Modular curve 80.288.7-40.fq.4.18

• Label: 80.288.7-40.fq.4.18
• Level: $80$
• Index: $288$
• Genus: $7$
• Cusps: $12$
• $\Q$-cusps: $4$

Invariants

Level:	$80$		$\SL_2$-level:	$80$		Newform level:	$160$
Index:	$288$		$\PSL_2$-index:	$144$
Genus:	$7 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$
Cusps:	$12$ (of which $4$ are rational)		Cusp widths	$1^{2}\cdot2\cdot4\cdot5^{2}\cdot8^{2}\cdot10\cdot20\cdot40^{2}$		Cusp orbits	$1^{4}\cdot2^{4}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	not computed
$\Q$-gonality:	$4 \le \gamma \le 7$
$\overline{\Q}$-gonality:	$4 \le \gamma \le 7$
Rational cusps:	$4$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:

40U7

Level structure

$\GL_2(\Z/80\Z)$-generators:	$\begin{bmatrix}1&20\\10&11\end{bmatrix}$, $\begin{bmatrix}25&26\\16&75\end{bmatrix}$, $\begin{bmatrix}44&39\\55&68\end{bmatrix}$, $\begin{bmatrix}45&8\\52&1\end{bmatrix}$, $\begin{bmatrix}71&58\\24&25\end{bmatrix}$, $\begin{bmatrix}73&6\\62&57\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 40.144.7.fq.4 for the level structure with $-I$)
Cyclic 80-isogeny field degree:	$2$
Cyclic 80-torsion field degree:	$32$
Full 80-torsion field degree:	$40960$

Models

Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations

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

Singular plane model Singular plane model

$ 0 $

$=$

$ - 5 x^{7} y^{2} - x^{7} z^{2} - 25 x^{6} y^{3} - 5 x^{6} y z^{2} - 45 x^{5} y^{4} - 9 x^{5} y^{2} z^{2} + \cdots + 5 y^{7} z^{2} $

Rational points

This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Canonical model

$(0:0:1:0:0:0:0)$, $(0:0:1:0:-2:1:1)$, $(0:0:0:0:0:-1:1)$, $(0:0:-2/3:0:2/3:1/3:1)$

Maps to other modular curves

$j$-invariant map of degree 144 from the canonical model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$

$=$

$\displaystyle -2^2\cdot3\,\frac{1934917632z^{12}-7739670528z^{11}v-23219011584z^{10}v^{2}+61917364224z^{9}v^{3}+371504185344z^{8}v^{4}+309586821120z^{7}v^{5}-3374496350208z^{6}v^{6}-15355506327552z^{5}v^{7}-13064563851264z^{4}v^{8}+152440550719488z^{3}v^{9}+781892475420672z^{2}v^{10}-120272597523zu^{11}-363106413711zu^{10}v+17740078020171zu^{9}v^{2}-72635338352817zu^{8}v^{3}-73922997043854zu^{7}v^{4}-174564082775430zu^{6}v^{5}+38995441267878zu^{5}v^{6}+1083415763387214zu^{4}v^{7}+2239684764676497zu^{3}v^{8}+1934138910616965zu^{2}v^{9}+226174741379103zuv^{10}-636987672331789zv^{11}+4383493277040640w^{2}v^{10}+7558272t^{12}-30233088t^{11}v-60466176t^{10}v^{2}+60466176t^{9}v^{3}-181398528t^{8}v^{4}+3869835264t^{7}v^{5}+24912064512t^{6}v^{6}+130969737216t^{5}v^{7}+773180992512t^{4}v^{8}+4627355516928t^{3}v^{9}+24568458381t^{2}u^{10}-1119969116658t^{2}u^{9}v+10223363381049t^{2}u^{8}v^{2}-18506194143192t^{2}u^{7}v^{3}+7083000912186t^{2}u^{6}v^{4}-102679332413580t^{2}u^{5}v^{5}-146806300089414t^{2}u^{4}v^{6}-64039005833112t^{2}u^{3}v^{7}+292322587833657t^{2}u^{2}v^{8}+636103265754606t^{2}uv^{9}-145028637817331t^{2}v^{10}-15919551351tu^{11}-447090389235tu^{10}v+13318243256331tu^{9}v^{2}-57894245845713tu^{8}v^{3}+11249172378378tu^{7}v^{4}-21925437777486tu^{6}v^{5}+229002873198534tu^{5}v^{6}+937796938619598tu^{4}v^{7}+1231187601083517tu^{3}v^{8}+242543775589713tu^{2}v^{9}+170710199372831tuv^{10}-636863837603341tv^{11}+22302216810u^{12}+1868415664050u^{11}v-16894097004906u^{10}v^{2}+11940027683742u^{9}v^{3}+90906139716228u^{8}v^{4}+348092318671380u^{7}v^{5}+800194321793484u^{6}v^{6}+815477211574812u^{5}v^{7}-539257098794382u^{4}v^{8}-3159385167551334u^{3}v^{9}-1880004157910978u^{2}v^{10}+437003627738086uv^{11}+30958682112v^{12}}{23219011584z^{2}v^{10}+8017850367zu^{11}-49928881950zu^{10}v-81723402657zu^{9}v^{2}+59177018232zu^{8}v^{3}+173243034414zu^{7}v^{4}+145405800540zu^{6}v^{5}+116891865126zu^{5}v^{6}+94110406632zu^{4}v^{7}-23687919693zu^{3}v^{8}-451825823790zu^{2}v^{9}+300814552731zuv^{10}-58866551920zv^{11}+115392348160w^{2}v^{10}-90699264t^{7}v^{5}-544195584t^{6}v^{6}-2902376448t^{5}v^{7}-17051461632t^{4}v^{8}-101673874944t^{3}v^{9}+3360065247t^{2}u^{10}-1525609647t^{2}u^{9}v-28221117984t^{2}u^{8}v^{2}-18397131480t^{2}u^{7}v^{3}+17443644678t^{2}u^{6}v^{4}+13819703058t^{2}u^{5}v^{5}-39912481080t^{2}u^{4}v^{6}-142795783008t^{2}u^{3}v^{7}-232717032189t^{2}u^{2}v^{8}-204661617195t^{2}uv^{9}+35647540336t^{2}v^{10}+4621588083tu^{11}-31784777622tu^{10}v-21854802537tu^{9}v^{2}+108958159584tu^{8}v^{3}+157070221902tu^{7}v^{4}+44663010228tu^{6}v^{5}-112357610514tu^{5}v^{6}-314630078112tu^{4}v^{7}-558518096097tu^{3}v^{8}-536867721198tu^{2}v^{9}+521073222299tuv^{10}-58866551920tv^{11}-7981653330u^{12}+2609099748u^{11}v+99527942430u^{10}v^{2}+187509635856u^{9}v^{3}+100733660316u^{8}v^{4}-82120125672u^{7}v^{5}-210521406996u^{6}v^{6}-254676181968u^{5}v^{7}-170588339082u^{4}v^{8}+81189176484u^{3}v^{9}+153720195382u^{2}v^{10}+30378794272uv^{11}}$

Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 40.144.7.fq.4 :

$\displaystyle X$	$=$	$\displaystyle x$
$\displaystyle Y$	$=$	$\displaystyle y$
$\displaystyle Z$	$=$	$\displaystyle z$

Equation of the image curve:

$0$

$=$

$ -5X^{7}Y^{2}-X^{7}Z^{2}-25X^{6}Y^{3}-5X^{6}YZ^{2}-45X^{5}Y^{4}-9X^{5}Y^{2}Z^{2}-25X^{4}Y^{5}+15X^{4}Y^{3}Z^{2}+4X^{4}YZ^{4}+25X^{3}Y^{6}+29X^{3}Y^{4}Z^{2}+4X^{3}Y^{2}Z^{4}+45X^{2}Y^{7}+25X^{2}Y^{5}Z^{2}+25XY^{8}+5XY^{6}Z^{2}+5Y^{9}+5Y^{7}Z^{2} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
80.144.3-40.bx.1.11	$80$	$2$	$2$	$3$	$?$
80.144.3-40.bx.1.18	$80$	$2$	$2$	$3$	$?$