Modular curve 280.384.11-56.s.2.38

• Label: 280.384.11-56.s.2.38
• Level: $280$
• Index: $384$
• Genus: $11$
• Cusps: $12$
• $\Q$-cusps: $8$

Invariants

Level:	$280$		$\SL_2$-level:	$56$		Newform level:	$56$
Index:	$384$		$\PSL_2$-index:	$192$
Genus:	$11 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$
Cusps:	$12$ (of which $8$ are rational)		Cusp widths	$2^{2}\cdot4^{3}\cdot8\cdot14^{2}\cdot28^{3}\cdot56$		Cusp orbits	$1^{8}\cdot2^{2}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	not computed
$\Q$-gonality:	$2 \le \gamma \le 11$
$\overline{\Q}$-gonality:	$2 \le \gamma \le 11$
Rational cusps:	$8$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:

56R11

Level structure

$\GL_2(\Z/280\Z)$-generators:	$\begin{bmatrix}55&28\\66&143\end{bmatrix}$, $\begin{bmatrix}103&0\\214&89\end{bmatrix}$, $\begin{bmatrix}121&140\\18&1\end{bmatrix}$, $\begin{bmatrix}129&196\\218&255\end{bmatrix}$, $\begin{bmatrix}241&84\\184&117\end{bmatrix}$, $\begin{bmatrix}241&196\\268&251\end{bmatrix}$, $\begin{bmatrix}271&28\\238&141\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 56.192.11.s.2 for the level structure with $-I$)
Cyclic 280-isogeny field degree:	$12$
Cyclic 280-torsion field degree:	$1152$
Full 280-torsion field degree:	$3870720$

Models

Canonical model in $\mathbb{P}^{ 10 }$ defined by 36 equations

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

Singular plane model Singular plane model

$ 0 $

$=$

$ - 4 x^{6} y^{4} z^{4} + 8 x^{5} y^{8} z - 4 x^{5} y^{6} z^{3} - 12 x^{5} y^{4} z^{5} + 8 x^{5} y^{2} z^{7} + \cdots + y^{2} z^{12} $

Rational points

This modular curve has 8 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Canonical model

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

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 28.96.5.b.1 :

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

Equation of the image curve:

$0$	$=$	$ XY+YZ+YW+ZT $
	$=$	$ Y^{2}-XW-2YT+T^{2} $
	$=$	$ Y^{2}+XZ+Z^{2}+XW+ZW-YT+T^{2} $

Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 56.192.11.s.2 :

$\displaystyle X$	$=$	$\displaystyle b$
$\displaystyle Y$	$=$	$\displaystyle 2x$
$\displaystyle Z$	$=$	$\displaystyle 2r$

Equation of the image curve:

$0$

$=$

$ 8X^{4}Y^{10}-8X^{2}Y^{12}+16Y^{14}+8X^{5}Y^{8}Z+8X^{3}Y^{10}Z+8XY^{12}Z+16X^{4}Y^{8}Z^{2}-24X^{2}Y^{10}Z^{2}-24Y^{12}Z^{2}-4X^{5}Y^{6}Z^{3}-34X^{3}Y^{8}Z^{3}-38XY^{10}Z^{3}-4X^{6}Y^{4}Z^{4}-30X^{4}Y^{6}Z^{4}-9X^{2}Y^{8}Z^{4}+39Y^{10}Z^{4}-12X^{5}Y^{4}Z^{5}+14X^{3}Y^{6}Z^{5}+95XY^{8}Z^{5}+16X^{4}Y^{4}Z^{6}+96X^{2}Y^{6}Z^{6}-56Y^{8}Z^{6}+8X^{5}Y^{2}Z^{7}+42X^{3}Y^{4}Z^{7}-118XY^{6}Z^{7}+6X^{4}Y^{2}Z^{8}-102X^{2}Y^{4}Z^{8}+40Y^{6}Z^{8}-38X^{3}Y^{2}Z^{9}+68XY^{4}Z^{9}-4X^{4}Z^{10}+40X^{2}Y^{2}Z^{10}-12Y^{4}Z^{10}+8X^{3}Z^{11}-16XY^{2}Z^{11}-5X^{2}Z^{12}+Y^{2}Z^{12}+XZ^{13} $

Modular covers

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

Factor curve	Level	Index	Degree	Genus	Rank
$X_0(7)$	$7$	$48$	$24$	$0$	$0$
40.48.0-8.e.1.5	$40$	$8$	$8$	$0$	$0$

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
40.48.0-8.e.1.5	$40$	$8$	$8$	$0$	$0$
280.192.5-28.b.1.18	$280$	$2$	$2$	$5$	$?$
280.192.5-28.b.1.37	$280$	$2$	$2$	$5$	$?$