Invariants
| Level: | $72$ | $\SL_2$-level: | $18$ | Newform level: | $1$ | ||
| Index: | $216$ | $\PSL_2$-index: | $216$ | ||||
| Genus: | $7 = 1 + \frac{ 216 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$ | ||||||
| Cusps: | $24$ (of which $2$ are rational) | Cusp widths | $6^{18}\cdot18^{6}$ | Cusp orbits | $1^{2}\cdot2^{2}\cdot6\cdot12$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 7$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 7$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 18N7 |
Level structure
| $\GL_2(\Z/72\Z)$-generators: | $\begin{bmatrix}10&39\\15&40\end{bmatrix}$, $\begin{bmatrix}16&27\\63&2\end{bmatrix}$, $\begin{bmatrix}44&39\\3&64\end{bmatrix}$, $\begin{bmatrix}47&30\\42&11\end{bmatrix}$, $\begin{bmatrix}50&9\\27&10\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 72.432.7-72.dj.1.1, 72.432.7-72.dj.1.2, 72.432.7-72.dj.1.3, 72.432.7-72.dj.1.4, 72.432.7-72.dj.1.5, 72.432.7-72.dj.1.6, 72.432.7-72.dj.1.7, 72.432.7-72.dj.1.8 |
| Cyclic 72-isogeny field degree: | $12$ |
| Cyclic 72-torsion field degree: | $288$ |
| Full 72-torsion field degree: | $27648$ |
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 |
|---|---|---|---|---|---|
| 18.108.2.b.1 | $18$ | $2$ | $2$ | $2$ | $0$ |
| 24.72.1.bn.1 | $24$ | $3$ | $3$ | $1$ | $0$ |
| 72.72.3.bp.1 | $72$ | $3$ | $3$ | $3$ | $?$ |
| 72.108.3.d.1 | $72$ | $2$ | $2$ | $3$ | $?$ |
| 72.108.4.k.1 | $72$ | $2$ | $2$ | $4$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 72.432.21.od.1 | $72$ | $2$ | $2$ | $21$ |
| 72.432.21.og.1 | $72$ | $2$ | $2$ | $21$ |
| 72.432.21.pf.1 | $72$ | $2$ | $2$ | $21$ |
| 72.432.21.pi.1 | $72$ | $2$ | $2$ | $21$ |
| 72.432.21.vr.1 | $72$ | $2$ | $2$ | $21$ |
| 72.432.21.vv.1 | $72$ | $2$ | $2$ | $21$ |
| 72.432.21.wt.1 | $72$ | $2$ | $2$ | $21$ |
| 72.432.21.wx.1 | $72$ | $2$ | $2$ | $21$ |