Invariants
| Level: | $72$ | $\SL_2$-level: | $36$ | 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 $2$ are rational) | Cusp widths | $6^{3}\cdot18^{3}$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 4$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 4$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 18D4 |
Level structure
| $\GL_2(\Z/72\Z)$-generators: | $\begin{bmatrix}29&54\\0&11\end{bmatrix}$, $\begin{bmatrix}48&65\\71&42\end{bmatrix}$, $\begin{bmatrix}59&52\\66&49\end{bmatrix}$, $\begin{bmatrix}66&71\\71&0\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 72.72.4.s.1 for the level structure with $-I$) |
| Cyclic 72-isogeny field degree: | $12$ |
| Cyclic 72-torsion field degree: | $288$ |
| Full 72-torsion field degree: | $41472$ |
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 |
|---|---|---|---|---|---|
| 24.48.0-24.cb.1.6 | $24$ | $3$ | $3$ | $0$ | $0$ |
| 36.72.2-18.c.1.3 | $36$ | $2$ | $2$ | $2$ | $0$ |
| 72.72.2-18.c.1.2 | $72$ | $2$ | $2$ | $2$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 72.288.9-72.cg.1.2 | $72$ | $2$ | $2$ | $9$ |
| 72.288.9-72.ci.1.4 | $72$ | $2$ | $2$ | $9$ |
| 72.288.9-72.cs.1.10 | $72$ | $2$ | $2$ | $9$ |
| 72.288.9-72.cu.1.8 | $72$ | $2$ | $2$ | $9$ |
| 72.288.9-72.ez.1.1 | $72$ | $2$ | $2$ | $9$ |
| 72.288.9-72.fa.1.4 | $72$ | $2$ | $2$ | $9$ |
| 72.288.9-72.fl.1.6 | $72$ | $2$ | $2$ | $9$ |
| 72.288.9-72.fm.1.10 | $72$ | $2$ | $2$ | $9$ |
| 72.432.10-72.ba.1.14 | $72$ | $3$ | $3$ | $10$ |
| 72.432.10-72.ba.2.10 | $72$ | $3$ | $3$ | $10$ |
| 72.432.10-72.bk.1.16 | $72$ | $3$ | $3$ | $10$ |
| 72.432.10-72.bo.1.6 | $72$ | $3$ | $3$ | $10$ |