Modular curve 280.144.3-40.be.1.5

• Label: 280.144.3-40.be.1.5
• Level: $280$
• Index: $144$
• Genus: $3$
• Cusps: $8$
• $\Q$-cusps: $0$

Invariants

Level:	$280$		$\SL_2$-level:	$20$		Newform level:	$1600$
Index:	$144$		$\PSL_2$-index:	$72$
Genus:	$3 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps:	$8$ (none of which are rational)		Cusp widths	$2^{2}\cdot4^{2}\cdot10^{2}\cdot20^{2}$		Cusp orbits	$2^{4}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	not computed
$\Q$-gonality:	$2 \le \gamma \le 4$
$\overline{\Q}$-gonality:	$2 \le \gamma \le 3$
Rational cusps:	$0$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:

20J3

Level structure

$\GL_2(\Z/280\Z)$-generators:	$\begin{bmatrix}69&220\\221&91\end{bmatrix}$, $\begin{bmatrix}93&150\\71&37\end{bmatrix}$, $\begin{bmatrix}117&90\\72&73\end{bmatrix}$, $\begin{bmatrix}257&200\\155&107\end{bmatrix}$, $\begin{bmatrix}263&230\\220&151\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 40.72.3.be.1 for the level structure with $-I$)
Cyclic 280-isogeny field degree:	$32$
Cyclic 280-torsion field degree:	$3072$
Full 280-torsion field degree:	$10321920$

Models

Embedded model Embedded model in $\mathbb{P}^{5}$

$ 0 $	$=$	$ - z u + w t $
	$=$	$x t + 3 y t - y u$
	$=$	$x z + 3 y z - y w$
	$=$	$x^{2} + 4 y^{2} - 3 z^{2} - 2 z w + w^{2}$
	$=$	$\cdots$

Singular plane model Singular plane model

$ 0 $

$=$

$ 1600 x^{4} y^{4} + 26000 x^{4} y^{2} z^{2} + 105625 x^{4} z^{4} + 4 x^{2} y^{6} + 80 x^{2} y^{4} z^{2} + \cdots + 100 z^{8} $

Geometric Weierstrass model Geometric Weierstrass model

$ 25 w^{2} $	$=$	$ 28 x^{4} - 96 x^{3} y - 132 x^{2} z^{2} + 72 x y z^{2} - 89 z^{4} $
$0$	$=$	$x^{2} + y^{2} + z^{2}$

Rational points

This modular curve has no real points and no $\Q_p$ points for $p=3$, and therefore no rational points.

Maps to other modular curves

$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$

$=$

$\displaystyle -\frac{1}{2}\cdot\frac{262143680000zw^{9}-155976132000zw^{7}u^{2}+40105498000zw^{5}u^{4}-30280937720zw^{3}u^{6}+17000223122zwu^{8}-65536000000w^{10}+47185952000w^{8}u^{2}-14286809200w^{6}u^{4}+8913143240w^{4}u^{6}-5228178696w^{2}u^{8}-46983897088t^{10}+53299819008t^{9}u+112037526272t^{8}u^{2}-180498856832t^{7}u^{3}-22703125008t^{6}u^{4}+173303599874t^{5}u^{5}-100513569914t^{4}u^{6}-13220920356t^{3}u^{7}+45791145491t^{2}u^{8}-17080485472tu^{9}+2371030225u^{10}}{1600zw^{5}u^{4}+2260zw^{3}u^{6}+958zwu^{8}-160w^{4}u^{6}-194w^{2}u^{8}+15925248t^{10}+2322432t^{9}u-9510912t^{8}u^{2}+3094272t^{7}u^{3}-728832t^{6}u^{4}+256496t^{5}u^{5}-14816t^{4}u^{6}-3009t^{3}u^{7}-591t^{2}u^{8}+17tu^{9}-65u^{10}}$

Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 40.72.3.be.1 :

$\displaystyle X$	$=$	$\displaystyle y$
$\displaystyle Y$	$=$	$\displaystyle 5w$
$\displaystyle Z$	$=$	$\displaystyle u$

Equation of the image curve:

$0$

$=$

$ 1600X^{4}Y^{4}+4X^{2}Y^{6}+26000X^{4}Y^{2}Z^{2}+80X^{2}Y^{4}Z^{2}+105625X^{4}Z^{4}+650X^{2}Y^{2}Z^{4}+Y^{4}Z^{4}+2500X^{2}Z^{6}+20Y^{2}Z^{6}+100Z^{8} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
140.72.1-20.a.1.7	$140$	$2$	$2$	$1$	$?$
280.24.0-40.j.1.1	$280$	$6$	$6$	$0$	$?$
280.72.1-20.a.1.2	$280$	$2$	$2$	$1$	$?$

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve	Level	Index	Degree	Genus
280.288.5-40.dg.1.11	$280$	$2$	$2$	$5$
280.288.5-40.dg.2.11	$280$	$2$	$2$	$5$
280.288.5-40.dh.1.5	$280$	$2$	$2$	$5$
280.288.5-40.dh.2.5	$280$	$2$	$2$	$5$
280.288.5-40.du.1.4	$280$	$2$	$2$	$5$
280.288.5-40.du.2.4	$280$	$2$	$2$	$5$
280.288.5-40.dv.1.1	$280$	$2$	$2$	$5$
280.288.5-40.dv.2.1	$280$	$2$	$2$	$5$
280.288.5-280.ku.1.7	$280$	$2$	$2$	$5$
280.288.5-280.ku.2.5	$280$	$2$	$2$	$5$
280.288.5-280.kv.1.2	$280$	$2$	$2$	$5$
280.288.5-280.kv.2.4	$280$	$2$	$2$	$5$
280.288.5-280.mk.1.8	$280$	$2$	$2$	$5$
280.288.5-280.mk.2.7	$280$	$2$	$2$	$5$
280.288.5-280.ml.1.1	$280$	$2$	$2$	$5$
280.288.5-280.ml.2.2	$280$	$2$	$2$	$5$