Modular curve 280.120.4-40.p.1.4

• Label: 280.120.4-40.p.1.4
• Level: $280$
• Index: $120$
• Genus: $4$
• Cusps: $4$
• $\Q$-cusps: $0$

Invariants

Level:	$280$		$\SL_2$-level:	$20$		Newform level:	$1600$
Index:	$120$		$\PSL_2$-index:	$60$
Genus:	$4 = 1 + \frac{ 60 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$
Cusps:	$4$ (none of which are rational)		Cusp widths	$10^{2}\cdot20^{2}$		Cusp orbits	$2^{2}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	not computed
$\Q$-gonality:	$3 \le \gamma \le 6$
$\overline{\Q}$-gonality:	$3 \le \gamma \le 4$
Rational cusps:	$0$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:

20A4

Level structure

$\GL_2(\Z/280\Z)$-generators:	$\begin{bmatrix}3&266\\248&127\end{bmatrix}$, $\begin{bmatrix}77&67\\244&247\end{bmatrix}$, $\begin{bmatrix}91&194\\38&43\end{bmatrix}$, $\begin{bmatrix}251&33\\122&99\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 40.60.4.p.1 for the level structure with $-I$)
Cyclic 280-isogeny field degree:	$192$
Cyclic 280-torsion field degree:	$18432$
Full 280-torsion field degree:	$12386304$

Models

Canonical model in $\mathbb{P}^{ 3 }$

$ 0 $	$=$	$ 35 x^{2} - 5 y^{2} + z^{2} - 2 z w + 2 w^{2} $
	$=$	$10 x^{3} + 10 x y^{2} - x z^{2} - y z^{2} + 2 y z w$

Singular plane model Singular plane model

$ 0 $

$=$

$ x^{6} + 2 x^{4} y^{2} - 20 x^{4} z^{2} + x^{2} y^{4} + 50 x^{2} y^{2} z^{2} + 100 x^{2} z^{4} + \cdots + 100 y^{2} z^{4} $

Rational points

This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.

Maps to other modular curves

$j$-invariant map of degree 60 from the canonical model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$

$=$

$\displaystyle -2^5\,\frac{426265xyz^{8}-4412870xyz^{7}w+15975400xyz^{6}w^{2}-30576840xyz^{5}w^{3}+37865100xyz^{4}w^{4}-31096520xyz^{3}w^{5}+15950480xyz^{2}w^{6}-4557280xyzw^{7}+229045y^{2}z^{8}-2051160y^{2}z^{7}w+8133460y^{2}z^{6}w^{2}-17882040y^{2}z^{5}w^{3}+24536700y^{2}z^{4}w^{4}-22360640y^{2}z^{3}w^{5}+13529920y^{2}z^{2}w^{6}-5208320y^{2}zw^{7}+1302080y^{2}w^{8}-4635z^{10}+106826z^{9}w-586950z^{8}w^{2}+1463488z^{7}w^{3}-2346068z^{6}w^{4}+2996696z^{5}w^{5}-2950984z^{4}w^{6}+2045696z^{3}w^{7}-977024z^{2}w^{8}+310400zw^{9}-62080w^{10}}{1015xyz^{8}-1400xyz^{7}w+770xyz^{6}w^{2}-9660xyz^{5}w^{3}+14700xyz^{4}w^{4}-7840xyz^{3}w^{5}+1960xyz^{2}w^{6}-560xyzw^{7}+665y^{2}z^{8}-1770y^{2}z^{7}w+670y^{2}z^{6}w^{2}+1920y^{2}z^{5}w^{3}-100y^{2}z^{4}w^{4}-1480y^{2}z^{3}w^{5}+1240y^{2}z^{2}w^{6}-640y^{2}zw^{7}+160y^{2}w^{8}-28z^{10}+130z^{9}w-534z^{8}w^{2}+1488z^{7}w^{3}-2812z^{6}w^{4}+3576z^{5}w^{5}-3208z^{4}w^{6}+2112z^{3}w^{7}-1008z^{2}w^{8}+320zw^{9}-64w^{10}}$

Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 40.60.4.p.1 :

$\displaystyle X$	$=$	$\displaystyle x$
$\displaystyle Y$	$=$	$\displaystyle y$
$\displaystyle Z$	$=$	$\displaystyle \frac{1}{10}z$

Equation of the image curve:

$0$

$=$

$ X^{6}+2X^{4}Y^{2}-20X^{4}Z^{2}+X^{2}Y^{4}+50X^{2}Y^{2}Z^{2}+100X^{2}Z^{4}-10Y^{4}Z^{2}+100Y^{2}Z^{4} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
140.60.2-20.a.1.4	$140$	$2$	$2$	$2$	$?$
280.24.0-40.g.1.1	$280$	$5$	$5$	$0$	$?$
280.60.2-20.a.1.2	$280$	$2$	$2$	$2$	$?$

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

Covering curve	Level	Index	Degree	Genus
280.360.10-40.bb.1.3	$280$	$3$	$3$	$10$
280.480.13-40.fz.1.1	$280$	$4$	$4$	$13$