Modular curve 280.336.9-28.b.1.9

• Label: 280.336.9-28.b.1.9
• Level: $280$
• Index: $336$
• Genus: $9$
• Cusps: $12$
• $\Q$-cusps: $1$

Invariants

Level:	$280$		$\SL_2$-level:	$56$		Newform level:	$784$
Index:	$336$		$\PSL_2$-index:	$168$
Genus:	$9 = 1 + \frac{ 168 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$
Cusps:	$12$ (of which $1$ is rational)		Cusp widths	$7^{8}\cdot28^{4}$		Cusp orbits	$1\cdot2\cdot3\cdot6$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	not computed
$\Q$-gonality:	$4 \le \gamma \le 9$
$\overline{\Q}$-gonality:	$4 \le \gamma \le 9$
Rational cusps:	$1$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:

28A9

Level structure

$\GL_2(\Z/280\Z)$-generators:	$\begin{bmatrix}25&26\\226&171\end{bmatrix}$, $\begin{bmatrix}126&109\\11&56\end{bmatrix}$, $\begin{bmatrix}134&229\\121&104\end{bmatrix}$, $\begin{bmatrix}166&267\\209&272\end{bmatrix}$, $\begin{bmatrix}192&107\\201&242\end{bmatrix}$, $\begin{bmatrix}279&202\\134&159\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 28.168.9.b.1 for the level structure with $-I$)
Cyclic 280-isogeny field degree:	$48$
Cyclic 280-torsion field degree:	$4608$
Full 280-torsion field degree:	$4423680$

Models

Canonical model in $\mathbb{P}^{ 8 }$ defined by 21 equations

$ 0 $	$=$	$ y v + t r $
	$=$	$x v + t v + t r + t s$
	$=$	$x r - y v - y r - y s$
	$=$	$x v + x s + z v + 2 z r + z s + w v + w s + t r + u v + u s$
	$=$	$\cdots$

Singular plane model Singular plane model

$ 0 $

$=$

$ 14700 x^{16} - 544880 x^{15} y + 4708508 x^{14} y^{2} + 3724 x^{14} z^{2} - 9545004 x^{13} y^{3} + \cdots + 400 y^{10} z^{6} $

Rational points

This modular curve has 1 rational cusp but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Canonical model

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

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 14.84.3.a.1 :

$\displaystyle X$	$=$	$\displaystyle -v+4r-s$
$\displaystyle Y$	$=$	$\displaystyle 4v-2r-3s$
$\displaystyle Z$	$=$	$\displaystyle -v-3r-s$

Equation of the image curve:

$0$

$=$

$ X^{2}Y^{2}+X^{3}Z+4XY^{2}Z+Y^{3}Z+2X^{2}Z^{2}-XYZ^{2}-2XZ^{3} $

Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 28.168.9.b.1 :

$\displaystyle X$	$=$	$\displaystyle x$
$\displaystyle Y$	$=$	$\displaystyle y$
$\displaystyle Z$	$=$	$\displaystyle 7v$

Equation of the image curve:

$0$

$=$

$ 14700X^{16}-544880X^{15}Y+4708508X^{14}Y^{2}+3724X^{14}Z^{2}-9545004X^{13}Y^{3}-60760X^{13}YZ^{2}-23422441X^{12}Y^{4}+408268X^{12}Y^{2}Z^{2}+196X^{12}Z^{4}+94646048X^{11}Y^{5}-1658944X^{11}Y^{3}Z^{2}-3136X^{11}YZ^{4}+2339946X^{10}Y^{6}+2830828X^{10}Y^{4}Z^{2}+24892X^{10}Y^{2}Z^{4}+4X^{10}Z^{6}-261537353X^{9}Y^{7}+8157128X^{9}Y^{5}Z^{2}-128772X^{9}Y^{3}Z^{4}-88X^{9}YZ^{6}+76905647X^{8}Y^{8}-28865312X^{8}Y^{6}Z^{2}+419881X^{8}Y^{4}Z^{4}+636X^{8}Y^{2}Z^{6}+335938561X^{7}Y^{9}-1545460X^{7}Y^{7}Z^{2}-813890X^{7}Y^{5}Z^{4}-1144X^{7}Y^{3}Z^{6}-24537877X^{6}Y^{10}+41913816X^{6}Y^{8}Z^{2}+567665X^{6}Y^{6}Z^{4}-4788X^{6}Y^{4}Z^{6}-140003682X^{5}Y^{11}-17966144X^{5}Y^{9}Z^{2}+1611806X^{5}Y^{7}Z^{4}+14616X^{5}Y^{5}Z^{6}-18288760X^{4}Y^{12}-19717404X^{4}Y^{10}Z^{2}-1010576X^{4}Y^{8}Z^{4}+10248X^{4}Y^{6}Z^{6}-1425851X^{3}Y^{13}+14508312X^{3}Y^{11}Z^{2}-169050X^{3}Y^{9}Z^{4}-29504X^{3}Y^{7}Z^{6}-212513X^{2}Y^{14}+2731456X^{2}Y^{12}Z^{2}+456925X^{2}Y^{10}Z^{4}+5956X^{2}Y^{8}Z^{6}+9849XY^{15}-2204412XY^{13}Z^{2}-183750XY^{11}Z^{4}+4240XY^{9}Z^{6}-98Y^{16}-350840Y^{14}Z^{2}+30625Y^{12}Z^{4}+400Y^{10}Z^{6} $

Modular covers

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

Factor curve	Level	Index	Degree	Genus	Rank
40.12.0-4.b.1.3	$40$	$28$	$28$	$0$	$0$
$X_{\mathrm{sp}}^+(7)$	$7$	$12$	$6$	$0$	$0$

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
40.12.0-4.b.1.3	$40$	$28$	$28$	$0$	$0$