Invariants
| Level: | $9$ | $\SL_2$-level: | $9$ | Newform level: | $81$ | ||
| Index: | $648$ | $\PSL_2$-index: | $324$ | ||||
| Genus: | $10 = 1 + \frac{ 324 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 36 }{2}$ | ||||||
| Cusps: | $36$ (of which $3$ are rational) | Cusp widths | $9^{36}$ | Cusp orbits | $1^{3}\cdot2^{3}\cdot3\cdot6^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $5 \le \gamma \le 6$ | ||||||
| $\overline{\Q}$-gonality: | $5 \le \gamma \le 6$ | ||||||
| Rational cusps: | $3$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 9A10 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 9.648.10.1 |
Level structure
| $\GL_2(\Z/9\Z)$-generators: | $\begin{bmatrix}5&0\\0&1\end{bmatrix}$ |
| $\GL_2(\Z/9\Z)$-subgroup: | $C_6$ |
| Contains $-I$: | no $\quad$ (see 9.324.10.a.1 for the level structure with $-I$) |
| Cyclic 9-isogeny field degree: | $1$ |
| Cyclic 9-torsion field degree: | $1$ |
| Full 9-torsion field degree: | $6$ |
Jacobian
| Conductor: | $3^{38}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{2}\cdot2^{2}\cdot4$ |
| Newforms: | 27.2.a.a$^{2}$, 81.2.a.a, 81.2.c.a, 81.2.c.b |
Models
Canonical model in $\mathbb{P}^{ 9 }$ defined by 28 equations
| $ 0 $ | $=$ | $ x s - y z - y t + z w + s^{2} $ |
| $=$ | $x y + x t + y w + y t - r a - s^{2}$ | |
| $=$ | $2 x v - x r - y u + y v + z v + z r + a^{2}$ | |
| $=$ | $x v - y u + y v + y r - z u - z v + r s + a^{2}$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 27 x^{18} - 81 x^{16} y z + 162 x^{14} y^{2} z^{2} - 243 x^{12} y^{3} z^{3} - 54 x^{12} z^{6} + \cdots - y^{3} z^{15} $ |
Rational points
This modular curve has 3 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(-1:-1:-1:0:1:-1:0:-1:0:1)$, $(-1:1:1:1:0:1:-1:0:0:1)$, $(-1:0:0:-1:-1:0:1:1:0:1)$ |
Maps to other modular curves
Map of degree 3 from the canonical model of this modular curve to the canonical model of the modular curve 9.108.4.a.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -y-z+w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w+t$ |
| $\displaystyle W$ | $=$ | $\displaystyle -u-v+r$ |
Equation of the image curve:
| $0$ | $=$ | $ 3X^{2}-Y^{2}+YZ-Z^{2} $ |
| $=$ | $ 3X^{3}+2XY^{2}+Y^{3}+3X^{2}Z-2XYZ+2Y^{2}Z+2XZ^{2}-5YZ^{2}-W^{3} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 9.324.10.a.1 :
| $\displaystyle X$ | $=$ | $\displaystyle r$ |
| $\displaystyle Y$ | $=$ | $\displaystyle s$ |
| $\displaystyle Z$ | $=$ | $\displaystyle a$ |
Equation of the image curve:
| $0$ | $=$ | $ 27X^{18}-81X^{16}YZ+162X^{14}Y^{2}Z^{2}-243X^{12}Y^{3}Z^{3}-54X^{10}Y^{4}Z^{4}-324X^{11}Y^{2}Z^{5}+135X^{8}Y^{5}Z^{5}-54X^{12}Z^{6}+18X^{9}Y^{3}Z^{6}-216X^{10}YZ^{7}+216X^{7}Y^{4}Z^{7}-27X^{4}Y^{7}Z^{7}+216X^{8}Y^{2}Z^{8}-108X^{5}Y^{5}Z^{8}+9X^{2}Y^{8}Z^{8}-81X^{6}Y^{3}Z^{9}+18X^{3}Y^{6}Z^{9}-Y^{9}Z^{9}+27X^{4}Y^{4}Z^{10}-9X^{2}Y^{5}Z^{11}+27X^{6}Z^{12}+Y^{6}Z^{12}-27X^{4}YZ^{13}+9X^{2}Y^{2}Z^{14}-Y^{3}Z^{15} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 9.216.1-9.a.1.1 | $9$ | $3$ | $3$ | $1$ | $0$ | $1\cdot2^{2}\cdot4$ |
| 9.216.1-9.a.2.1 | $9$ | $3$ | $3$ | $1$ | $0$ | $1\cdot2^{2}\cdot4$ |
| 9.216.4-9.a.1.1 | $9$ | $3$ | $3$ | $4$ | $0$ | $1^{2}\cdot4$ |
| 9.216.4-9.b.1.1 | $9$ | $3$ | $3$ | $4$ | $0$ | $1^{2}\cdot2^{2}$ |
| 9.216.4-9.c.1.1 | $9$ | $3$ | $3$ | $4$ | $0$ | $2\cdot4$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 18.1296.37-18.h.1.1 | $18$ | $2$ | $2$ | $37$ | $1$ | $1^{9}\cdot2^{9}$ |
| 18.1296.37-18.u.1.1 | $18$ | $2$ | $2$ | $37$ | $1$ | $1^{9}\cdot2^{9}$ |
| 18.1944.46-18.a.1.1 | $18$ | $3$ | $3$ | $46$ | $1$ | $1^{10}\cdot2^{11}\cdot4$ |
| 27.1944.46-27.a.1.1 | $27$ | $3$ | $3$ | $46$ | $0$ | $12^{3}$ |
| 27.1944.46-27.e.1.2 | $27$ | $3$ | $3$ | $46$ | $0$ | $12^{3}$ |
| 27.1944.46-27.g.1.1 | $27$ | $3$ | $3$ | $46$ | $6$ | $6^{2}\cdot12^{2}$ |
| 27.1944.46-27.h.1.1 | $27$ | $3$ | $3$ | $46$ | $0$ | $12^{3}$ |
| 27.1944.55-27.g.1.1 | $27$ | $3$ | $3$ | $55$ | $4$ | $1^{3}\cdot2^{6}\cdot3^{2}\cdot4^{3}\cdot6^{2}$ |
| 27.1944.55-27.g.2.1 | $27$ | $3$ | $3$ | $55$ | $4$ | $1^{3}\cdot2^{6}\cdot3^{2}\cdot4^{3}\cdot6^{2}$ |
| 27.1944.64-27.c.1.1 | $27$ | $3$ | $3$ | $64$ | $6$ | $6^{3}\cdot12^{3}$ |
| 36.1296.37-36.bf.1.3 | $36$ | $2$ | $2$ | $37$ | $7$ | $1^{9}\cdot2^{9}$ |
| 36.1296.37-36.bw.1.1 | $36$ | $2$ | $2$ | $37$ | $7$ | $1^{9}\cdot2^{9}$ |
| 36.2592.91-36.ez.1.1 | $36$ | $4$ | $4$ | $91$ | $9$ | $1^{19}\cdot2^{17}\cdot4^{7}$ |
| 45.3240.118-45.a.1.1 | $45$ | $5$ | $5$ | $118$ | $15$ | $1^{16}\cdot2^{10}\cdot4^{6}\cdot8^{6}$ |
| 45.3888.127-45.a.1.1 | $45$ | $6$ | $6$ | $127$ | $10$ | $1^{19}\cdot2^{21}\cdot3^{2}\cdot4^{5}\cdot6^{5}$ |
| 45.6480.235-45.e.1.1 | $45$ | $10$ | $10$ | $235$ | $29$ | $1^{35}\cdot2^{31}\cdot3^{2}\cdot4^{11}\cdot6^{5}\cdot8^{6}$ |
| 63.5184.181-63.e.1.2 | $63$ | $8$ | $8$ | $181$ | $14$ | $1^{19}\cdot2^{17}\cdot3^{6}\cdot4^{5}\cdot6^{12}\cdot8$ |
| 63.13608.514-63.a.1.1 | $63$ | $21$ | $21$ | $514$ | $69$ | $1^{18}\cdot2^{27}\cdot3^{4}\cdot4^{17}\cdot5^{2}\cdot6^{2}\cdot8^{9}\cdot10^{5}\cdot12^{2}\cdot16^{2}\cdot24^{5}\cdot32$ |
| 63.18144.685-63.e.1.1 | $63$ | $28$ | $28$ | $685$ | $83$ | $1^{37}\cdot2^{44}\cdot3^{10}\cdot4^{22}\cdot5^{2}\cdot6^{14}\cdot8^{10}\cdot10^{5}\cdot12^{2}\cdot16^{2}\cdot24^{5}\cdot32$ |