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Properties

Label 36.162.10.d.1
Level $36$
Index $162$
Genus $10$
Analytic rank $0$
Cusps $9$
$\Q$-cusps $0$

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Invariants

Level: $36$ $\SL_2$-level: $36$ Newform level: $324$
Index: $162$ $\PSL_2$-index:$162$
Genus: $10 = 1 + \frac{ 162 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 9 }{2}$
Cusps: $9$ (none of which are rational) Cusp widths $9^{6}\cdot36^{3}$ Cusp orbits $3^{3}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $3 \le \gamma \le 6$
$\overline{\Q}$-gonality: $3 \le \gamma \le 6$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 36M10
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 36.162.10.1

Level structure

$\GL_2(\Z/36\Z)$-generators: $\begin{bmatrix}9&22\\28&19\end{bmatrix}$, $\begin{bmatrix}13&31\\16&3\end{bmatrix}$, $\begin{bmatrix}23&20\\16&17\end{bmatrix}$, $\begin{bmatrix}23&35\\4&21\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 36.324.10-36.d.1.1, 36.324.10-36.d.1.2, 36.324.10-36.d.1.3, 36.324.10-36.d.1.4, 72.324.10-36.d.1.1, 72.324.10-36.d.1.2, 72.324.10-36.d.1.3, 72.324.10-36.d.1.4, 72.324.10-36.d.1.5, 72.324.10-36.d.1.6, 72.324.10-36.d.1.7, 72.324.10-36.d.1.8, 72.324.10-36.d.1.9, 72.324.10-36.d.1.10, 72.324.10-36.d.1.11, 72.324.10-36.d.1.12, 180.324.10-36.d.1.1, 180.324.10-36.d.1.2, 180.324.10-36.d.1.3, 180.324.10-36.d.1.4, 252.324.10-36.d.1.1, 252.324.10-36.d.1.2, 252.324.10-36.d.1.3, 252.324.10-36.d.1.4
Cyclic 36-isogeny field degree: $12$
Cyclic 36-torsion field degree: $144$
Full 36-torsion field degree: $2304$

Jacobian

Conductor: $2^{12}\cdot3^{36}$
Simple: no
Squarefree: no
Decomposition: $1^{10}$
Newforms: 54.2.a.a$^{2}$, 54.2.a.b$^{2}$, 162.2.a.c$^{2}$, 162.2.a.d$^{2}$, 324.2.a.b, 324.2.a.c

Models

Canonical model in $\mathbb{P}^{ 9 }$ defined by 28 equations

$ 0 $ $=$ $ y^{2} + y v - y s - 2 v s - v a + s^{2} - a^{2} $
$=$ $2 x v - x s - x a - y z + y t - y u + y r - u v$
$=$ $x y + x v + 2 x a + y w - y u + y r + w v - w s - t s - u v + v r$
$=$ $x y - x v + x a - z a - w s + 2 w a + t a - u a + r s + r a$
$=$$\cdots$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 101250 x^{12} - 273375 x^{11} z - 307800 x^{10} y^{2} + 303750 x^{10} z^{2} + 883710 x^{9} y^{2} z + \cdots + 50 y^{6} z^{6} $
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Rational points

This modular curve has no $\Q_p$ points for $p=5$, and therefore no rational points.

Maps between models of this curve

Birational map from canonical model to plane model:

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle y$
$\displaystyle Z$ $=$ $\displaystyle z$

Maps to other modular curves

Map of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve 18.81.4.a.1 :

$\displaystyle X$ $=$ $\displaystyle -y$
$\displaystyle Y$ $=$ $\displaystyle -v$
$\displaystyle Z$ $=$ $\displaystyle s$
$\displaystyle W$ $=$ $\displaystyle -y-s-a$

Equation of the image curve:

$0$ $=$ $ 2XY+3XZ+YZ+2XW-YW-2ZW-W^{2} $
$=$ $ X^{3}+11X^{2}Y+17XY^{2}-3Y^{3}-7XYZ-5Y^{2}Z+8YZ^{2}-Z^{3}-4X^{2}W-6XYW-4Y^{2}W+2XZW-4Z^{2}W-2XW^{2}-2YW^{2}+2ZW^{2} $

Modular covers

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Cover information

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The following modular covers realize this modular curve as a fiber product over $X(1)$.

Factor curve Level Index Degree Genus Rank Kernel decomposition
$X_0(4)$ $4$ $27$ $27$ $0$ $0$ full Jacobian
9.27.0.a.1 $9$ $6$ $6$ $0$ $0$ full Jacobian

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
$X_0(4)$ $4$ $27$ $27$ $0$ $0$ full Jacobian
18.81.4.a.1 $18$ $2$ $2$ $4$ $0$ $1^{6}$

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
36.324.22.b.1 $36$ $2$ $2$ $22$ $2$ $1^{10}\cdot2$
36.324.22.t.1 $36$ $2$ $2$ $22$ $6$ $1^{10}\cdot2$
36.324.22.ba.1 $36$ $2$ $2$ $22$ $3$ $1^{10}\cdot2$
36.324.22.bb.1 $36$ $2$ $2$ $22$ $5$ $1^{10}\cdot2$
36.486.28.c.1 $36$ $3$ $3$ $28$ $3$ $1^{18}$
36.648.37.bd.1 $36$ $4$ $4$ $37$ $3$ $1^{21}\cdot2^{3}$
72.324.22.n.1 $72$ $2$ $2$ $22$ $?$ not computed
72.324.22.cg.1 $72$ $2$ $2$ $22$ $?$ not computed
72.324.22.da.1 $72$ $2$ $2$ $22$ $?$ not computed
72.324.22.dd.1 $72$ $2$ $2$ $22$ $?$ not computed
72.324.22.ds.1 $72$ $2$ $2$ $22$ $?$ not computed
72.324.22.dt.1 $72$ $2$ $2$ $22$ $?$ not computed
72.324.22.du.1 $72$ $2$ $2$ $22$ $?$ not computed
72.324.22.dv.1 $72$ $2$ $2$ $22$ $?$ not computed
72.324.22.dw.1 $72$ $2$ $2$ $22$ $?$ not computed
72.324.22.dx.1 $72$ $2$ $2$ $22$ $?$ not computed
72.324.22.dy.1 $72$ $2$ $2$ $22$ $?$ not computed
72.324.22.dz.1 $72$ $2$ $2$ $22$ $?$ not computed
180.324.22.ba.1 $180$ $2$ $2$ $22$ $?$ not computed
180.324.22.bb.1 $180$ $2$ $2$ $22$ $?$ not computed
180.324.22.be.1 $180$ $2$ $2$ $22$ $?$ not computed
180.324.22.bf.1 $180$ $2$ $2$ $22$ $?$ not computed
252.324.22.ba.1 $252$ $2$ $2$ $22$ $?$ not computed
252.324.22.bb.1 $252$ $2$ $2$ $22$ $?$ not computed
252.324.22.be.1 $252$ $2$ $2$ $22$ $?$ not computed
252.324.22.bf.1 $252$ $2$ $2$ $22$ $?$ not computed