Invariants
| Level: | $9$ | $\SL_2$-level: | $9$ | ||||
| Index: | $9$ | $\PSL_2$-index: | $9$ | ||||
| Genus: | $0 = 1 + \frac{ 9 }{12} - \frac{ 5 }{4} - \frac{ 0 }{3} - \frac{ 1 }{2}$ | ||||||
| Cusps: | $1$ (which is rational) | Cusp widths | $9$ | Cusp orbits | $1$ | ||
| Elliptic points: | $5$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $1$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-8,-11$) |
Other labels
| Cummins and Pauli (CP) label: | 9A0 |
| Sutherland and Zywina (SZ) label: | 9A0-9a |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 9.9.0.1 |
Level structure
| $\GL_2(\Z/9\Z)$-generators: | $\begin{bmatrix}7&4\\1&2\end{bmatrix}$, $\begin{bmatrix}8&8\\1&8\end{bmatrix}$ |
| $\GL_2(\Z/9\Z)$-subgroup: | $C_3^3:\SD_{16}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 9-isogeny field degree: | $12$ |
| Cyclic 9-torsion field degree: | $72$ |
| Full 9-torsion field degree: | $432$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 15 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 9 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^9}\cdot\frac{x^{9}(x^{3}+36xy^{2}-48y^{3})^{3}}{y^{9}x^{9}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(3)$ | $3$ | $3$ | $3$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 9.18.0.b.1 | $9$ | $2$ | $2$ | $0$ |
| 9.18.0.d.1 | $9$ | $2$ | $2$ | $0$ |
| 9.81.1.a.1 | $9$ | $9$ | $9$ | $1$ |
| 18.18.1.b.1 | $18$ | $2$ | $2$ | $1$ |
| 18.18.1.d.1 | $18$ | $2$ | $2$ | $1$ |
| 18.18.2.a.1 | $18$ | $2$ | $2$ | $2$ |
| 18.18.2.b.1 | $18$ | $2$ | $2$ | $2$ |
| 18.27.1.a.1 | $18$ | $3$ | $3$ | $1$ |
| 36.18.0.a.1 | $36$ | $2$ | $2$ | $0$ |
| 36.18.0.b.1 | $36$ | $2$ | $2$ | $0$ |
| 36.18.1.a.1 | $36$ | $2$ | $2$ | $1$ |
| 36.18.1.b.1 | $36$ | $2$ | $2$ | $1$ |
| 36.18.2.a.1 | $36$ | $2$ | $2$ | $2$ |
| 36.18.2.b.1 | $36$ | $2$ | $2$ | $2$ |
| 36.36.1.e.1 | $36$ | $4$ | $4$ | $1$ |
| 45.18.0.a.1 | $45$ | $2$ | $2$ | $0$ |
| 45.18.0.b.1 | $45$ | $2$ | $2$ | $0$ |
| 45.45.3.a.1 | $45$ | $5$ | $5$ | $3$ |
| 45.54.2.c.1 | $45$ | $6$ | $6$ | $2$ |
| 45.90.5.a.1 | $45$ | $10$ | $10$ | $5$ |
| 63.18.0.d.1 | $63$ | $2$ | $2$ | $0$ |
| 63.18.0.e.1 | $63$ | $2$ | $2$ | $0$ |
| 63.72.6.a.1 | $63$ | $8$ | $8$ | $6$ |
| 63.189.9.a.1 | $63$ | $21$ | $21$ | $9$ |
| 63.252.15.a.1 | $63$ | $28$ | $28$ | $15$ |
| 72.18.0.a.1 | $72$ | $2$ | $2$ | $0$ |
| 72.18.0.b.1 | $72$ | $2$ | $2$ | $0$ |
| 72.18.0.c.1 | $72$ | $2$ | $2$ | $0$ |
| 72.18.0.d.1 | $72$ | $2$ | $2$ | $0$ |
| 72.18.1.a.1 | $72$ | $2$ | $2$ | $1$ |
| 72.18.1.b.1 | $72$ | $2$ | $2$ | $1$ |
| 72.18.1.c.1 | $72$ | $2$ | $2$ | $1$ |
| 72.18.1.d.1 | $72$ | $2$ | $2$ | $1$ |
| 72.18.2.a.1 | $72$ | $2$ | $2$ | $2$ |
| 72.18.2.b.1 | $72$ | $2$ | $2$ | $2$ |
| 72.18.2.c.1 | $72$ | $2$ | $2$ | $2$ |
| 72.18.2.d.1 | $72$ | $2$ | $2$ | $2$ |
| 90.18.1.a.1 | $90$ | $2$ | $2$ | $1$ |
| 90.18.1.b.1 | $90$ | $2$ | $2$ | $1$ |
| 90.18.2.a.1 | $90$ | $2$ | $2$ | $2$ |
| 90.18.2.b.1 | $90$ | $2$ | $2$ | $2$ |
| 99.18.0.a.1 | $99$ | $2$ | $2$ | $0$ |
| 99.18.0.b.1 | $99$ | $2$ | $2$ | $0$ |
| 99.108.9.a.1 | $99$ | $12$ | $12$ | $9$ |
| 117.18.0.d.1 | $117$ | $2$ | $2$ | $0$ |
| 117.18.0.e.1 | $117$ | $2$ | $2$ | $0$ |
| 117.126.8.a.1 | $117$ | $14$ | $14$ | $8$ |
| 126.18.1.c.1 | $126$ | $2$ | $2$ | $1$ |
| 126.18.1.d.1 | $126$ | $2$ | $2$ | $1$ |
| 126.18.2.a.1 | $126$ | $2$ | $2$ | $2$ |
| 126.18.2.b.1 | $126$ | $2$ | $2$ | $2$ |
| 153.18.0.a.1 | $153$ | $2$ | $2$ | $0$ |
| 153.18.0.b.1 | $153$ | $2$ | $2$ | $0$ |
| 153.162.11.a.1 | $153$ | $18$ | $18$ | $11$ |
| 171.18.0.d.1 | $171$ | $2$ | $2$ | $0$ |
| 171.18.0.e.1 | $171$ | $2$ | $2$ | $0$ |
| 171.180.15.a.1 | $171$ | $20$ | $20$ | $15$ |
| 180.18.0.a.1 | $180$ | $2$ | $2$ | $0$ |
| 180.18.0.b.1 | $180$ | $2$ | $2$ | $0$ |
| 180.18.1.a.1 | $180$ | $2$ | $2$ | $1$ |
| 180.18.1.b.1 | $180$ | $2$ | $2$ | $1$ |
| 180.18.2.a.1 | $180$ | $2$ | $2$ | $2$ |
| 180.18.2.b.1 | $180$ | $2$ | $2$ | $2$ |
| 198.18.1.a.1 | $198$ | $2$ | $2$ | $1$ |
| 198.18.1.b.1 | $198$ | $2$ | $2$ | $1$ |
| 198.18.2.a.1 | $198$ | $2$ | $2$ | $2$ |
| 198.18.2.b.1 | $198$ | $2$ | $2$ | $2$ |
| 207.18.0.a.1 | $207$ | $2$ | $2$ | $0$ |
| 207.18.0.b.1 | $207$ | $2$ | $2$ | $0$ |
| 207.216.18.a.1 | $207$ | $24$ | $24$ | $18$ |
| 234.18.1.c.1 | $234$ | $2$ | $2$ | $1$ |
| 234.18.1.d.1 | $234$ | $2$ | $2$ | $1$ |
| 234.18.2.a.1 | $234$ | $2$ | $2$ | $2$ |
| 234.18.2.b.1 | $234$ | $2$ | $2$ | $2$ |
| 252.18.0.a.1 | $252$ | $2$ | $2$ | $0$ |
| 252.18.0.b.1 | $252$ | $2$ | $2$ | $0$ |
| 252.18.1.a.1 | $252$ | $2$ | $2$ | $1$ |
| 252.18.1.b.1 | $252$ | $2$ | $2$ | $1$ |
| 252.18.2.a.1 | $252$ | $2$ | $2$ | $2$ |
| 252.18.2.b.1 | $252$ | $2$ | $2$ | $2$ |
| 261.18.0.a.1 | $261$ | $2$ | $2$ | $0$ |
| 261.18.0.b.1 | $261$ | $2$ | $2$ | $0$ |
| 261.270.20.a.1 | $261$ | $30$ | $30$ | $20$ |
| 279.18.0.d.1 | $279$ | $2$ | $2$ | $0$ |
| 279.18.0.e.1 | $279$ | $2$ | $2$ | $0$ |
| 279.288.24.a.1 | $279$ | $32$ | $32$ | $24$ |
| 306.18.1.a.1 | $306$ | $2$ | $2$ | $1$ |
| 306.18.1.b.1 | $306$ | $2$ | $2$ | $1$ |
| 306.18.2.a.1 | $306$ | $2$ | $2$ | $2$ |
| 306.18.2.b.1 | $306$ | $2$ | $2$ | $2$ |
| 315.18.0.a.1 | $315$ | $2$ | $2$ | $0$ |
| 315.18.0.b.1 | $315$ | $2$ | $2$ | $0$ |
| 333.18.0.d.1 | $333$ | $2$ | $2$ | $0$ |
| 333.18.0.e.1 | $333$ | $2$ | $2$ | $0$ |