Modular curve 80.288.7-40.fo.2.18

• Label: 80.288.7-40.fo.2.18
• Level: $80$
• Index: $288$
• Genus: $7$
• Cusps: $12$
• $\Q$-cusps: $4$

Invariants

Level:	$80$		$\SL_2$-level:	$80$		Newform level:	$800$
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:	$3 \le \gamma \le 7$
$\overline{\Q}$-gonality:	$3 \le \gamma \le 7$
Rational cusps:	$4$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:

40V7

Level structure

$\GL_2(\Z/80\Z)$-generators:	$\begin{bmatrix}4&59\\21&2\end{bmatrix}$, $\begin{bmatrix}5&12\\32&65\end{bmatrix}$, $\begin{bmatrix}7&40\\38&49\end{bmatrix}$, $\begin{bmatrix}25&12\\56&61\end{bmatrix}$, $\begin{bmatrix}38&41\\47&72\end{bmatrix}$, $\begin{bmatrix}78&35\\65&48\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 40.144.7.fo.2 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 $	$=$	$ y w - y v - z u $
	$=$	$ - y v + z w - z t$
	$=$	$x w - x v - y u$
	$=$	$ - x v + y w - y t$
	$=$	$\cdots$

Singular plane model Singular plane model

$ 0 $

$=$

$ 5 x^{6} y z^{2} + 10 x^{5} y^{2} z^{2} - 100 x^{5} z^{4} + x^{4} y^{5} + 15 x^{4} y^{3} 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

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

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\,\frac{141926535z^{2}u^{10}+248644980z^{2}u^{9}v+11363403330z^{2}u^{8}v^{2}+9266078970z^{2}u^{7}v^{3}+180334889880z^{2}u^{6}v^{4}+775866815760z^{2}u^{5}v^{5}+3136244502400z^{2}u^{4}v^{6}+12570331978400z^{2}u^{3}v^{7}+53347631066400z^{2}u^{2}v^{8}+227686883933120z^{2}uv^{9}+858060381097920z^{2}v^{10}-340271846061088wtv^{10}-94844548wu^{11}-19004817wu^{10}v-2315218443wu^{9}v^{2}-14555191004wu^{8}v^{3}-40285725146wu^{7}v^{4}-193867844160wu^{6}v^{5}-787820601792wu^{5}v^{6}-3203948724176wu^{4}v^{7}-13503143231808wu^{3}v^{8}-51967887253280wu^{2}v^{9}+126074723445248wuv^{10}-171612074221632wv^{11}+128t^{12}-3584t^{11}v-41472t^{10}v^{2}+1924096t^{9}v^{3}-4928512t^{8}v^{4}-306214912t^{7}v^{5}+2818287616t^{6}v^{6}+6162292736t^{5}v^{7}-228456564736t^{4}v^{8}+1405662437376t^{3}v^{9}-65703591t^{2}u^{10}+141447598t^{2}u^{9}v-2598315761t^{2}u^{8}v^{2}-5038349050t^{2}u^{7}v^{3}-31041515232t^{2}u^{6}v^{4}-163706252488t^{2}u^{5}v^{5}-630620926464t^{2}u^{4}v^{6}-2560523916624t^{2}u^{3}v^{7}-10909983255136t^{2}u^{2}v^{8}-44523216401024t^{2}uv^{9}+169267121842176t^{2}v^{10}+110429389tu^{11}-575888498tu^{10}v+3796277918tu^{9}v^{2}+6460078028tu^{8}v^{3}+20718549534tu^{7}v^{4}+132285636744tu^{6}v^{5}+463687142064tu^{5}v^{6}+1927389573424tu^{4}v^{7}+8733573305792tu^{3}v^{8}+31143502416992tu^{2}v^{9}-210839518610656tuv^{10}+170582260541664tv^{11}-12725926u^{12}+157468227u^{11}v-1242055330u^{10}v^{2}-1342598766u^{9}v^{3}-196716280u^{8}v^{4}-34617497406u^{7}v^{5}-106685666328u^{6}v^{6}-461625424096u^{5}v^{7}-2041850467424u^{4}v^{8}-7274114078208u^{3}v^{9}-33838014870848u^{2}v^{10}-126074697443200uv^{11}+171612076221632v^{12}}{318750z^{2}u^{10}-5641500z^{2}u^{9}v+21605790z^{2}u^{8}v^{2}-25321080z^{2}u^{7}v^{3}-7530700z^{2}u^{6}v^{4}-30596660z^{2}u^{5}v^{5}-531501880z^{2}u^{4}v^{6}-1394571840z^{2}u^{3}v^{7}-3122425600z^{2}u^{2}v^{8}-2854431680z^{2}uv^{9}-38178534400z^{2}v^{10}+16854303552wtv^{10}-189875wu^{11}+476925wu^{10}v+2107928wu^{9}v^{2}-8099590wu^{8}v^{3}+7287034wu^{7}v^{4}+23284332wu^{6}v^{5}+43596028wu^{5}v^{6}+364953272wu^{4}v^{7}+432593440wu^{3}v^{8}+3121048576wu^{2}v^{9}-7064820544wuv^{10}+7635706880wv^{11}-512t^{10}v^{2}+13312t^{9}v^{3}-184832t^{8}v^{4}+1742848t^{7}v^{5}-12172288t^{6}v^{6}+65177600t^{5}v^{7}-268657664t^{4}v^{8}+828817408t^{3}v^{9}-271125t^{2}u^{10}+1565225t^{2}u^{9}v-2036212t^{2}u^{8}v^{2}-2819004t^{2}u^{7}v^{3}+9951326t^{2}u^{6}v^{4}+13149744t^{2}u^{5}v^{5}+69798216t^{2}u^{4}v^{6}+122436608t^{2}u^{3}v^{7}+502141408t^{2}u^{2}v^{8}+1123341312t^{2}uv^{9}-8792233856t^{2}v^{10}+370250tu^{11}-2402350tu^{10}v+3314678tu^{9}v^{2}+6601386tu^{8}v^{3}-19866416tu^{7}v^{4}-15406248tu^{6}v^{5}-35555236tu^{5}v^{6}-190059560tu^{4}v^{7}+371641472tu^{3}v^{8}-2126168320tu^{2}v^{9}+11041234432tuv^{10}-8386110528tv^{11}-99125u^{12}+665125u^{11}v-812604u^{10}v^{2}-3657910u^{9}v^{3}+11072356u^{8}v^{4}-2614752u^{7}v^{5}+7138504u^{6}v^{6}+50354836u^{5}v^{7}-127677560u^{4}v^{8}+1019496448u^{3}v^{9}+696804864u^{2}v^{10}+7064820544uv^{11}-7635706880v^{12}}$

Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 40.144.7.fo.2 :

$\displaystyle X$	$=$	$\displaystyle x$
$\displaystyle Y$	$=$	$\displaystyle y$
$\displaystyle Z$	$=$	$\displaystyle \frac{1}{5}w$

Equation of the image curve:

$0$

$=$

$ 5X^{6}YZ^{2}+10X^{5}Y^{2}Z^{2}-100X^{5}Z^{4}+X^{4}Y^{5}+15X^{4}Y^{3}Z^{2}+100X^{3}Y^{2}Z^{4}-2X^{2}Y^{7}+15X^{2}Y^{5}Z^{2}-10XY^{6}Z^{2}+Y^{9}+5Y^{7}Z^{2} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
80.48.0-40.ca.1.8	$80$	$6$	$6$	$0$	$?$
80.144.3-40.bx.1.3	$80$	$2$	$2$	$3$	$?$
80.144.3-40.bx.1.18	$80$	$2$	$2$	$3$	$?$