↑

Properties

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

Related objects

Downloads

Learn more

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 $6^{4}\cdot12\cdot18^{4}\cdot36$ 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: 36H8
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 36.144.8.3

Level structure

$\GL_2(\Z/36\Z)$-generators: $\begin{bmatrix}1&32\\30&31\end{bmatrix}$, $\begin{bmatrix}5&12\\18&1\end{bmatrix}$, $\begin{bmatrix}5&26\\6&1\end{bmatrix}$, $\begin{bmatrix}7&28\\12&17\end{bmatrix}$, $\begin{bmatrix}35&32\\30&23\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 36.288.8-36.b.2.1, 36.288.8-36.b.2.2, 36.288.8-36.b.2.3, 36.288.8-36.b.2.4, 36.288.8-36.b.2.5, 36.288.8-36.b.2.6, 36.288.8-36.b.2.7, 36.288.8-36.b.2.8, 36.288.8-36.b.2.9, 36.288.8-36.b.2.10, 36.288.8-36.b.2.11, 36.288.8-36.b.2.12, 36.288.8-36.b.2.13, 36.288.8-36.b.2.14, 36.288.8-36.b.2.15, 36.288.8-36.b.2.16, 72.288.8-36.b.2.1, 72.288.8-36.b.2.2, 72.288.8-36.b.2.3, 72.288.8-36.b.2.4, 72.288.8-36.b.2.5, 72.288.8-36.b.2.6, 72.288.8-36.b.2.7, 72.288.8-36.b.2.8, 72.288.8-36.b.2.9, 72.288.8-36.b.2.10, 72.288.8-36.b.2.11, 72.288.8-36.b.2.12, 72.288.8-36.b.2.13, 72.288.8-36.b.2.14, 72.288.8-36.b.2.15, 72.288.8-36.b.2.16, 180.288.8-36.b.2.1, 180.288.8-36.b.2.2, 180.288.8-36.b.2.3, 180.288.8-36.b.2.4, 180.288.8-36.b.2.5, 180.288.8-36.b.2.6, 180.288.8-36.b.2.7, 180.288.8-36.b.2.8, 180.288.8-36.b.2.9, 180.288.8-36.b.2.10, 180.288.8-36.b.2.11, 180.288.8-36.b.2.12, 180.288.8-36.b.2.13, 180.288.8-36.b.2.14, 180.288.8-36.b.2.15, 180.288.8-36.b.2.16, 252.288.8-36.b.2.1, 252.288.8-36.b.2.2, 252.288.8-36.b.2.3, 252.288.8-36.b.2.4, 252.288.8-36.b.2.5, 252.288.8-36.b.2.6, 252.288.8-36.b.2.7, 252.288.8-36.b.2.8, 252.288.8-36.b.2.9, 252.288.8-36.b.2.10, 252.288.8-36.b.2.11, 252.288.8-36.b.2.12, 252.288.8-36.b.2.13, 252.288.8-36.b.2.14, 252.288.8-36.b.2.15, 252.288.8-36.b.2.16
Cyclic 36-isogeny field degree: $6$
Cyclic 36-torsion field degree: $72$
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 $ $=$ $ z r + w v $
$=$ $t v + t r - u r$
$=$ $z t - w t + w u$
$=$ $x v + w t + w u$
$=$$\cdots$
Copy content Toggle raw display

Singular plane model Singular plane model

$ 0 $ $=$ $ - 9 x^{8} y^{3} + 6 x^{6} y^{3} z^{2} + 8 x^{4} y^{3} z^{4} + 3 x^{3} z^{8} - 6 x^{2} y^{3} z^{6} + \cdots + y^{3} z^{8} $
 Copy content Toggle raw display

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
$(0:0:0:0:0:0:0:1)$, $(0:0:0:0:-1:1:0:0)$, $(0:0:-1/3:1/3:0:0:1:1)$, $(0:0:1/3:-1/3:0:0:1:1)$

Maps between models of this curve

Birational map from canonical model to plane model:

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle \frac{3}{2}z$
$\displaystyle Z$ $=$ $\displaystyle t$

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 18.72.4.b.1 :

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle -x+y$
$\displaystyle Z$ $=$ $\displaystyle z-w$
$\displaystyle W$ $=$ $\displaystyle -w$

Equation of the image curve:

$0$ $=$ $ XZ-XW-YW $
$=$ $ X^{2}Y+XY^{2}+2Z^{3}-3Z^{2}W-3ZW^{2}+2W^{3} $

Modular covers

Sorry, your browser does not support the nearby lattice.

Cover information

Click on a modular curve in the diagram to see information about it.
See a full page version of the diagram

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
12.48.0.a.2 $12$ $3$ $3$ $0$ $0$ full Jacobian
18.72.4.b.1 $18$ $2$ $2$ $4$ $0$ $4$

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.a.1 $36$ $2$ $2$ $17$ $2$ $1^{5}\cdot4$
36.288.17.b.1 $36$ $2$ $2$ $17$ $0$ $1^{5}\cdot4$
36.288.17.b.4 $36$ $2$ $2$ $17$ $0$ $1^{5}\cdot4$
36.288.17.c.1 $36$ $2$ $2$ $17$ $2$ $1^{5}\cdot4$
36.288.17.c.2 $36$ $2$ $2$ $17$ $2$ $1^{5}\cdot4$
36.288.17.d.1 $36$ $2$ $2$ $17$ $0$ $1^{5}\cdot4$
36.288.17.d.2 $36$ $2$ $2$ $17$ $0$ $1^{5}\cdot4$
36.288.19.o.2 $36$ $2$ $2$ $19$ $2$ $1^{5}\cdot2\cdot4$
36.288.19.p.2 $36$ $2$ $2$ $19$ $0$ $1^{5}\cdot2\cdot4$
36.288.19.q.1 $36$ $2$ $2$ $19$ $2$ $1^{5}\cdot2\cdot4$
36.288.19.r.1 $36$ $2$ $2$ $19$ $0$ $1^{5}\cdot2\cdot4$
36.432.22.a.1 $36$ $3$ $3$ $22$ $0$ $1^{6}\cdot2^{2}\cdot4$
36.432.22.e.1 $36$ $3$ $3$ $22$ $0$ $2^{3}\cdot8$
36.432.22.f.2 $36$ $3$ $3$ $22$ $0$ $2^{3}\cdot8$
36.432.22.g.1 $36$ $3$ $3$ $22$ $0$ $1^{6}\cdot8$
72.288.17.j.2 $72$ $2$ $2$ $17$ $?$ not computed
72.288.17.j.4 $72$ $2$ $2$ $17$ $?$ not computed
72.288.17.l.2 $72$ $2$ $2$ $17$ $?$ not computed
72.288.17.l.4 $72$ $2$ $2$ $17$ $?$ not computed
72.288.17.n.2 $72$ $2$ $2$ $17$ $?$ not computed
72.288.17.n.4 $72$ $2$ $2$ $17$ $?$ not computed
72.288.17.p.2 $72$ $2$ $2$ $17$ $?$ not computed
72.288.17.p.4 $72$ $2$ $2$ $17$ $?$ not computed
72.288.19.cj.2 $72$ $2$ $2$ $19$ $?$ not computed
72.288.19.cm.2 $72$ $2$ $2$ $19$ $?$ not computed
72.288.19.cp.2 $72$ $2$ $2$ $19$ $?$ not computed
72.288.19.cs.2 $72$ $2$ $2$ $19$ $?$ not computed
180.288.17.m.2 $180$ $2$ $2$ $17$ $?$ not computed
180.288.17.m.4 $180$ $2$ $2$ $17$ $?$ not computed
180.288.17.n.2 $180$ $2$ $2$ $17$ $?$ not computed
180.288.17.n.4 $180$ $2$ $2$ $17$ $?$ not computed
180.288.17.o.1 $180$ $2$ $2$ $17$ $?$ not computed
180.288.17.o.3 $180$ $2$ $2$ $17$ $?$ not computed
180.288.17.p.2 $180$ $2$ $2$ $17$ $?$ not computed
180.288.17.p.3 $180$ $2$ $2$ $17$ $?$ not computed
180.288.19.bk.1 $180$ $2$ $2$ $19$ $?$ not computed
180.288.19.bl.2 $180$ $2$ $2$ $19$ $?$ not computed
180.288.19.bm.1 $180$ $2$ $2$ $19$ $?$ not computed
180.288.19.bn.1 $180$ $2$ $2$ $19$ $?$ not computed
252.288.17.m.1 $252$ $2$ $2$ $17$ $?$ not computed
252.288.17.m.4 $252$ $2$ $2$ $17$ $?$ not computed
252.288.17.n.1 $252$ $2$ $2$ $17$ $?$ not computed
252.288.17.n.3 $252$ $2$ $2$ $17$ $?$ not computed
252.288.17.o.3 $252$ $2$ $2$ $17$ $?$ not computed
252.288.17.o.4 $252$ $2$ $2$ $17$ $?$ not computed
252.288.17.p.2 $252$ $2$ $2$ $17$ $?$ not computed
252.288.17.p.4 $252$ $2$ $2$ $17$ $?$ not computed
252.288.19.bk.2 $252$ $2$ $2$ $19$ $?$ not computed
252.288.19.bl.2 $252$ $2$ $2$ $19$ $?$ not computed
252.288.19.bm.1 $252$ $2$ $2$ $19$ $?$ not computed
252.288.19.bn.2 $252$ $2$ $2$ $19$ $?$ not computed
252.432.22.q.1 $252$ $3$ $3$ $22$ $?$ not computed
252.432.22.r.1 $252$ $3$ $3$ $22$ $?$ not computed
252.432.22.s.1 $252$ $3$ $3$ $22$ $?$ not computed
252.432.22.t.1 $252$ $3$ $3$ $22$ $?$ not computed
252.432.22.u.1 $252$ $3$ $3$ $22$ $?$ not computed
252.432.22.v.1 $252$ $3$ $3$ $22$ $?$ not computed