Invariants
| Level: | $180$ | $\SL_2$-level: | $36$ | Newform level: | $1$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $4 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $3^{2}\cdot9^{2}\cdot12\cdot36$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 4$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 4$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 36G4 |
Level structure
| $\GL_2(\Z/180\Z)$-generators: | $\begin{bmatrix}7&132\\118&119\end{bmatrix}$, $\begin{bmatrix}41&178\\24&79\end{bmatrix}$, $\begin{bmatrix}60&127\\79&168\end{bmatrix}$, $\begin{bmatrix}75&88\\88&69\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 180.72.4.f.1 for the level structure with $-I$) |
| Cyclic 180-isogeny field degree: | $36$ |
| Cyclic 180-torsion field degree: | $1728$ |
| Full 180-torsion field degree: | $1244160$ |
Rational points
This modular curve has 2 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 |
|---|---|---|---|---|---|
| 36.72.2-18.c.1.3 | $36$ | $2$ | $2$ | $2$ | $0$ |
| 60.48.0-60.p.1.5 | $60$ | $3$ | $3$ | $0$ | $0$ |
| 90.72.2-18.c.1.4 | $90$ | $2$ | $2$ | $2$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 180.288.9-180.a.1.13 | $180$ | $2$ | $2$ | $9$ |
| 180.288.9-180.i.1.6 | $180$ | $2$ | $2$ | $9$ |
| 180.288.9-180.ch.1.4 | $180$ | $2$ | $2$ | $9$ |
| 180.288.9-180.cl.1.6 | $180$ | $2$ | $2$ | $9$ |
| 180.288.9-180.ds.1.6 | $180$ | $2$ | $2$ | $9$ |
| 180.288.9-180.du.1.8 | $180$ | $2$ | $2$ | $9$ |
| 180.288.9-180.dx.1.4 | $180$ | $2$ | $2$ | $9$ |
| 180.288.9-180.dy.1.10 | $180$ | $2$ | $2$ | $9$ |
| 180.432.10-180.d.1.14 | $180$ | $3$ | $3$ | $10$ |
| 180.432.10-180.l.1.3 | $180$ | $3$ | $3$ | $10$ |
| 180.432.10-180.l.2.4 | $180$ | $3$ | $3$ | $10$ |
| 180.432.10-180.r.1.4 | $180$ | $3$ | $3$ | $10$ |