Modular curve 80.360.10-40.cj.1.2

• Label: 80.360.10-40.cj.1.2
• Level: $80$
• Index: $360$
• Genus: $10$
• Cusps: $12$
• $\Q$-cusps: $4$

Invariants

Level:	$80$		$\SL_2$-level:	$80$		Newform level:	$200$
Index:	$360$		$\PSL_2$-index:	$180$
Genus:	$10 = 1 + \frac{ 180 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$
Cusps:	$12$ (of which $4$ are rational)		Cusp widths	$5^{6}\cdot10^{3}\cdot40^{3}$		Cusp orbits	$1^{4}\cdot2^{4}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	not computed
$\Q$-gonality:	$5 \le \gamma \le 10$
$\overline{\Q}$-gonality:	$5 \le \gamma \le 10$
Rational cusps:	$4$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:

40D10

Level structure

$\GL_2(\Z/80\Z)$-generators:	$\begin{bmatrix}2&45\\15&8\end{bmatrix}$, $\begin{bmatrix}8&5\\35&26\end{bmatrix}$, $\begin{bmatrix}15&72\\32&15\end{bmatrix}$, $\begin{bmatrix}22&15\\35&66\end{bmatrix}$, $\begin{bmatrix}30&3\\29&20\end{bmatrix}$, $\begin{bmatrix}43&20\\50&53\end{bmatrix}$, $\begin{bmatrix}60&43\\27&20\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 40.180.10.cj.1 for the level structure with $-I$)
Cyclic 80-isogeny field degree:	$4$
Cyclic 80-torsion field degree:	$64$
Full 80-torsion field degree:	$32768$

Models

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

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

Singular plane model Singular plane model

$ 0 $

$=$

$ 4 x^{10} y^{2} - 12 x^{10} y z + 4 x^{10} z^{2} - 4 x^{8} y^{4} - 6 x^{8} y^{3} z + 34 x^{8} y^{2} z^{2} + \cdots - 2 z^{12} $

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

$(0:0:0:0:0:0:0:-1:1:0)$, $(1:1:1:2:0:-2:1:-1:1:1)$, $(0:0:0:1:-1:0:0:0:1:0)$, $(-1:-1:-1:2:0:-2:-1:-1:1:1)$

Maps to other modular curves

Map of degree 3 from the canonical model of this modular curve to the canonical model of the modular curve 40.60.4.bl.1 :

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

Equation of the image curve:

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

Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 40.180.10.cj.1 :

$\displaystyle X$	$=$	$\displaystyle y$
$\displaystyle Y$	$=$	$\displaystyle s$
$\displaystyle Z$	$=$	$\displaystyle a$

Equation of the image curve:

$0$

$=$

$ 4X^{10}Y^{2}-4X^{8}Y^{4}+X^{6}Y^{6}-12X^{10}YZ-6X^{8}Y^{3}Z+8X^{6}Y^{5}Z-X^{4}Y^{7}Z+4X^{10}Z^{2}+34X^{8}Y^{2}Z^{2}-X^{6}Y^{4}Z^{2}-3X^{4}Y^{6}Z^{2}-38X^{8}YZ^{3}+11X^{6}Y^{3}Z^{3}-5X^{4}Y^{5}Z^{3}+2X^{2}Y^{7}Z^{3}+16X^{8}Z^{4}+40X^{6}Y^{2}Z^{4}-11X^{4}Y^{4}Z^{4}+3X^{2}Y^{6}Z^{4}-77X^{6}YZ^{5}+75X^{4}Y^{3}Z^{5}-28X^{2}Y^{5}Z^{5}+26X^{6}Z^{6}+29X^{4}Y^{2}Z^{6}-17X^{2}Y^{4}Z^{6}+2Y^{6}Z^{6}-155X^{4}YZ^{7}+109X^{2}Y^{3}Z^{7}-2Y^{5}Z^{7}+67X^{4}Z^{8}-26X^{2}Y^{2}Z^{8}-8Y^{4}Z^{8}-141X^{2}YZ^{9}+6Y^{3}Z^{9}+94X^{2}Z^{10}+8Y^{2}Z^{10}-2YZ^{11}-2Z^{12} $

Modular covers

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

Factor curve	Level	Index	Degree	Genus	Rank
$X_{\mathrm{sp}}^+(5)$	$5$	$24$	$12$	$0$	$0$
16.24.0-8.n.1.8	$16$	$15$	$15$	$0$	$0$

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
80.120.4-40.bl.1.5	$80$	$3$	$3$	$4$	$?$