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: | $17$ | ||||||
| $\Q$-gonality: | $9 \le \gamma \le 24$ | ||||||
| $\overline{\Q}$-gonality: | $9 \le \gamma \le 24$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-3,-12$) |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 66.1320.46.10 |
Level structure
| $\GL_2(\Z/66\Z)$-generators: | $\begin{bmatrix}13&54\\0&31\end{bmatrix}$, $\begin{bmatrix}23&54\\6&43\end{bmatrix}$, $\begin{bmatrix}37&33\\42&29\end{bmatrix}$, $\begin{bmatrix}47&47\\12&41\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 66.660.46.b.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: | $2^{18}\cdot3^{30}\cdot11^{92}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{6}\cdot2^{16}\cdot4^{2}$ |
| Newforms: | 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 2 rational CM points but no rational cusps or other known 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 |
| $X_{\mathrm{ns}}^+(11)$ | $11$ | $24$ | $12$ | $1$ | $1$ | $1^{5}\cdot2^{16}\cdot4^{2}$ |
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 | $6$ | $55$ | $55$ | $0$ | $0$ | full Jacobian |
| 66.440.14-33.a.1.7 | $66$ | $3$ | $3$ | $14$ | $6$ | $1^{4}\cdot2^{12}\cdot4$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 66.2640.91-66.a.1.3 | $66$ | $2$ | $2$ | $91$ | $26$ | $1^{29}\cdot2^{8}$ |
| 66.2640.91-66.b.1.3 | $66$ | $2$ | $2$ | $91$ | $50$ | $1^{29}\cdot2^{8}$ |
| 66.2640.91-66.c.1.2 | $66$ | $2$ | $2$ | $91$ | $25$ | $1^{29}\cdot2^{8}$ |
| 66.2640.91-66.d.1.2 | $66$ | $2$ | $2$ | $91$ | $36$ | $1^{29}\cdot2^{8}$ |
| 66.2640.96-66.b.1.6 | $66$ | $2$ | $2$ | $96$ | $30$ | $1^{4}\cdot2^{21}\cdot4$ |
| 66.2640.96-66.e.1.8 | $66$ | $2$ | $2$ | $96$ | $42$ | $1^{4}\cdot2^{21}\cdot4$ |
| 66.2640.96-66.g.1.8 | $66$ | $2$ | $2$ | $96$ | $34$ | $1^{4}\cdot2^{21}\cdot4$ |
| 66.2640.96-66.h.1.5 | $66$ | $2$ | $2$ | $96$ | $38$ | $1^{4}\cdot2^{21}\cdot4$ |
| 66.2640.96-66.m.1.1 | $66$ | $2$ | $2$ | $96$ | $39$ | $1^{30}\cdot2^{10}$ |
| 66.2640.96-66.n.1.3 | $66$ | $2$ | $2$ | $96$ | $27$ | $1^{30}\cdot2^{10}$ |
| 66.2640.96-66.q.1.4 | $66$ | $2$ | $2$ | $96$ | $37$ | $1^{30}\cdot2^{10}$ |
| 66.2640.96-66.s.1.2 | $66$ | $2$ | $2$ | $96$ | $35$ | $1^{30}\cdot2^{10}$ |
| 66.3960.136-66.a.1.3 | $66$ | $3$ | $3$ | $136$ | $42$ | $1^{34}\cdot2^{24}\cdot4^{2}$ |
| 66.3960.146-66.b.1.7 | $66$ | $3$ | $3$ | $146$ | $53$ | $1^{8}\cdot2^{34}\cdot4^{6}$ |