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Properties

Label 36.144.8.f.2
Level $36$
Index $144$
Genus $8$
Analytic rank $0$
Cusps $10$
$\Q$-cusps $4$

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Invariants

Level: $36$ $\SL_2$-level: $36$ Newform level: $108$
Index: $144$ $\PSL_2$-index:$144$
Genus: $8 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$
Cusps: $10$ (of which $4$ are rational) Cusp widths $3^{2}\cdot6\cdot9^{2}\cdot12^{2}\cdot18\cdot36^{2}$ Cusp orbits $1^{4}\cdot2^{3}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $3$
$\overline{\Q}$-gonality: $3$
Rational cusps: $4$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 36J8
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 36.144.8.9

Level structure

$\GL_2(\Z/36\Z)$-generators: $\begin{bmatrix}5&4\\24&23\end{bmatrix}$, $\begin{bmatrix}5&33\\0&29\end{bmatrix}$, $\begin{bmatrix}23&6\\0&11\end{bmatrix}$, $\begin{bmatrix}29&24\\0&7\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 36.288.8-36.f.2.1, 36.288.8-36.f.2.2, 36.288.8-36.f.2.3, 36.288.8-36.f.2.4, 36.288.8-36.f.2.5, 36.288.8-36.f.2.6, 36.288.8-36.f.2.7, 36.288.8-36.f.2.8, 72.288.8-36.f.2.1, 72.288.8-36.f.2.2, 72.288.8-36.f.2.3, 72.288.8-36.f.2.4, 72.288.8-36.f.2.5, 72.288.8-36.f.2.6, 72.288.8-36.f.2.7, 72.288.8-36.f.2.8, 72.288.8-36.f.2.9, 72.288.8-36.f.2.10, 72.288.8-36.f.2.11, 72.288.8-36.f.2.12, 72.288.8-36.f.2.13, 72.288.8-36.f.2.14, 72.288.8-36.f.2.15, 72.288.8-36.f.2.16, 72.288.8-36.f.2.17, 72.288.8-36.f.2.18, 72.288.8-36.f.2.19, 72.288.8-36.f.2.20, 72.288.8-36.f.2.21, 72.288.8-36.f.2.22, 72.288.8-36.f.2.23, 72.288.8-36.f.2.24, 180.288.8-36.f.2.1, 180.288.8-36.f.2.2, 180.288.8-36.f.2.3, 180.288.8-36.f.2.4, 180.288.8-36.f.2.5, 180.288.8-36.f.2.6, 180.288.8-36.f.2.7, 180.288.8-36.f.2.8, 252.288.8-36.f.2.1, 252.288.8-36.f.2.2, 252.288.8-36.f.2.3, 252.288.8-36.f.2.4, 252.288.8-36.f.2.5, 252.288.8-36.f.2.6, 252.288.8-36.f.2.7, 252.288.8-36.f.2.8
Cyclic 36-isogeny field degree: $3$
Cyclic 36-torsion field degree: $36$
Full 36-torsion field degree: $2592$

Jacobian

Conductor: $2^{12}\cdot3^{24}$
Simple: no
Squarefree: no
Decomposition: $1^{4}\cdot4$
Newforms: 54.2.a.a$^{2}$, 54.2.a.b$^{2}$, 108.2.b.a

Models

Canonical model in $\mathbb{P}^{ 7 }$ defined by 20 equations

$ 0 $ $=$ $ y v - z u $
$=$ $x v - y v - w u$
$=$ $x v + y t$
$=$ $x u - x r - w t$
$=$$\cdots$
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Singular plane model Singular plane model

$ 0 $ $=$ $ - 9 x^{8} y^{3} + x^{7} z^{4} - 18 x^{6} y^{3} z^{2} - x^{5} z^{6} - 8 x^{4} y^{3} z^{4} + \cdots + y^{3} z^{8} $
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Rational points

This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Canonical model
$(0:0:0:0:0:1:0:1)$, $(0:0:0:0:1:-1:1:1)$, $(0:0:0:0:-1:-1:-1:1)$, $(0:0:1:0:0:0:0:0)$

Maps between models of this curve

Birational map from canonical model to plane model:

$\displaystyle X$ $=$ $\displaystyle u$
$\displaystyle Y$ $=$ $\displaystyle \frac{1}{4}w$
$\displaystyle Z$ $=$ $\displaystyle v$

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 36.72.4.f.1 :

$\displaystyle X$ $=$ $\displaystyle -x$
$\displaystyle Y$ $=$ $\displaystyle y$
$\displaystyle Z$ $=$ $\displaystyle -t$
$\displaystyle W$ $=$ $\displaystyle v$

Equation of the image curve:

$0$ $=$ $ YZ+XW $
$=$ $ X^{3}-9XY^{2}-Z^{2}W+W^{3} $

Modular covers

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Cover information

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This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
12.48.0.c.3 $12$ $3$ $3$ $0$ $0$ full Jacobian
36.72.4.f.1 $36$ $2$ $2$ $4$ $0$ $4$

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
36.288.17.b.3 $36$ $2$ $2$ $17$ $0$ $1^{5}\cdot4$
36.288.17.j.2 $36$ $2$ $2$ $17$ $2$ $1^{5}\cdot4$
36.288.17.k.2 $36$ $2$ $2$ $17$ $0$ $1^{5}\cdot4$
36.288.17.n.1 $36$ $2$ $2$ $17$ $2$ $1^{5}\cdot4$
36.432.22.o.4 $36$ $3$ $3$ $22$ $0$ $1^{6}\cdot2^{2}\cdot4$
36.432.22.s.1 $36$ $3$ $3$ $22$ $0$ $2^{3}\cdot8$
36.432.22.t.1 $36$ $3$ $3$ $22$ $0$ $2^{3}\cdot8$
36.432.22.u.3 $36$ $3$ $3$ $22$ $0$ $1^{6}\cdot8$
72.288.17.bf.4 $72$ $2$ $2$ $17$ $?$ not computed
72.288.17.ch.4 $72$ $2$ $2$ $17$ $?$ not computed
72.288.17.ej.4 $72$ $2$ $2$ $17$ $?$ not computed
72.288.17.en.4 $72$ $2$ $2$ $17$ $?$ not computed
72.288.17.eo.2 $72$ $2$ $2$ $17$ $?$ not computed
72.288.17.er.2 $72$ $2$ $2$ $17$ $?$ not computed
72.288.17.es.3 $72$ $2$ $2$ $17$ $?$ not computed
72.288.17.ev.3 $72$ $2$ $2$ $17$ $?$ not computed
72.288.17.ex.3 $72$ $2$ $2$ $17$ $?$ not computed
72.288.17.ey.3 $72$ $2$ $2$ $17$ $?$ not computed
72.288.17.fb.2 $72$ $2$ $2$ $17$ $?$ not computed
72.288.17.fc.2 $72$ $2$ $2$ $17$ $?$ not computed
72.288.19.ld.2 $72$ $2$ $2$ $19$ $?$ not computed
72.288.19.le.2 $72$ $2$ $2$ $19$ $?$ not computed
72.288.19.lh.3 $72$ $2$ $2$ $19$ $?$ not computed
72.288.19.li.3 $72$ $2$ $2$ $19$ $?$ not computed
72.288.19.lk.3 $72$ $2$ $2$ $19$ $?$ not computed
72.288.19.ln.3 $72$ $2$ $2$ $19$ $?$ not computed
72.288.19.lo.2 $72$ $2$ $2$ $19$ $?$ not computed
72.288.19.lr.2 $72$ $2$ $2$ $19$ $?$ not computed
180.288.17.ba.3 $180$ $2$ $2$ $17$ $?$ not computed
180.288.17.bb.3 $180$ $2$ $2$ $17$ $?$ not computed
180.288.17.bc.4 $180$ $2$ $2$ $17$ $?$ not computed
180.288.17.bd.3 $180$ $2$ $2$ $17$ $?$ not computed
252.288.17.ba.4 $252$ $2$ $2$ $17$ $?$ not computed
252.288.17.bb.4 $252$ $2$ $2$ $17$ $?$ not computed
252.288.17.bc.2 $252$ $2$ $2$ $17$ $?$ not computed
252.288.17.bd.4 $252$ $2$ $2$ $17$ $?$ not computed
252.432.22.dc.2 $252$ $3$ $3$ $22$ $?$ not computed
252.432.22.dd.2 $252$ $3$ $3$ $22$ $?$ not computed
252.432.22.dg.2 $252$ $3$ $3$ $22$ $?$ not computed
252.432.22.dh.2 $252$ $3$ $3$ $22$ $?$ not computed
252.432.22.dk.3 $252$ $3$ $3$ $22$ $?$ not computed
252.432.22.dl.3 $252$ $3$ $3$ $22$ $?$ not computed