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Properties

Label 9.36.1.a.1
Level $9$
Index $36$
Genus $1$
Analytic rank $0$
Cusps $6$
$\Q$-cusps $2$

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Invariants

Level: $9$ $\SL_2$-level: $9$ Newform level: $27$
Index: $36$ $\PSL_2$-index:$36$
Genus: $1 = 1 + \frac{ 36 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$
Cusps: $6$ (of which $2$ are rational) Cusp widths $3^{3}\cdot9^{3}$ Cusp orbits $1^{2}\cdot2^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $2$
Rational CM points: yes $\quad(D =$ $-3$)

Other labels

Cummins and Pauli (CP) label: 9C1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 9.36.1.1

Level structure

$\GL_2(\Z/9\Z)$-generators: $\begin{bmatrix}2&6\\0&8\end{bmatrix}$, $\begin{bmatrix}5&0\\0&7\end{bmatrix}$
$\GL_2(\Z/9\Z)$-subgroup: $C_3^2\times D_6$
Contains $-I$: yes
Quadratic refinements: 9.72.1-9.a.1.1, 9.72.1-9.a.1.2, 18.72.1-9.a.1.1, 18.72.1-9.a.1.2, 36.72.1-9.a.1.1, 36.72.1-9.a.1.2, 36.72.1-9.a.1.3, 36.72.1-9.a.1.4, 45.72.1-9.a.1.1, 45.72.1-9.a.1.2, 63.72.1-9.a.1.1, 63.72.1-9.a.1.2, 72.72.1-9.a.1.1, 72.72.1-9.a.1.2, 72.72.1-9.a.1.3, 72.72.1-9.a.1.4, 72.72.1-9.a.1.5, 72.72.1-9.a.1.6, 72.72.1-9.a.1.7, 72.72.1-9.a.1.8, 90.72.1-9.a.1.1, 90.72.1-9.a.1.2, 99.72.1-9.a.1.1, 99.72.1-9.a.1.2, 117.72.1-9.a.1.1, 117.72.1-9.a.1.2, 126.72.1-9.a.1.1, 126.72.1-9.a.1.2, 153.72.1-9.a.1.1, 153.72.1-9.a.1.2, 171.72.1-9.a.1.1, 171.72.1-9.a.1.2, 180.72.1-9.a.1.1, 180.72.1-9.a.1.2, 180.72.1-9.a.1.3, 180.72.1-9.a.1.4, 198.72.1-9.a.1.1, 198.72.1-9.a.1.2, 207.72.1-9.a.1.1, 207.72.1-9.a.1.2, 234.72.1-9.a.1.1, 234.72.1-9.a.1.2, 252.72.1-9.a.1.1, 252.72.1-9.a.1.2, 252.72.1-9.a.1.3, 252.72.1-9.a.1.4, 261.72.1-9.a.1.1, 261.72.1-9.a.1.2, 279.72.1-9.a.1.1, 279.72.1-9.a.1.2, 306.72.1-9.a.1.1, 306.72.1-9.a.1.2, 315.72.1-9.a.1.1, 315.72.1-9.a.1.2, 333.72.1-9.a.1.1, 333.72.1-9.a.1.2
Cyclic 9-isogeny field degree: $1$
Cyclic 9-torsion field degree: $6$
Full 9-torsion field degree: $108$

Jacobian

Conductor: $3^{3}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 27.2.a.a

Models

Weierstrass model Weierstrass model

$ y^{2} + y $ $=$ $ x^{3} - 7 $
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Rational points

This modular curve has 2 rational cusps and 1 rational CM point, but no other known rational points. The following are the known rational points on this modular curve (one row per $j$-invariant).

Elliptic curve CM $j$-invariant $j$-heightWeierstrass model
27.a3 $-3$$0$ $0.000$$(3:-5:1)$
no$\infty$ $0.000$$(3:4:1)$, $(0:1:0)$

Maps to other modular curves

$j$-invariant map of degree 36 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{(y+5z)^{12}(y^{3}+231y^{2}z+291yz^{2}+1637z^{3})^{3}}{z(y-4z)^{9}(y+5z)^{9}(y^{2}+yz+7z^{2})}$

Modular covers

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Cover information

Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
$X_{\mathrm{sp}}(3)$ $3$ $3$ $3$ $0$ $0$ full Jacobian
$X_0(9)$ $9$ $3$ $3$ $0$ $0$ full Jacobian
9.12.0.b.1 $9$ $3$ $3$ $0$ $0$ full Jacobian
9.12.1.a.1 $9$ $3$ $3$ $1$ $0$ dimension zero

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve Level Index Degree Genus Rank Kernel decomposition
9.108.1.a.1 $9$ $3$ $3$ $1$ $0$ dimension zero
9.108.1.a.2 $9$ $3$ $3$ $1$ $0$ dimension zero
9.108.1.b.1 $9$ $3$ $3$ $1$ $0$ dimension zero
$X_{\mathrm{sp}}(9)$ $9$ $3$ $3$ $4$ $0$ $1\cdot2$
18.72.4.c.1 $18$ $2$ $2$ $4$ $0$ $1^{3}$
18.72.4.m.1 $18$ $2$ $2$ $4$ $0$ $1^{3}$
18.108.4.c.1 $18$ $3$ $3$ $4$ $0$ $1^{3}$
27.108.4.a.1 $27$ $3$ $3$ $4$ $0$ $1\cdot2$
27.108.7.a.1 $27$ $3$ $3$ $7$ $1$ $1^{2}\cdot2^{2}$
27.108.7.b.1 $27$ $3$ $3$ $7$ $3$ $3^{2}$
36.72.4.h.1 $36$ $2$ $2$ $4$ $1$ $1^{3}$
36.72.4.r.1 $36$ $2$ $2$ $4$ $1$ $1^{3}$
36.144.10.w.1 $36$ $4$ $4$ $10$ $1$ $1^{9}$
45.180.13.a.1 $45$ $5$ $5$ $13$ $3$ $1^{10}\cdot2$
45.216.13.a.1 $45$ $6$ $6$ $13$ $1$ $1^{8}\cdot2^{2}$
45.360.25.a.1 $45$ $10$ $10$ $25$ $6$ $1^{18}\cdot2^{3}$
63.108.1.a.1 $63$ $3$ $3$ $1$ $0$ dimension zero
63.108.1.a.2 $63$ $3$ $3$ $1$ $0$ dimension zero
63.108.1.b.1 $63$ $3$ $3$ $1$ $0$ dimension zero
63.108.1.b.2 $63$ $3$ $3$ $1$ $0$ dimension zero
63.108.1.c.1 $63$ $3$ $3$ $1$ $0$ dimension zero
63.108.1.c.2 $63$ $3$ $3$ $1$ $0$ dimension zero
63.288.19.a.1 $63$ $8$ $8$ $19$ $2$ $1^{10}\cdot2^{4}$
63.756.55.a.1 $63$ $21$ $21$ $55$ $20$ $1^{10}\cdot2^{11}\cdot3^{2}\cdot4^{4}$
63.1008.73.a.1 $63$ $28$ $28$ $73$ $22$ $1^{20}\cdot2^{15}\cdot3^{2}\cdot4^{4}$
72.72.4.i.1 $72$ $2$ $2$ $4$ $?$ not computed
72.72.4.j.1 $72$ $2$ $2$ $4$ $?$ not computed
72.72.4.y.1 $72$ $2$ $2$ $4$ $?$ not computed
72.72.4.z.1 $72$ $2$ $2$ $4$ $?$ not computed
90.72.4.a.1 $90$ $2$ $2$ $4$ $?$ not computed
90.72.4.b.1 $90$ $2$ $2$ $4$ $?$ not computed
117.108.1.a.1 $117$ $3$ $3$ $1$ $?$ dimension zero
117.108.1.a.2 $117$ $3$ $3$ $1$ $?$ dimension zero
117.108.1.b.1 $117$ $3$ $3$ $1$ $?$ dimension zero
117.108.1.b.2 $117$ $3$ $3$ $1$ $?$ dimension zero
117.108.1.c.1 $117$ $3$ $3$ $1$ $?$ dimension zero
117.108.1.c.2 $117$ $3$ $3$ $1$ $?$ dimension zero
126.72.4.m.1 $126$ $2$ $2$ $4$ $?$ not computed
126.72.4.p.1 $126$ $2$ $2$ $4$ $?$ not computed
171.108.1.a.1 $171$ $3$ $3$ $1$ $?$ dimension zero
171.108.1.a.2 $171$ $3$ $3$ $1$ $?$ dimension zero
171.108.1.b.1 $171$ $3$ $3$ $1$ $?$ dimension zero
171.108.1.b.2 $171$ $3$ $3$ $1$ $?$ dimension zero
171.108.1.c.1 $171$ $3$ $3$ $1$ $?$ dimension zero
171.108.1.c.2 $171$ $3$ $3$ $1$ $?$ dimension zero
180.72.4.a.1 $180$ $2$ $2$ $4$ $?$ not computed
180.72.4.b.1 $180$ $2$ $2$ $4$ $?$ not computed
198.72.4.b.1 $198$ $2$ $2$ $4$ $?$ not computed
198.72.4.c.1 $198$ $2$ $2$ $4$ $?$ not computed
234.72.4.n.1 $234$ $2$ $2$ $4$ $?$ not computed
234.72.4.p.1 $234$ $2$ $2$ $4$ $?$ not computed
252.72.4.f.1 $252$ $2$ $2$ $4$ $?$ not computed
252.72.4.i.1 $252$ $2$ $2$ $4$ $?$ not computed
279.108.1.a.1 $279$ $3$ $3$ $1$ $?$ dimension zero
279.108.1.a.2 $279$ $3$ $3$ $1$ $?$ dimension zero
279.108.1.b.1 $279$ $3$ $3$ $1$ $?$ dimension zero
279.108.1.b.2 $279$ $3$ $3$ $1$ $?$ dimension zero
279.108.1.c.1 $279$ $3$ $3$ $1$ $?$ dimension zero
279.108.1.c.2 $279$ $3$ $3$ $1$ $?$ dimension zero
306.72.4.a.1 $306$ $2$ $2$ $4$ $?$ not computed
306.72.4.b.1 $306$ $2$ $2$ $4$ $?$ not computed
333.108.1.a.1 $333$ $3$ $3$ $1$ $?$ dimension zero
333.108.1.a.2 $333$ $3$ $3$ $1$ $?$ dimension zero
333.108.1.b.1 $333$ $3$ $3$ $1$ $?$ dimension zero
333.108.1.b.2 $333$ $3$ $3$ $1$ $?$ dimension zero
333.108.1.c.1 $333$ $3$ $3$ $1$ $?$ dimension zero
333.108.1.c.2 $333$ $3$ $3$ $1$ $?$ dimension zero