Invariants
| Level: | $72$ | $\SL_2$-level: | $72$ | Newform level: | $1$ | ||
| Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $9 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $4$ are rational) | Cusp widths | $3^{2}\cdot6\cdot9^{2}\cdot18\cdot24\cdot72$ | Cusp orbits | $1^{4}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 9$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 9$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 72D9 |
Level structure
| $\GL_2(\Z/72\Z)$-generators: | $\begin{bmatrix}1&6\\0&67\end{bmatrix}$, $\begin{bmatrix}3&20\\46&53\end{bmatrix}$, $\begin{bmatrix}14&63\\11&34\end{bmatrix}$, $\begin{bmatrix}67&14\\0&65\end{bmatrix}$, $\begin{bmatrix}68&57\\27&2\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 72.144.9.dk.1 for the level structure with $-I$) |
| Cyclic 72-isogeny field degree: | $6$ |
| Cyclic 72-torsion field degree: | $72$ |
| Full 72-torsion field degree: | $20736$ |
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 |
|---|---|---|---|---|---|
| 24.96.1-24.iu.1.18 | $24$ | $3$ | $3$ | $1$ | $0$ |
| 36.144.4-36.f.1.8 | $36$ | $2$ | $2$ | $4$ | $0$ |
| 72.144.4-36.f.1.11 | $72$ | $2$ | $2$ | $4$ | $?$ |