Modular curve 280.192.5-28.i.1.5

• Label: 280.192.5-28.i.1.5
• Level: $280$
• Index: $192$
• Genus: $5$
• Cusps: $8$
• $\Q$-cusps: $0$

Invariants

Level:	$280$		$\SL_2$-level:	$56$		Newform level:	$784$
Index:	$192$		$\PSL_2$-index:	$96$
Genus:	$5 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps:	$8$ (none of which are rational)		Cusp widths	$2^{2}\cdot4^{2}\cdot14^{2}\cdot28^{2}$		Cusp orbits	$2^{4}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	not computed
$\Q$-gonality:	$3 \le \gamma \le 8$
$\overline{\Q}$-gonality:	$3 \le \gamma \le 5$
Rational cusps:	$0$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:

28E5

Level structure

$\GL_2(\Z/280\Z)$-generators:	$\begin{bmatrix}5&84\\172&269\end{bmatrix}$, $\begin{bmatrix}43&98\\255&59\end{bmatrix}$, $\begin{bmatrix}129&210\\189&57\end{bmatrix}$, $\begin{bmatrix}215&238\\268&41\end{bmatrix}$, $\begin{bmatrix}237&126\\230&243\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 28.96.5.i.1 for the level structure with $-I$)
Cyclic 280-isogeny field degree:	$24$
Cyclic 280-torsion field degree:	$2304$
Full 280-torsion field degree:	$7741440$

Models

Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations

$ 0 $	$=$	$ x y - x z + x w - y t $
	$=$	$2 x^{2} - x t + 2 y^{2} - y z - y w + z^{2} + 2 z w - w^{2} + t^{2}$
	$=$	$5 x^{2} + x t - 2 y^{2} + y z - z^{2} - t^{2}$

Singular plane model Singular plane model

$ 0 $

$=$

$ 98 x^{6} - 49 x^{5} z + 16 x^{4} y^{2} + 49 x^{4} z^{2} + 33 x^{3} y^{2} z + 2 x^{2} y^{4} + \cdots + 4 y^{2} z^{4} $

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 96 from the canonical model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$

$=$

$\displaystyle \frac{1}{2^4\cdot7^4}\cdot\frac{21506200187788231776xzw^{9}t+510563477623476935168xzw^{7}t^{3}+2043659366272677005160xzw^{5}t^{5}+1924525863030996255002xzw^{3}t^{7}+113569497127102282343xzwt^{9}-15280658393452016320xw^{10}t-492615919907265281680xw^{8}t^{3}-2441899412733439333920xw^{6}t^{5}-3156187624763869524290xw^{4}t^{7}-1142668320365742070390xw^{2}t^{9}-257327438486058780770xt^{11}-28473505099200yz^{11}-169891913758560yz^{9}t^{2}+21481361363673120yz^{7}t^{4}-23632435807580530yz^{5}t^{6}-8912406232625887530yz^{3}t^{8}+57855442153174054900yzt^{10}-315395281544443456yw^{11}-29084601271389334832yw^{9}t^{2}+65719435589848923988yw^{7}t^{4}+1534239752462076567068yw^{5}t^{6}+2388444602239985528169yw^{3}t^{8}+515802307953273073558ywt^{10}+14553124828480z^{12}+116266812488400z^{10}t^{2}-6443079645530640z^{8}t^{4}-176293855121202525z^{6}t^{6}+2516936353927556365z^{4}t^{8}-344779088607544192z^{2}w^{10}-10218508433580598816z^{2}w^{8}t^{2}+51706203265961812680z^{2}w^{6}t^{4}+376859517953697807476z^{2}w^{4}t^{6}+491995821042074836144z^{2}w^{2}t^{8}+101087715881266658460z^{2}t^{10}+975569651696431104zw^{11}+84817032862907852192zw^{9}t^{2}+709661989345921270680zw^{7}t^{4}+1327541775638612752078zw^{5}t^{6}+632356725144590137307zw^{3}t^{8}+142790082339543467500zwt^{10}-415199442414827328w^{12}-37330910039228062880w^{10}t^{2}-309508074676103910628w^{8}t^{4}-432047548548055652051w^{6}t^{6}+203029264007253020449w^{4}t^{8}+350546473784073087332w^{2}t^{10}+98318095370876371410t^{12}}{134873088xzw^{9}t-23381514240xzw^{7}t^{3}-1442759203424xzw^{5}t^{5}-9948216272512xzw^{3}t^{7}-14778819611206xzwt^{9}+97341440xw^{10}t+41972224000xw^{8}t^{3}+1182797936320xw^{6}t^{5}+6902875260680xw^{4}t^{7}+10512644044000xw^{2}t^{9}-1762760849780xt^{11}+790601280yz^{7}t^{4}-1185901920yz^{5}t^{6}-106335872160yz^{3}t^{8}+1506597799630yzt^{10}-401408yw^{11}+857909248yw^{9}t^{2}+47177161664yw^{7}t^{4}+113952653248yw^{5}t^{6}-2186361875096yw^{3}t^{8}-9950686418911ywt^{10}-263533760z^{8}t^{4}-6390693680z^{6}t^{6}+57648010000z^{4}t^{8}-2809856z^{2}w^{10}-1945825280z^{2}w^{8}t^{2}-14803117952z^{2}w^{6}t^{4}+757565786544z^{2}w^{4}t^{6}+2773223169192z^{2}w^{2}t^{8}+209315806595z^{2}t^{10}+3612672zw^{11}-360765440zw^{9}t^{2}-182143438336zw^{7}t^{4}-2639140938448zw^{5}t^{6}-6288911938204zw^{3}t^{8}-302221457160zwt^{10}-1204224w^{12}+571604992w^{10}t^{2}+85857712832w^{8}t^{4}+1006433828400w^{6}t^{6}+1973607189260w^{4}t^{8}-325407251484w^{2}t^{10}+145540636675t^{12}}$

Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 28.96.5.i.1 :

$\displaystyle X$	$=$	$\displaystyle x$
$\displaystyle Y$	$=$	$\displaystyle w$
$\displaystyle Z$	$=$	$\displaystyle t$

Equation of the image curve:

$0$

$=$

$ 98X^{6}+16X^{4}Y^{2}+2X^{2}Y^{4}-49X^{5}Z+33X^{3}Y^{2}Z-XY^{4}Z+49X^{4}Z^{2}-29X^{2}Y^{2}Z^{2}+Y^{4}Z^{2}-8XY^{2}Z^{3}+4Y^{2}Z^{4} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
280.24.0-28.e.1.3	$280$	$8$	$8$	$0$	$?$
280.96.2-28.b.1.13	$280$	$2$	$2$	$2$	$?$
280.96.2-28.b.1.15	$280$	$2$	$2$	$2$	$?$

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

Covering curve	Level	Index	Degree	Genus
280.384.11-56.ec.1.1	$280$	$2$	$2$	$11$
280.384.11-56.ec.1.5	$280$	$2$	$2$	$11$
280.384.11-56.ec.2.1	$280$	$2$	$2$	$11$
280.384.11-56.ec.2.5	$280$	$2$	$2$	$11$
280.384.11-56.ed.1.3	$280$	$2$	$2$	$11$
280.384.11-56.ed.1.7	$280$	$2$	$2$	$11$
280.384.11-56.ed.2.3	$280$	$2$	$2$	$11$
280.384.11-56.ed.2.7	$280$	$2$	$2$	$11$
280.384.11-280.je.1.4	$280$	$2$	$2$	$11$
280.384.11-280.je.1.9	$280$	$2$	$2$	$11$
280.384.11-280.je.2.4	$280$	$2$	$2$	$11$
280.384.11-280.je.2.9	$280$	$2$	$2$	$11$
280.384.11-280.jf.1.3	$280$	$2$	$2$	$11$
280.384.11-280.jf.1.10	$280$	$2$	$2$	$11$
280.384.11-280.jf.2.3	$280$	$2$	$2$	$11$
280.384.11-280.jf.2.10	$280$	$2$	$2$	$11$