Invariants
| Level: | $180$ | $\SL_2$-level: | $36$ | Newform level: | $1$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $3 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $4$ are rational) | Cusp widths | $2^{3}\cdot4^{3}\cdot18\cdot36$ | Cusp orbits | $1^{4}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 3$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 3$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 36G3 |
Level structure
| $\GL_2(\Z/180\Z)$-generators: | $\begin{bmatrix}97&178\\126&143\end{bmatrix}$, $\begin{bmatrix}121&148\\108&137\end{bmatrix}$, $\begin{bmatrix}128&99\\3&170\end{bmatrix}$, $\begin{bmatrix}153&82\\100&177\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 180.72.3.bf.1 for the level structure with $-I$) |
| Cyclic 180-isogeny field degree: | $12$ |
| Cyclic 180-torsion field degree: | $576$ |
| Full 180-torsion field degree: | $1244160$ |
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 |
|---|---|---|---|---|---|
| 36.72.0-18.a.1.12 | $36$ | $2$ | $2$ | $0$ | $0$ |
| 60.48.1-60.w.1.11 | $60$ | $3$ | $3$ | $1$ | $0$ |
| 90.72.0-18.a.1.3 | $90$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 180.288.5-180.c.1.10 | $180$ | $2$ | $2$ | $5$ |
| 180.288.5-180.g.1.7 | $180$ | $2$ | $2$ | $5$ |
| 180.288.5-180.m.1.6 | $180$ | $2$ | $2$ | $5$ |
| 180.288.5-180.p.1.10 | $180$ | $2$ | $2$ | $5$ |
| 180.288.5-180.q.1.6 | $180$ | $2$ | $2$ | $5$ |
| 180.288.5-180.t.1.7 | $180$ | $2$ | $2$ | $5$ |
| 180.288.5-180.u.1.6 | $180$ | $2$ | $2$ | $5$ |
| 180.288.5-180.x.1.6 | $180$ | $2$ | $2$ | $5$ |
| 180.432.11-180.ff.1.3 | $180$ | $3$ | $3$ | $11$ |
| 180.432.11-180.ff.2.6 | $180$ | $3$ | $3$ | $11$ |
| 180.432.11-180.fj.1.5 | $180$ | $3$ | $3$ | $11$ |
| 180.432.13-180.bn.1.13 | $180$ | $3$ | $3$ | $13$ |