Invariants
| Level: | $36$ | $\SL_2$-level: | $36$ | Newform level: | $108$ | ||
| Index: | $864$ | $\PSL_2$-index: | $432$ | ||||
| Genus: | $22 = 1 + \frac{ 432 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 30 }{2}$ | ||||||
| Cusps: | $30$ (of which $4$ are rational) | Cusp widths | $3^{6}\cdot6^{3}\cdot9^{6}\cdot12^{6}\cdot18^{3}\cdot36^{6}$ | Cusp orbits | $1^{4}\cdot2^{9}\cdot4^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $6 \le \gamma \le 9$ | ||||||
| $\overline{\Q}$-gonality: | $6 \le \gamma \le 9$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 36L22 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 36.864.22.62 |
Level structure
| $\GL_2(\Z/36\Z)$-generators: | $\begin{bmatrix}5&27\\0&13\end{bmatrix}$, $\begin{bmatrix}19&18\\0&13\end{bmatrix}$, $\begin{bmatrix}25&3\\0&31\end{bmatrix}$ |
| $\GL_2(\Z/36\Z)$-subgroup: | $C_{12}:C_6^2$ |
| Contains $-I$: | no $\quad$ (see 36.432.22.o.4 for the level structure with $-I$) |
| Cyclic 36-isogeny field degree: | $1$ |
| Cyclic 36-torsion field degree: | $12$ |
| Full 36-torsion field degree: | $432$ |
Jacobian
| Conductor: | $2^{34}\cdot3^{60}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{10}\cdot2^{2}\cdot4^{2}$ |
| Newforms: | 27.2.a.a$^{3}$, 36.2.a.a$^{2}$, 36.2.b.a$^{2}$, 54.2.a.a$^{2}$, 54.2.a.b$^{2}$, 108.2.a.a, 108.2.b.a, 108.2.b.b |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.288.3-12.c.1.7 | $12$ | $3$ | $3$ | $3$ | $0$ | $1^{9}\cdot2\cdot4^{2}$ |
| 36.288.3-36.c.2.5 | $36$ | $3$ | $3$ | $3$ | $0$ | $1^{9}\cdot2\cdot4^{2}$ |
| 36.288.8-36.e.2.4 | $36$ | $3$ | $3$ | $8$ | $0$ | $1^{6}\cdot2^{2}\cdot4$ |
| 36.288.8-36.f.2.4 | $36$ | $3$ | $3$ | $8$ | $0$ | $1^{6}\cdot2^{2}\cdot4$ |
| 36.432.10-36.g.1.2 | $36$ | $2$ | $2$ | $10$ | $0$ | $2^{2}\cdot4^{2}$ |
| 36.432.10-36.g.1.5 | $36$ | $2$ | $2$ | $10$ | $0$ | $2^{2}\cdot4^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 36.1728.49-36.bf.4.2 | $36$ | $2$ | $2$ | $49$ | $1$ | $1^{15}\cdot2^{2}\cdot4^{2}$ |
| 36.1728.49-36.ed.2.2 | $36$ | $2$ | $2$ | $49$ | $3$ | $1^{15}\cdot2^{2}\cdot4^{2}$ |
| 36.1728.49-36.fk.1.1 | $36$ | $2$ | $2$ | $49$ | $3$ | $1^{15}\cdot2^{2}\cdot4^{2}$ |
| 36.1728.49-36.fl.2.2 | $36$ | $2$ | $2$ | $49$ | $1$ | $1^{15}\cdot2^{2}\cdot4^{2}$ |
| 36.2592.64-36.g.1.1 | $36$ | $3$ | $3$ | $64$ | $0$ | $2^{9}\cdot8^{3}$ |
| 36.2592.64-36.h.3.4 | $36$ | $3$ | $3$ | $64$ | $0$ | $2^{9}\cdot8^{3}$ |
| 36.2592.64-36.i.3.4 | $36$ | $3$ | $3$ | $64$ | $2$ | $1^{18}\cdot8^{3}$ |
| 36.2592.79-36.u.2.2 | $36$ | $3$ | $3$ | $79$ | $3$ | $1^{21}\cdot2^{4}\cdot4^{3}\cdot8^{2}$ |