Invariants
| Level: | $9$ | $\SL_2$-level: | $9$ | ||||
| Index: | $72$ | $\PSL_2$-index: | $36$ | ||||
| Genus: | $0 = 1 + \frac{ 36 }{12} - \frac{ 0 }{4} - \frac{ 3 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $3$ are rational) | Cusp widths | $3^{3}\cdot9^{3}$ | Cusp orbits | $1^{3}\cdot3$ | ||
| Elliptic points: | $0$ of order $2$ and $3$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $3$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 9J0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 9.72.0.14 |
Level structure
| $\GL_2(\Z/9\Z)$-generators: | $\begin{bmatrix}4&7\\6&1\end{bmatrix}$, $\begin{bmatrix}5&5\\6&4\end{bmatrix}$ |
| $\GL_2(\Z/9\Z)$-subgroup: | $C_3^2:C_6$ |
| Contains $-I$: | no $\quad$ (see 9.36.0.f.2 for the level structure with $-I$) |
| Cyclic 9-isogeny field degree: | $3$ |
| Cyclic 9-torsion field degree: | $6$ |
| Full 9-torsion field degree: | $54$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 5 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 36 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^9}\cdot\frac{x^{36}(x^{2}-6xy+12y^{2})^{3}(x^{3}-36xy^{2}+72y^{3})(x^{9}-18x^{8}y+108x^{7}y^{2}-120x^{6}y^{3}-1296x^{5}y^{4}+5760x^{4}y^{5}-10368x^{3}y^{6}+14976x^{2}y^{7}-27648xy^{8}+26112y^{9})^{3}}{y^{9}x^{36}(x-4y)^{9}(x-2y)^{9}(x^{3}-6x^{2}y+24y^{3})^{3}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 9.24.0-9.b.1.1 | $9$ | $3$ | $3$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.