Invariants
| Level: | $72$ | $\SL_2$-level: | $18$ | Newform level: | $1$ | ||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $2 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $6\cdot18$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 18B2 |
Level structure
| $\GL_2(\Z/72\Z)$-generators: | $\begin{bmatrix}1&44\\64&51\end{bmatrix}$, $\begin{bmatrix}54&35\\19&32\end{bmatrix}$, $\begin{bmatrix}67&29\\33&32\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 72.24.2.a.1 for the level structure with $-I$) |
| Cyclic 72-isogeny field degree: | $36$ |
| Cyclic 72-torsion field degree: | $864$ |
| Full 72-torsion field degree: | $124416$ |
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 |
|---|---|---|---|---|---|
| 36.24.1-9.a.1.4 | $36$ | $2$ | $2$ | $1$ | $0$ |
| 72.24.1-9.a.1.4 | $72$ | $2$ | $2$ | $1$ | $?$ |
| 24.16.0-24.a.1.2 | $24$ | $3$ | $3$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 72.144.4-72.a.1.9 | $72$ | $3$ | $3$ | $4$ |
| 72.144.4-72.i.1.4 | $72$ | $3$ | $3$ | $4$ |
| 72.144.4-72.k.1.6 | $72$ | $3$ | $3$ | $4$ |
| 72.144.4-72.k.2.5 | $72$ | $3$ | $3$ | $4$ |
| 72.144.4-72.m.1.6 | $72$ | $3$ | $3$ | $4$ |
| 72.192.7-72.a.1.5 | $72$ | $4$ | $4$ | $7$ |