Invariants
| Level: | $72$ | $\SL_2$-level: | $36$ | Newform level: | $1$ | ||
| Index: | $48$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $4 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $12\cdot36$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 4$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 4$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 36A4 |
Level structure
| $\GL_2(\Z/72\Z)$-generators: | $\begin{bmatrix}18&61\\71&43\end{bmatrix}$, $\begin{bmatrix}58&41\\37&21\end{bmatrix}$, $\begin{bmatrix}58&51\\37&53\end{bmatrix}$, $\begin{bmatrix}59&0\\26&31\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 72.96.4-72.a.1.1, 72.96.4-72.a.1.2, 72.96.4-72.a.1.3, 72.96.4-72.a.1.4, 72.96.4-72.a.1.5, 72.96.4-72.a.1.6, 72.96.4-72.a.1.7, 72.96.4-72.a.1.8 |
| 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 |
|---|---|---|---|---|---|
| 18.24.2.a.1 | $18$ | $2$ | $2$ | $2$ | $0$ |
| 24.16.0.a.1 | $24$ | $3$ | $3$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 72.144.10.a.1 | $72$ | $3$ | $3$ | $10$ |
| 72.144.10.m.2 | $72$ | $3$ | $3$ | $10$ |
| 72.144.10.q.1 | $72$ | $3$ | $3$ | $10$ |
| 72.144.10.r.2 | $72$ | $3$ | $3$ | $10$ |
| 72.144.10.s.1 | $72$ | $3$ | $3$ | $10$ |
| 72.144.10.u.2 | $72$ | $3$ | $3$ | $10$ |
| 72.144.10.v.1 | $72$ | $3$ | $3$ | $10$ |
| 72.144.10.w.2 | $72$ | $3$ | $3$ | $10$ |
| 72.144.10.y.2 | $72$ | $3$ | $3$ | $10$ |
| 72.144.10.z.2 | $72$ | $3$ | $3$ | $10$ |
| 72.144.10.ba.1 | $72$ | $3$ | $3$ | $10$ |
| 72.144.10.bb.1 | $72$ | $3$ | $3$ | $10$ |
| 72.144.10.bc.2 | $72$ | $3$ | $3$ | $10$ |
| 72.192.13.a.1 | $72$ | $4$ | $4$ | $13$ |