Invariants
| Level: | $36$ | $\SL_2$-level: | $36$ | Newform level: | $216$ | ||
| Index: | $576$ | $\PSL_2$-index: | $288$ | ||||
| Genus: | $17 = 1 + \frac{ 288 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (none of which are rational) | Cusp widths | $6^{4}\cdot12^{4}\cdot18^{4}\cdot36^{4}$ | Cusp orbits | $2^{8}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $6$ | ||||||
| $\overline{\Q}$-gonality: | $6$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 36B17 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 36.576.17.106 |
Level structure
| $\GL_2(\Z/36\Z)$-generators: | $\begin{bmatrix}5&0\\24&11\end{bmatrix}$, $\begin{bmatrix}9&14\\14&33\end{bmatrix}$, $\begin{bmatrix}23&28\\30&1\end{bmatrix}$ |
| $\GL_2(\Z/36\Z)$-subgroup: | $C_{18}:C_6^2$ |
| Contains $-I$: | no $\quad$ (see 36.288.17.h.2 for the level structure with $-I$) |
| Cyclic 36-isogeny field degree: | $6$ |
| Cyclic 36-torsion field degree: | $36$ |
| Full 36-torsion field degree: | $648$ |
Jacobian
| Conductor: | $2^{29}\cdot3^{50}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{9}\cdot4^{2}$ |
| Newforms: | 27.2.a.a$^{4}$, 72.2.a.a, 108.2.a.a$^{2}$, 108.2.b.b$^{2}$, 216.2.a.a, 216.2.a.d |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.192.1-12.d.1.8 | $12$ | $3$ | $3$ | $1$ | $0$ | $1^{8}\cdot4^{2}$ |
| 36.288.8-36.a.1.9 | $36$ | $2$ | $2$ | $8$ | $0$ | $1^{5}\cdot4$ |
| 36.288.8-36.a.1.16 | $36$ | $2$ | $2$ | $8$ | $0$ | $1^{5}\cdot4$ |
| 36.288.8-36.a.2.6 | $36$ | $2$ | $2$ | $8$ | $0$ | $1^{5}\cdot4$ |
| 36.288.8-36.a.2.9 | $36$ | $2$ | $2$ | $8$ | $0$ | $1^{5}\cdot4$ |
| 36.288.9-36.d.1.3 | $36$ | $2$ | $2$ | $9$ | $1$ | $4^{2}$ |
| 36.288.9-36.d.1.4 | $36$ | $2$ | $2$ | $9$ | $1$ | $4^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 36.1152.37-36.g.1.2 | $36$ | $2$ | $2$ | $37$ | $3$ | $1^{10}\cdot2^{3}\cdot4$ |
| 36.1152.37-36.g.2.4 | $36$ | $2$ | $2$ | $37$ | $3$ | $1^{10}\cdot2^{3}\cdot4$ |
| 36.1152.37-36.h.2.4 | $36$ | $2$ | $2$ | $37$ | $3$ | $1^{10}\cdot2^{3}\cdot4$ |
| 36.1152.37-36.h.4.1 | $36$ | $2$ | $2$ | $37$ | $3$ | $1^{10}\cdot2^{3}\cdot4$ |
| 36.1728.49-36.bo.2.8 | $36$ | $3$ | $3$ | $49$ | $1$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 36.1728.49-36.de.2.3 | $36$ | $3$ | $3$ | $49$ | $1$ | $2^{6}\cdot4\cdot8^{2}$ |
| 36.1728.49-36.de.4.6 | $36$ | $3$ | $3$ | $49$ | $1$ | $2^{6}\cdot4\cdot8^{2}$ |
| 36.1728.49-36.di.2.8 | $36$ | $3$ | $3$ | $49$ | $5$ | $1^{12}\cdot2^{2}\cdot8^{2}$ |