Invariants
| Level: | $72$ | $\SL_2$-level: | $36$ | Newform level: | $1$ | ||
| Index: | $288$ | $\PSL_2$-index: | $288$ | ||||
| Genus: | $17 = 1 + \frac{ 288 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (of which $2$ are rational) | Cusp widths | $6^{4}\cdot12^{4}\cdot18^{4}\cdot36^{4}$ | Cusp orbits | $1^{2}\cdot2^{3}\cdot4^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 17$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 17$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 36B17 |
Level structure
| $\GL_2(\Z/72\Z)$-generators: | $\begin{bmatrix}7&58\\30&7\end{bmatrix}$, $\begin{bmatrix}19&62\\6&47\end{bmatrix}$, $\begin{bmatrix}31&24\\54&43\end{bmatrix}$, $\begin{bmatrix}53&16\\30&59\end{bmatrix}$, $\begin{bmatrix}55&6\\18&13\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 72-isogeny field degree: | $12$ |
| Cyclic 72-torsion field degree: | $144$ |
| 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 | $24$ | $3$ | $3$ | $1$ | $1$ |
| 36.144.8.a.1 | $36$ | $2$ | $2$ | $8$ | $0$ |
| 72.144.8.a.2 | $72$ | $2$ | $2$ | $8$ | $?$ |
| 72.144.9.d.1 | $72$ | $2$ | $2$ | $9$ | $?$ |