Invariants
| Level: | $72$ | $\SL_2$-level: | $18$ | Newform level: | $1$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $4 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $3$ are rational) | Cusp widths | $6^{3}\cdot18^{3}$ | Cusp orbits | $1^{3}\cdot3$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 4$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 4$ | ||||||
| Rational cusps: | $3$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 18I4 |
Level structure
| $\GL_2(\Z/72\Z)$-generators: | $\begin{bmatrix}13&8\\46&63\end{bmatrix}$, $\begin{bmatrix}66&35\\23&54\end{bmatrix}$, $\begin{bmatrix}69&2\\49&29\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 72.72.4.k.2 for the level structure with $-I$) |
| Cyclic 72-isogeny field degree: | $36$ |
| Cyclic 72-torsion field degree: | $864$ |
| Full 72-torsion field degree: | $41472$ |
Rational points
This modular curve has 3 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 |
|---|---|---|---|---|---|
| 36.72.1-9.b.2.4 | $36$ | $2$ | $2$ | $1$ | $0$ |
| 72.48.2-72.a.1.5 | $72$ | $3$ | $3$ | $2$ | $?$ |
| 72.72.1-9.b.2.4 | $72$ | $2$ | $2$ | $1$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 72.432.10-72.b.1.7 | $72$ | $3$ | $3$ | $10$ |
| 72.432.10-72.u.1.6 | $72$ | $3$ | $3$ | $10$ |