Invariants
| Level: | $9$ | $\SL_2$-level: | $9$ | ||||
| Index: | $18$ | $\PSL_2$-index: | $18$ | ||||
| Genus: | $0 = 1 + \frac{ 18 }{12} - \frac{ 6 }{4} - \frac{ 0 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $9^{2}$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $6$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-8,-11$) |
Other labels
| Cummins and Pauli (CP) label: | 9D0 |
| Sutherland and Zywina (SZ) label: | 9D0-9a |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 9.18.0.2 |
Level structure
| $\GL_2(\Z/9\Z)$-generators: | $\begin{bmatrix}3&7\\1&3\end{bmatrix}$, $\begin{bmatrix}8&0\\0&7\end{bmatrix}$ |
| $\GL_2(\Z/9\Z)$-subgroup: | $C_3^3:D_4$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 9-isogeny field degree: | $6$ |
| Cyclic 9-torsion field degree: | $36$ |
| Full 9-torsion field degree: | $216$ |
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 18 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{18}(x^{3}-9y^{3})^{3}(x^{3}+3y^{3})^{3}}{y^{9}x^{27}}$ |
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 |
|---|---|---|---|---|---|
| $X_{\mathrm{sp}}^+(3)$ | $3$ | $3$ | $3$ | $0$ | $0$ |
| 9.9.0.a.1 | $9$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 9.36.2.a.1 | $9$ | $2$ | $2$ | $2$ |
| $X_{\mathrm{sp}}^+(9)$ | $9$ | $3$ | $3$ | $1$ |
| 18.36.0.b.1 | $18$ | $2$ | $2$ | $0$ |
| 18.36.3.e.1 | $18$ | $2$ | $2$ | $3$ |
| 18.36.3.i.1 | $18$ | $2$ | $2$ | $3$ |
| 18.54.2.e.1 | $18$ | $3$ | $3$ | $2$ |
| 36.36.0.c.1 | $36$ | $2$ | $2$ | $0$ |
| 36.36.2.b.1 | $36$ | $2$ | $2$ | $2$ |
| 36.36.3.h.1 | $36$ | $2$ | $2$ | $3$ |
| 36.36.3.p.1 | $36$ | $2$ | $2$ | $3$ |
| 36.72.3.bc.1 | $36$ | $4$ | $4$ | $3$ |
| 45.36.2.a.1 | $45$ | $2$ | $2$ | $2$ |
| 45.90.6.g.1 | $45$ | $5$ | $5$ | $6$ |
| 45.108.5.f.1 | $45$ | $6$ | $6$ | $5$ |
| 45.180.11.h.1 | $45$ | $10$ | $10$ | $11$ |
| 63.36.2.a.1 | $63$ | $2$ | $2$ | $2$ |
| 63.144.11.m.1 | $63$ | $8$ | $8$ | $11$ |
| 63.378.22.h.1 | $63$ | $21$ | $21$ | $22$ |
| 63.504.33.i.1 | $63$ | $28$ | $28$ | $33$ |
| 72.36.0.e.1 | $72$ | $2$ | $2$ | $0$ |
| 72.36.0.f.1 | $72$ | $2$ | $2$ | $0$ |
| 72.36.2.c.1 | $72$ | $2$ | $2$ | $2$ |
| 72.36.2.d.1 | $72$ | $2$ | $2$ | $2$ |
| 72.36.3.v.1 | $72$ | $2$ | $2$ | $3$ |
| 72.36.3.bb.1 | $72$ | $2$ | $2$ | $3$ |
| 72.36.3.bx.1 | $72$ | $2$ | $2$ | $3$ |
| 72.36.3.cd.1 | $72$ | $2$ | $2$ | $3$ |
| 90.36.0.a.1 | $90$ | $2$ | $2$ | $0$ |
| 90.36.3.o.1 | $90$ | $2$ | $2$ | $3$ |
| 90.36.3.p.1 | $90$ | $2$ | $2$ | $3$ |
| 99.36.2.a.1 | $99$ | $2$ | $2$ | $2$ |
| 99.216.17.e.1 | $99$ | $12$ | $12$ | $17$ |
| 117.36.2.a.1 | $117$ | $2$ | $2$ | $2$ |
| 117.252.17.n.1 | $117$ | $14$ | $14$ | $17$ |
| 126.36.0.a.1 | $126$ | $2$ | $2$ | $0$ |
| 126.36.3.bd.1 | $126$ | $2$ | $2$ | $3$ |
| 126.36.3.be.1 | $126$ | $2$ | $2$ | $3$ |
| 153.36.2.a.1 | $153$ | $2$ | $2$ | $2$ |
| 153.324.23.h.1 | $153$ | $18$ | $18$ | $23$ |
| 171.36.2.a.1 | $171$ | $2$ | $2$ | $2$ |
| 180.36.0.c.1 | $180$ | $2$ | $2$ | $0$ |
| 180.36.2.c.1 | $180$ | $2$ | $2$ | $2$ |
| 180.36.3.bk.1 | $180$ | $2$ | $2$ | $3$ |
| 180.36.3.bn.1 | $180$ | $2$ | $2$ | $3$ |
| 198.36.0.a.1 | $198$ | $2$ | $2$ | $0$ |
| 198.36.3.o.1 | $198$ | $2$ | $2$ | $3$ |
| 198.36.3.p.1 | $198$ | $2$ | $2$ | $3$ |
| 207.36.2.a.1 | $207$ | $2$ | $2$ | $2$ |
| 234.36.0.a.1 | $234$ | $2$ | $2$ | $0$ |
| 234.36.3.bd.1 | $234$ | $2$ | $2$ | $3$ |
| 234.36.3.be.1 | $234$ | $2$ | $2$ | $3$ |
| 252.36.0.c.1 | $252$ | $2$ | $2$ | $0$ |
| 252.36.2.c.1 | $252$ | $2$ | $2$ | $2$ |
| 252.36.3.bw.1 | $252$ | $2$ | $2$ | $3$ |
| 252.36.3.bz.1 | $252$ | $2$ | $2$ | $3$ |
| 261.36.2.a.1 | $261$ | $2$ | $2$ | $2$ |
| 279.36.2.a.1 | $279$ | $2$ | $2$ | $2$ |
| 306.36.0.a.1 | $306$ | $2$ | $2$ | $0$ |
| 306.36.3.o.1 | $306$ | $2$ | $2$ | $3$ |
| 306.36.3.p.1 | $306$ | $2$ | $2$ | $3$ |
| 315.36.2.a.1 | $315$ | $2$ | $2$ | $2$ |
| 333.36.2.a.1 | $333$ | $2$ | $2$ | $2$ |