Invariants
| Level: | $36$ | $\SL_2$-level: | $36$ | Newform level: | $36$ | ||
| Index: | $864$ | $\PSL_2$-index: | $432$ | ||||
| Genus: | $17 = 1 + \frac{ 432 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 40 }{2}$ | ||||||
| Cusps: | $40$ (of which $9$ are rational) | Cusp widths | $1^{6}\cdot2^{3}\cdot3^{4}\cdot4^{6}\cdot6^{2}\cdot9^{6}\cdot12^{4}\cdot18^{3}\cdot36^{6}$ | Cusp orbits | $1^{9}\cdot2^{6}\cdot3\cdot4\cdot6^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $5 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $5 \le \gamma \le 8$ | ||||||
| Rational cusps: | $9$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 36L17 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 36.864.17.227 |
Level structure
| $\GL_2(\Z/36\Z)$-generators: | $\begin{bmatrix}1&22\\0&29\end{bmatrix}$, $\begin{bmatrix}1&32\\0&31\end{bmatrix}$, $\begin{bmatrix}19&21\\0&25\end{bmatrix}$ |
| $\GL_2(\Z/36\Z)$-subgroup: | $D_{36}:C_6$ |
| Contains $-I$: | no $\quad$ (see 36.432.17.t.1 for the level structure with $-I$) |
| Cyclic 36-isogeny field degree: | $1$ |
| Cyclic 36-torsion field degree: | $1$ |
| Full 36-torsion field degree: | $432$ |
Jacobian
| Conductor: | $2^{30}\cdot3^{34}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1\cdot2^{4}\cdot8$ |
| Newforms: | 18.2.c.a$^{2}$, 36.2.a.a, 36.2.b.a, 36.2.e.a, 36.2.h.a |
Rational points
This modular curve has 9 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 |
|---|---|---|---|---|---|---|
| $X_1(4)$ | $4$ | $72$ | $72$ | $0$ | $0$ | full Jacobian |
| $X_1(9)$ | $9$ | $12$ | $12$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 36.288.3-36.c.1.1 | $36$ | $3$ | $3$ | $3$ | $0$ | $2^{3}\cdot8$ |
| 36.432.7-36.u.1.1 | $36$ | $2$ | $2$ | $7$ | $0$ | $2\cdot8$ |
| 36.432.7-36.u.1.2 | $36$ | $2$ | $2$ | $7$ | $0$ | $2\cdot8$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_1(2,36)$ | $36$ | $2$ | $2$ | $41$ | $0$ | $1^{4}\cdot2^{4}\cdot4\cdot8$ |
| 36.1728.41-36.bf.2.2 | $36$ | $2$ | $2$ | $41$ | $0$ | $1^{4}\cdot2^{4}\cdot4\cdot8$ |
| 36.1728.41-36.bj.4.3 | $36$ | $2$ | $2$ | $41$ | $0$ | $1^{4}\cdot2^{4}\cdot4\cdot8$ |
| 36.1728.41-36.bk.2.1 | $36$ | $2$ | $2$ | $41$ | $0$ | $1^{4}\cdot2^{4}\cdot4\cdot8$ |
| 36.2592.64-36.h.4.2 | $36$ | $3$ | $3$ | $64$ | $0$ | $1^{9}\cdot2^{7}\cdot4^{2}\cdot8^{2}$ |