2^{34}\cdot4^{6}$ |
| 66.7920.271-66.c.1.2 | $66$ | $5$ | $5$ | $271$ | $73$ | $1^{88}\cdot2^{56}\cdot4^{4}$ |