Invariants
| Level: | $72$ | $\SL_2$-level: | $36$ | Newform level: | $1$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $3 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $4$ are rational) | Cusp widths | $2^{3}\cdot4^{3}\cdot18\cdot36$ | Cusp orbits | $1^{4}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 3$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 3$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 36G3 |
Level structure
| $\GL_2(\Z/72\Z)$-generators: | $\begin{bmatrix}36&5\\5&12\end{bmatrix}$, $\begin{bmatrix}44&53\\19&60\end{bmatrix}$, $\begin{bmatrix}48&49\\13&54\end{bmatrix}$, $\begin{bmatrix}68&57\\67&16\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 72.72.3.cn.1 for the level structure with $-I$) |
| Cyclic 72-isogeny field degree: | $4$ |
| Cyclic 72-torsion field degree: | $96$ |
| Full 72-torsion field degree: | $41472$ |
Rational points
This modular curve has 4 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 |
|---|---|---|---|---|---|
| 24.48.1-24.er.1.10 | $24$ | $3$ | $3$ | $1$ | $0$ |
| 36.72.0-18.a.1.12 | $36$ | $2$ | $2$ | $0$ | $0$ |
| 72.72.0-18.a.1.3 | $72$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 72.288.5-72.c.1.11 | $72$ | $2$ | $2$ | $5$ |
| 72.288.5-72.g.1.4 | $72$ | $2$ | $2$ | $5$ |
| 72.288.5-72.s.1.7 | $72$ | $2$ | $2$ | $5$ |
| 72.288.5-72.t.1.7 | $72$ | $2$ | $2$ | $5$ |
| 72.288.5-72.y.1.12 | $72$ | $2$ | $2$ | $5$ |
| 72.288.5-72.z.1.6 | $72$ | $2$ | $2$ | $5$ |
| 72.288.5-72.bh.1.17 | $72$ | $2$ | $2$ | $5$ |
| 72.288.5-72.bi.1.7 | $72$ | $2$ | $2$ | $5$ |
| 72.432.11-72.ij.1.15 | $72$ | $3$ | $3$ | $11$ |
| 72.432.11-72.il.1.7 | $72$ | $3$ | $3$ | $11$ |
| 72.432.11-72.il.2.15 | $72$ | $3$ | $3$ | $11$ |
| 72.432.13-72.cs.1.8 | $72$ | $3$ | $3$ | $13$ |
| 216.432.11-216.f.1.15 | $216$ | $3$ | $3$ | $11$ |
| 216.432.13-216.bd.1.16 | $216$ | $3$ | $3$ | $13$ |
| 216.432.15-216.b.1.2 | $216$ | $3$ | $3$ | $15$ |