Invariants
| Level: | $72$ | $\SL_2$-level: | $18$ | ||||
| Index: | $72$ | $\PSL_2$-index: | $36$ | ||||
| Genus: | $0 = 1 + \frac{ 36 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $3^{6}\cdot9^{2}$ | Cusp orbits | $2\cdot3^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 9H0 |
Level structure
| $\GL_2(\Z/72\Z)$-generators: | $\begin{bmatrix}9&44\\47&45\end{bmatrix}$, $\begin{bmatrix}34&69\\39&52\end{bmatrix}$, $\begin{bmatrix}47&54\\36&65\end{bmatrix}$, $\begin{bmatrix}57&41\\47&6\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 9.36.0.c.1 for the level structure with $-I$) |
| Cyclic 72-isogeny field degree: | $36$ |
| Cyclic 72-torsion field degree: | $864$ |
| Full 72-torsion field degree: | $82944$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 4 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 36 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 3^9\,\frac{y^{3}x^{3}(3x-y)^{3}(3x+y)^{3}(3x+2y)^{39}(6x+y)^{3}(2916x^{6}+7290x^{5}y+5589x^{4}y^{2}+756x^{3}y^{3}-189x^{2}y^{4}-18xy^{5}+4y^{6})^{3}}{(3x+2y)^{36}(9x^{2}+3xy+y^{2})^{9}(27x^{3}+27x^{2}y-y^{3})^{3}(27x^{3}+54x^{2}y+9xy^{2}-y^{3})^{3}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.24.0-3.a.1.4 | $24$ | $3$ | $3$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 72.144.3-18.e.1.1 | $72$ | $2$ | $2$ | $3$ |
| 72.144.3-36.g.1.1 | $72$ | $2$ | $2$ | $3$ |
| 72.144.3-18.n.1.1 | $72$ | $2$ | $2$ | $3$ |
| 72.144.3-36.q.1.1 | $72$ | $2$ | $2$ | $3$ |
| 72.144.3-72.r.1.1 | $72$ | $2$ | $2$ | $3$ |
| 72.144.3-72.x.1.3 | $72$ | $2$ | $2$ | $3$ |
| 72.144.3-72.by.1.4 | $72$ | $2$ | $2$ | $3$ |
| 72.144.3-72.ce.1.1 | $72$ | $2$ | $2$ | $3$ |
| 72.216.1-9.b.1.5 | $72$ | $3$ | $3$ | $1$ |
| 72.216.1-9.b.1.7 | $72$ | $3$ | $3$ | $1$ |
| 72.216.2-18.c.1.3 | $72$ | $3$ | $3$ | $2$ |
| 72.216.4-9.d.1.3 | $72$ | $3$ | $3$ | $4$ |
| 72.216.4-9.e.1.3 | $72$ | $3$ | $3$ | $4$ |
| 72.288.9-36.cl.1.1 | $72$ | $4$ | $4$ | $9$ |