Invariants
| Level: | $9$ | $\SL_2$-level: | $9$ | Newform level: | $81$ | ||
| Index: | $216$ | $\PSL_2$-index: | $108$ | ||||
| Genus: | $4 = 1 + \frac{ 108 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (of which $3$ are rational) | Cusp widths | $9^{12}$ | Cusp orbits | $1^{3}\cdot3\cdot6$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $3$ | ||||||
| $\overline{\Q}$-gonality: | $3$ | ||||||
| Rational cusps: | $3$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 9A4 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 9.216.4.1 |
Level structure
| $\GL_2(\Z/9\Z)$-generators: | $\begin{bmatrix}1&6\\3&1\end{bmatrix}$, $\begin{bmatrix}5&0\\0&1\end{bmatrix}$ |
| $\GL_2(\Z/9\Z)$-subgroup: | $C_3\times S_3$ |
| Contains $-I$: | no $\quad$ (see 9.108.4.b.1 for the level structure with $-I$) |
| Cyclic 9-isogeny field degree: | $3$ |
| Cyclic 9-torsion field degree: | $3$ |
| Full 9-torsion field degree: | $18$ |
Jacobian
| Conductor: | $3^{16}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $4$ |
| Newforms: | 81.2.c.b |
Models
Canonical model in $\mathbb{P}^{ 3 }$
| $ 0 $ | $=$ | $ x y - x z - 2 y w - z w $ |
| $=$ | $2 x^{3} - 3 x^{2} w - 3 x w^{2} - y^{3} - 6 y^{2} z - 3 y z^{2} + z^{3} + 2 w^{3}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - x^{3} y^{3} + 6 x^{3} z^{3} - 9 x^{2} z^{4} + 3 x y^{3} z^{2} - 9 x z^{5} - y^{3} z^{3} + 6 z^{6} $ |
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 |
|---|
| $(1/2:0:0:1)$, $(2:0:0:1)$, $(-1:0:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 108 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{3^3}{2^9}\cdot\frac{1844658653184x^{2}z^{15}w+472111349760x^{2}z^{12}w^{4}+35340862464x^{2}z^{9}w^{7}+2166563520x^{2}z^{6}w^{10}-5199192x^{2}z^{3}w^{13}+786429x^{2}w^{16}+58114847232xz^{15}w^{2}-971059968xz^{12}w^{5}+7335004896xz^{9}w^{8}-303023088xz^{6}w^{11}+35105454xz^{3}w^{14}-786429xw^{17}-1971240834048y^{3}z^{15}-542911712512y^{3}z^{12}w^{3}-52162252800y^{3}z^{9}w^{6}-582867024y^{3}z^{6}w^{9}-100464260y^{3}z^{3}w^{12}+917505y^{3}w^{15}+811107846144y^{2}z^{16}+403823078400y^{2}z^{13}w^{3}+37487015424y^{2}z^{10}w^{6}+11963676096y^{2}z^{7}w^{9}-690686328y^{2}z^{4}w^{12}+14876676y^{2}zw^{15}+1524392838144yz^{17}-668798323968yz^{14}w^{3}-289323668736yz^{11}w^{6}-4269012624yz^{8}w^{9}-2659307676yz^{5}w^{12}+72613881yz^{2}w^{15}-364269333504z^{18}-1879794109184z^{15}w^{3}-479899540992z^{12}w^{6}-28422660144z^{9}w^{9}-2466122908z^{6}w^{12}+42401799z^{3}w^{15}+196614w^{18}}{z^{2}(2448x^{2}z^{13}w+227628x^{2}z^{10}w^{4}+1053243x^{2}z^{7}w^{7}+444672x^{2}z^{4}w^{10}+9216x^{2}zw^{13}+31032xz^{13}w^{2}+1016334xz^{10}w^{5}+2513349xz^{7}w^{8}+623232xz^{4}w^{11}+6912xzw^{14}+48y^{3}z^{13}+21704y^{3}z^{10}w^{3}+224535y^{3}z^{7}w^{6}+204672y^{3}z^{4}w^{9}+12928y^{3}zw^{12}+288y^{2}z^{14}+139080y^{2}z^{11}w^{3}+1575900y^{2}z^{8}w^{6}+1611648y^{2}z^{5}w^{9}+136320y^{2}z^{2}w^{12}+144yz^{15}+129408yz^{12}w^{3}+2456415yz^{9}w^{6}+4242816yz^{6}w^{9}+720768yz^{3}w^{12}+3456yw^{15}-48z^{16}+13720z^{13}w^{3}+684537z^{10}w^{6}+1404042z^{7}w^{9}+177152z^{4}w^{12}-2304zw^{15})}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 9.108.4.b.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y-z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ -X^{3}Y^{3}+6X^{3}Z^{3}-9X^{2}Z^{4}+3XY^{3}Z^{2}-9XZ^{5}-Y^{3}Z^{3}+6Z^{6} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 9.72.0-9.a.1.1 | $9$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
| 9.72.0-9.b.1.1 | $9$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{\mathrm{arith}}(9)$ | $9$ | $3$ | $3$ | $10$ | $0$ | $1^{2}\cdot2^{2}$ |
| 18.432.13-18.e.1.1 | $18$ | $2$ | $2$ | $13$ | $1$ | $1^{3}\cdot2^{3}$ |
| 18.432.13-18.m.1.1 | $18$ | $2$ | $2$ | $13$ | $1$ | $1^{3}\cdot2^{3}$ |
| 18.648.16-18.b.1.1 | $18$ | $3$ | $3$ | $16$ | $0$ | $1^{2}\cdot2^{3}\cdot4$ |
| 27.648.16-27.b.1.1 | $27$ | $3$ | $3$ | $16$ | $0$ | $12$ |
| 27.648.16-27.f.1.2 | $27$ | $3$ | $3$ | $16$ | $0$ | $12$ |
| 27.648.16-27.h.1.1 | $27$ | $3$ | $3$ | $16$ | $0$ | $12$ |
| 36.432.13-36.k.1.1 | $36$ | $2$ | $2$ | $13$ | $3$ | $1^{3}\cdot2^{3}$ |
| 36.432.13-36.y.1.1 | $36$ | $2$ | $2$ | $13$ | $3$ | $1^{3}\cdot2^{3}$ |
| 36.864.31-36.dr.1.1 | $36$ | $4$ | $4$ | $31$ | $3$ | $1^{5}\cdot2^{5}\cdot4^{3}$ |
| 45.1080.40-45.b.1.1 | $45$ | $5$ | $5$ | $40$ | $5$ | $1^{4}\cdot4^{4}\cdot8^{2}$ |
| 45.1296.43-45.b.1.1 | $45$ | $6$ | $6$ | $43$ | $4$ | $1^{5}\cdot2^{5}\cdot3^{2}\cdot4^{3}\cdot6$ |
| 45.2160.79-45.b.1.1 | $45$ | $10$ | $10$ | $79$ | $9$ | $1^{9}\cdot2^{5}\cdot3^{2}\cdot4^{7}\cdot6\cdot8^{2}$ |
| 63.1728.61-63.b.1.2 | $63$ | $8$ | $8$ | $61$ | $6$ | $1^{3}\cdot2^{3}\cdot3^{4}\cdot4\cdot6^{4}\cdot8$ |
| 63.4536.172-63.b.1.1 | $63$ | $21$ | $21$ | $172$ | $23$ | $1^{6}\cdot2^{7}\cdot5^{2}\cdot8^{4}\cdot10\cdot12^{2}\cdot16\cdot24\cdot32$ |
| 63.6048.229-63.b.1.1 | $63$ | $28$ | $28$ | $229$ | $29$ | $1^{9}\cdot2^{10}\cdot3^{4}\cdot4\cdot5^{2}\cdot6^{4}\cdot8^{5}\cdot10\cdot12^{2}\cdot16\cdot24\cdot32$ |
| 72.432.13-72.be.1.1 | $72$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 72.432.13-72.bf.1.1 | $72$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 72.432.13-72.cw.1.1 | $72$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 72.432.13-72.cx.1.1 | $72$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 90.432.13-90.o.1.1 | $90$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 90.432.13-90.q.1.1 | $90$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 126.432.13-126.y.1.1 | $126$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 126.432.13-126.ba.1.1 | $126$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 180.432.13-180.ba.1.1 | $180$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 180.432.13-180.bc.1.1 | $180$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 198.432.13-198.g.1.1 | $198$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 198.432.13-198.i.1.1 | $198$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 234.432.13-234.y.1.1 | $234$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 234.432.13-234.ba.1.1 | $234$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 252.432.13-252.be.1.1 | $252$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 252.432.13-252.bg.1.1 | $252$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 306.432.13-306.i.1.1 | $306$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 306.432.13-306.k.1.1 | $306$ | $2$ | $2$ | $13$ | $?$ | not computed |