Invariants
| Level: | $72$ | $\SL_2$-level: | $36$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 12 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $12\cdot36$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $12$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 36A0 |
Level structure
| $\GL_2(\Z/72\Z)$-generators: | $\begin{bmatrix}1&53\\61&24\end{bmatrix}$, $\begin{bmatrix}17&66\\18&41\end{bmatrix}$, $\begin{bmatrix}60&65\\71&45\end{bmatrix}$, $\begin{bmatrix}71&64\\18&7\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 72.96.0-72.a.1.1, 72.96.0-72.a.1.2, 72.96.0-72.a.1.3, 72.96.0-72.a.1.4, 72.96.0-72.a.1.5, 72.96.0-72.a.1.6, 72.96.0-72.a.1.7, 72.96.0-72.a.1.8 |
| Cyclic 72-isogeny field degree: | $36$ |
| Cyclic 72-torsion field degree: | $864$ |
| Full 72-torsion field degree: | $124416$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points but none with conductor small enough to be contained within the database of elliptic curves over $\Q$.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 18.24.0.b.1 | $18$ | $2$ | $2$ | $0$ | $0$ |
| 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 |
|---|---|---|---|---|
| 72.144.6.a.2 | $72$ | $3$ | $3$ | $6$ |
| 72.144.6.c.1 | $72$ | $3$ | $3$ | $6$ |
| 72.144.6.d.2 | $72$ | $3$ | $3$ | $6$ |
| 72.144.6.e.1 | $72$ | $3$ | $3$ | $6$ |
| 72.144.6.g.2 | $72$ | $3$ | $3$ | $6$ |
| 72.144.6.h.1 | $72$ | $3$ | $3$ | $6$ |
| 72.144.6.i.2 | $72$ | $3$ | $3$ | $6$ |
| 72.144.6.j.1 | $72$ | $3$ | $3$ | $6$ |
| 72.144.6.k.2 | $72$ | $3$ | $3$ | $6$ |
| 72.144.10.b.2 | $72$ | $3$ | $3$ | $10$ |
| 72.144.10.m.2 | $72$ | $3$ | $3$ | $10$ |
| 72.144.10.o.2 | $72$ | $3$ | $3$ | $10$ |
| 72.144.10.p.1 | $72$ | $3$ | $3$ | $10$ |
| 72.192.9.c.1 | $72$ | $4$ | $4$ | $9$ |
| 216.144.4.a.2 | $216$ | $3$ | $3$ | $4$ |
| 216.144.8.a.2 | $216$ | $3$ | $3$ | $8$ |
| 216.144.12.a.2 | $216$ | $3$ | $3$ | $12$ |