Invariants
| Level: | $72$ | $\SL_2$-level: | $18$ | Newform level: | $1$ | ||
| Index: | $72$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $5 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $18^{4}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 5$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 5$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 18A5 |
Level structure
| $\GL_2(\Z/72\Z)$-generators: | $\begin{bmatrix}4&25\\33&26\end{bmatrix}$, $\begin{bmatrix}7&61\\15&20\end{bmatrix}$, $\begin{bmatrix}29&12\\6&47\end{bmatrix}$, $\begin{bmatrix}31&66\\45&25\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 72.144.5-72.b.1.1, 72.144.5-72.b.1.2, 72.144.5-72.b.1.3, 72.144.5-72.b.1.4 |
| Cyclic 72-isogeny field degree: | $36$ |
| Cyclic 72-torsion field degree: | $864$ |
| Full 72-torsion field degree: | $82944$ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points.
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.2.0.a.1 | $8$ | $36$ | $36$ | $0$ | $0$ |
| 9.36.2.a.1 | $9$ | $2$ | $2$ | $2$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 9.36.2.a.1 | $9$ | $2$ | $2$ | $2$ | $0$ |
| 24.24.1.bx.1 | $24$ | $3$ | $3$ | $1$ | $0$ |
| 72.36.0.f.1 | $72$ | $2$ | $2$ | $0$ | $?$ |
| 72.36.3.v.1 | $72$ | $2$ | $2$ | $3$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 72.216.13.c.1 | $72$ | $3$ | $3$ | $13$ |
| 72.216.13.bc.1 | $72$ | $3$ | $3$ | $13$ |
| 72.216.13.bh.1 | $72$ | $3$ | $3$ | $13$ |
| 72.216.13.ch.1 | $72$ | $3$ | $3$ | $13$ |
| 72.288.21.gv.1 | $72$ | $4$ | $4$ | $21$ |