Invariants
| Level: | $72$ | $\SL_2$-level: | $36$ | 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: | $8$ (none of which are rational) | Cusp widths | $6^{2}\cdot12^{2}\cdot18^{2}\cdot36^{2}$ | Cusp orbits | $2^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 16$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 9$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 36C9 |
Level structure
| $\GL_2(\Z/72\Z)$-generators: | $\begin{bmatrix}17&52\\48&31\end{bmatrix}$, $\begin{bmatrix}61&29\\22&33\end{bmatrix}$, $\begin{bmatrix}69&26\\58&53\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 72.144.9.fn.1 for the level structure with $-I$) |
| Cyclic 72-isogeny field degree: | $12$ |
| Cyclic 72-torsion field degree: | $288$ |
| Full 72-torsion field degree: | $20736$ |
Rational points
This modular curve has no $\Q_p$ points for $p=7$, and therefore no 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.jh.1.5 | $24$ | $3$ | $3$ | $1$ | $0$ |
| 36.144.4-36.m.1.1 | $36$ | $2$ | $2$ | $4$ | $1$ |
| 72.144.4-36.m.1.2 | $72$ | $2$ | $2$ | $4$ | $?$ |
| 72.144.4-72.x.1.2 | $72$ | $2$ | $2$ | $4$ | $?$ |
| 72.144.4-72.x.1.3 | $72$ | $2$ | $2$ | $4$ | $?$ |
| 72.144.5-72.bh.1.2 | $72$ | $2$ | $2$ | $5$ | $?$ |
| 72.144.5-72.bh.1.13 | $72$ | $2$ | $2$ | $5$ | $?$ |