Invariants
| Level: | $72$ | $\SL_2$-level: | $36$ | Newform level: | $1$ | ||
| Index: | $216$ | $\PSL_2$-index: | $216$ | ||||
| Genus: | $11 = 1 + \frac{ 216 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (of which $4$ are rational) | Cusp widths | $6^{6}\cdot12^{6}\cdot18^{2}\cdot36^{2}$ | Cusp orbits | $1^{4}\cdot6^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 11$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 11$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 36I11 |
Level structure
| $\GL_2(\Z/72\Z)$-generators: | $\begin{bmatrix}9&14\\44&39\end{bmatrix}$, $\begin{bmatrix}21&34\\58&3\end{bmatrix}$, $\begin{bmatrix}43&27\\60&61\end{bmatrix}$, $\begin{bmatrix}59&30\\6&5\end{bmatrix}$, $\begin{bmatrix}63&29\\50&45\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 72.432.11-72.ge.1.1, 72.432.11-72.ge.1.2, 72.432.11-72.ge.1.3, 72.432.11-72.ge.1.4, 72.432.11-72.ge.1.5, 72.432.11-72.ge.1.6, 72.432.11-72.ge.1.7, 72.432.11-72.ge.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 4 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 |
|---|---|---|---|---|---|
| 8.6.0.f.1 | $8$ | $36$ | $36$ | $0$ | $0$ |
| 9.36.0.b.1 | $9$ | $6$ | $6$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 18.108.2.b.1 | $18$ | $2$ | $2$ | $2$ | $0$ |
| 24.72.3.ug.1 | $24$ | $3$ | $3$ | $3$ | $0$ |
| 72.108.5.d.1 | $72$ | $2$ | $2$ | $5$ | $?$ |
| 72.108.6.c.1 | $72$ | $2$ | $2$ | $6$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 72.432.21.e.1 | $72$ | $2$ | $2$ | $21$ |
| 72.432.21.ck.1 | $72$ | $2$ | $2$ | $21$ |
| 72.432.21.ir.1 | $72$ | $2$ | $2$ | $21$ |
| 72.432.21.iu.1 | $72$ | $2$ | $2$ | $21$ |
| 72.432.21.wm.1 | $72$ | $2$ | $2$ | $21$ |
| 72.432.21.wn.1 | $72$ | $2$ | $2$ | $21$ |
| 72.432.21.wt.1 | $72$ | $2$ | $2$ | $21$ |
| 72.432.21.wu.1 | $72$ | $2$ | $2$ | $21$ |