Invariants
| Level: | $36$ | $\SL_2$-level: | $36$ | Newform level: | $1296$ | ||
| Index: | $1296$ | $\PSL_2$-index: | $648$ | ||||
| Genus: | $46 = 1 + \frac{ 648 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 18 }{2}$ | ||||||
| Cusps: | $18$ (none of which are rational) | Cusp widths | $36^{18}$ | Cusp orbits | $3^{2}\cdot6^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $10$ | ||||||
| $\Q$-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: | 36.1296.46.11 |
Level structure
| $\GL_2(\Z/36\Z)$-generators: | $\begin{bmatrix}1&26\\24&17\end{bmatrix}$, $\begin{bmatrix}3&34\\8&17\end{bmatrix}$ |
| $\GL_2(\Z/36\Z)$-subgroup: | $C_6\times \GL(2,3)$ |
| Contains $-I$: | no $\quad$ (see 36.648.46.g.1 for the level structure with $-I$) |
| Cyclic 36-isogeny field degree: | $12$ |
| Cyclic 36-torsion field degree: | $144$ |
| Full 36-torsion field degree: | $288$ |
Jacobian
| Conductor: | $2^{112}\cdot3^{168}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{40}\cdot2^{3}$ |
| Newforms: | 54.2.a.a$^{4}$, 54.2.a.b$^{4}$, 162.2.a.a, 162.2.a.b, 162.2.a.c$^{3}$, 162.2.a.d$^{3}$, 216.2.a.b$^{2}$, 216.2.a.c$^{2}$, 324.2.a.a, 324.2.a.b$^{2}$, 324.2.a.c$^{2}$, 324.2.a.d, 432.2.a.b, 432.2.a.c, 432.2.a.f, 432.2.a.g, 648.2.a.a, 648.2.a.b, 648.2.a.c, 648.2.a.d, 648.2.a.f, 648.2.a.g, 1296.2.a.b, 1296.2.a.c, 1296.2.a.f, 1296.2.a.h, 1296.2.a.i, 1296.2.a.j, 1296.2.a.p |
Rational points
This modular curve has no $\Q_p$ points for $p=5,11,59,67,83,131,229$, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.48.0-12.c.1.3 | $12$ | $27$ | $27$ | $0$ | $0$ | full Jacobian |
| 36.648.22-36.b.1.4 | $36$ | $2$ | $2$ | $22$ | $2$ | $1^{20}\cdot2^{2}$ |
| 36.648.22-36.b.1.5 | $36$ | $2$ | $2$ | $22$ | $2$ | $1^{20}\cdot2^{2}$ |
| 36.648.22-36.c.1.1 | $36$ | $2$ | $2$ | $22$ | $5$ | $1^{20}\cdot2^{2}$ |
| 36.648.22-36.c.1.4 | $36$ | $2$ | $2$ | $22$ | $5$ | $1^{20}\cdot2^{2}$ |
| 36.648.22-36.d.1.1 | $36$ | $2$ | $2$ | $22$ | $3$ | $1^{20}\cdot2^{2}$ |
| 36.648.22-36.d.1.3 | $36$ | $2$ | $2$ | $22$ | $3$ | $1^{20}\cdot2^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 36.3888.136-36.e.1.4 | $36$ | $3$ | $3$ | $136$ | $33$ | $1^{78}\cdot2^{6}$ |
| 36.5184.181-36.cm.1.5 | $36$ | $4$ | $4$ | $181$ | $43$ | $1^{93}\cdot2^{21}$ |