Modular curve 280.120.4-40.m.1.3

• Label: 280.120.4-40.m.1.3
• 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:	$\begin{bmatrix}41&126\\226&129\end{bmatrix}$, $\begin{bmatrix}59&129\\140&113\end{bmatrix}$, $\begin{bmatrix}101&43\\134&9\end{bmatrix}$, $\begin{bmatrix}169&7\\208&191\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 40.60.4.m.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 $	$=$	$ 7 x^{2} + y^{2} - 2 y w - z^{2} + 2 w^{2} $
	$=$	$2 x^{3} - x y^{2} + 2 x z^{2} + y^{2} z - 2 y z w$

Singular plane model Singular plane model

$ 0 $

$=$

$ - x^{6} + 4 x^{4} y^{2} - 2 x^{4} z^{2} - 4 x^{2} y^{4} - 10 x^{2} y^{2} z^{2} - x^{2} z^{4} + \cdots + 2 y^{2} z^{4} $

Rational points

This modular curve has no $\Q_p$ points for $p=3$, and therefore no 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 $X(1)$ :

$\displaystyle j$

$=$

$\displaystyle 2^7\cdot3^3\,\frac{1413001576xyz^{7}w-124827512124xyz^{5}w^{3}+183938099198xyz^{3}w^{5}-32985497797xyzw^{7}-1126118336xz^{9}+88663171856xz^{7}w^{2}-95250518104xz^{5}w^{4}-4929675828xz^{3}w^{6}+5804613178xzw^{8}+7038405216y^{3}z^{6}w-32876523360y^{3}z^{4}w^{3}+14474543400y^{3}z^{2}w^{5}+91232064y^{3}w^{7}-2138402624y^{2}z^{8}+12708501216y^{2}z^{6}w^{2}+74316697040y^{2}z^{4}w^{4}-69149247976y^{2}z^{2}w^{6}+3597996240y^{2}w^{8}+733087144yz^{8}w-49080569916yz^{6}w^{3}+5113499054yz^{4}w^{5}+44375579147yz^{2}w^{7}-1664402976yw^{9}+665512128z^{10}+12051720864z^{8}w^{2}-58256913456z^{6}w^{4}+72929369592z^{4}w^{6}-22244993280z^{2}w^{8}+1846867104w^{10}}{434203616xyz^{7}w+1912501920xyz^{5}w^{3}-1148674121xyz^{3}w^{5}+85281952xyzw^{7}+1196018432xz^{9}-6844588352xz^{7}w^{2}+4678878232xz^{5}w^{4}-416049438xz^{3}w^{6}-86142784xzw^{8}+708179040y^{3}z^{6}w-794733840y^{3}z^{4}w^{3}+144208944y^{3}z^{2}w^{5}+1096704y^{3}w^{7}+184196096y^{2}z^{8}-3128525088y^{2}z^{6}w^{2}+2548728760y^{2}z^{4}w^{4}-415366832y^{2}z^{2}w^{6}-905472y^{2}w^{8}-371398624yz^{8}w+3994302144yz^{6}w^{3}-2885906537yz^{4}w^{5}+534402976yz^{2}w^{7}-382464yw^{9}+250129152z^{10}-1256240448z^{8}w^{2}+1693322856z^{6}w^{4}-273470976z^{4}w^{6}-179780256z^{2}w^{8}+2575872w^{10}}$

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

$\displaystyle X$	$=$	$\displaystyle x$
$\displaystyle Y$	$=$	$\displaystyle \frac{1}{2}y$
$\displaystyle Z$	$=$	$\displaystyle z$

Equation of the image curve:

$0$

$=$

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

Modular covers

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

Factor curve	Level	Index	Degree	Genus	Rank
$X_{S_4}(5)$	$5$	$24$	$12$	$0$	$0$
56.24.0-8.g.1.2	$56$	$5$	$5$	$0$	$0$

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
56.24.0-8.g.1.2	$56$	$5$	$5$	$0$	$0$
140.60.2-20.a.1.3	$140$	$2$	$2$	$2$	$?$
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.y.1.7	$280$	$3$	$3$	$10$
280.480.13-40.eq.1.1	$280$	$4$	$4$	$13$