Invariants
| Level: | $72$ | $\SL_2$-level: | $36$ | Newform level: | $1$ | ||
| Index: | $288$ | $\PSL_2$-index: | $288$ | ||||
| Genus: | $9 = 1 + \frac{ 288 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 32 }{2}$ | ||||||
| Cusps: | $32$ (of which $2$ are rational) | Cusp widths | $2^{12}\cdot4^{12}\cdot18^{4}\cdot36^{4}$ | Cusp orbits | $1^{2}\cdot2^{5}\cdot4^{3}\cdot8$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 9$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 9$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 36Q9 |
Level structure
| $\GL_2(\Z/72\Z)$-generators: | $\begin{bmatrix}1&18\\48&61\end{bmatrix}$, $\begin{bmatrix}13&66\\26&35\end{bmatrix}$, $\begin{bmatrix}17&18\\50&59\end{bmatrix}$, $\begin{bmatrix}37&54\\6&61\end{bmatrix}$, $\begin{bmatrix}67&66\\24&35\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 72-isogeny field degree: | $4$ |
| Cyclic 72-torsion field degree: | $48$ |
| Full 72-torsion field degree: | $20736$ |
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 |
|---|---|---|---|---|---|
| 24.96.1.cp.2 | $24$ | $3$ | $3$ | $1$ | $1$ |
| 36.144.3.a.2 | $36$ | $2$ | $2$ | $3$ | $0$ |
| 72.144.3.a.1 | $72$ | $2$ | $2$ | $3$ | $?$ |
| 72.144.5.d.1 | $72$ | $2$ | $2$ | $5$ | $?$ |