Modular curve 72.324.10-36.d.1.5

• Label: 72.324.10-36.d.1.5
• Level: $72$
• Index: $324$
• Genus: $10$
• Cusps: $9$
• $\Q$-cusps: $0$

Invariants

Level:	$72$		$\SL_2$-level:	$72$		Newform level:	$324$
Index:	$324$		$\PSL_2$-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:	not computed
$\Q$-gonality:	$3 \le \gamma \le 18$
$\overline{\Q}$-gonality:	$3 \le \gamma \le 10$
Rational cusps:	$0$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:

36M10

Level structure

$\GL_2(\Z/72\Z)$-generators:	$\begin{bmatrix}18&1\\19&4\end{bmatrix}$, $\begin{bmatrix}50&5\\69&58\end{bmatrix}$, $\begin{bmatrix}51&64\\68&3\end{bmatrix}$, $\begin{bmatrix}56&61\\11&54\end{bmatrix}$, $\begin{bmatrix}67&2\\32&41\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 36.162.10.d.1 for the level structure with $-I$)
Cyclic 72-isogeny field degree:	$24$
Cyclic 72-torsion field degree:	$576$
Full 72-torsion field degree:	$18432$

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} $
	$=$	$2 x v - x s - x a - y z + y t - y u + y r - u v$
	$=$	$x y + x v + 2 x a + y w - y u + y r + w v - w s - t s - u v + v r$
	$=$	$x y - x v + x a - z a - w s + 2 w a + t a - u a + r s + r a$
	$=$	$\cdots$

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} $

Rational points

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

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} $

Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 36.162.10.d.1 :

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

Equation of the image curve:

$0$

$=$

$ 101250X^{12}-273375X^{11}Z-307800X^{10}Y^{2}+303750X^{10}Z^{2}+883710X^{9}Y^{2}Z-162405X^{9}Z^{3}-37908X^{8}Y^{4}-924777X^{8}Y^{2}Z^{2}+18468X^{8}Z^{4}+424008X^{7}Y^{4}Z+346545X^{7}Y^{2}Z^{3}+26568X^{7}Z^{5}-217224X^{6}Y^{6}-786807X^{6}Y^{4}Z^{2}+57645X^{6}Y^{2}Z^{4}-13122X^{6}Z^{6}+501714X^{5}Y^{6}Z+509058X^{5}Y^{4}Z^{3}-68499X^{5}Y^{2}Z^{5}+729X^{5}Z^{7}+20898X^{4}Y^{8}-404703X^{4}Y^{6}Z^{2}-70425X^{4}Y^{4}Z^{4}+7623X^{4}Y^{2}Z^{6}+810X^{4}Z^{8}-42201X^{3}Y^{8}Z+141189X^{3}Y^{6}Z^{3}-42372X^{3}Y^{4}Z^{5}+2853X^{3}Y^{2}Z^{7}-150X^{3}Z^{9}+12240X^{2}Y^{10}+12969X^{2}Y^{8}Z^{2}-14769X^{2}Y^{6}Z^{4}+10809X^{2}Y^{4}Z^{6}-450X^{2}Y^{2}Z^{8}-9312XY^{10}Z+7533XY^{8}Z^{3}-3393XY^{6}Z^{5}+150XY^{4}Z^{7}-2048Y^{12}+2448Y^{10}Z^{2}-1191Y^{8}Z^{4}+50Y^{6}Z^{6} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
24.12.0-4.c.1.5	$24$	$27$	$27$	$0$	$0$