Modular curve 80.288.7-40.fs.1.10

• Label: 80.288.7-40.fs.1.10
• Level: $80$
• Index: $288$
• Genus: $7$
• Cusps: $12$
• $\Q$-cusps: $8$

Invariants

Level:	$80$		$\SL_2$-level:	$80$		Newform level:	$40$
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 $8$ are rational)		Cusp widths	$1^{2}\cdot2\cdot4\cdot5^{2}\cdot8^{2}\cdot10\cdot20\cdot40^{2}$		Cusp orbits	$1^{8}\cdot2^{2}$
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:	$8$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:

40V7

Level structure

$\GL_2(\Z/80\Z)$-generators:	$\begin{bmatrix}4&13\\37&20\end{bmatrix}$, $\begin{bmatrix}13&36\\66&23\end{bmatrix}$, $\begin{bmatrix}13&50\\24&39\end{bmatrix}$, $\begin{bmatrix}16&31\\57&30\end{bmatrix}$, $\begin{bmatrix}29&52\\78&43\end{bmatrix}$, $\begin{bmatrix}39&64\\42&61\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 40.144.7.fs.1 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 z - z w - w u $
	$=$	$x u + z^{2} + z t + w u$
	$=$	$x^{2} + x y - x z - y z - z t$
	$=$	$x y - x w + x u + y^{2} - y w - z u - u v$
	$=$	$\cdots$

Singular plane model Singular plane model

$ 0 $

$=$

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

Rational points

This modular curve has 8 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Canonical model

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

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 \frac{2^5}{3^{14}}\cdot\frac{477651648706909855186603581xtu^{10}+192729188872432503506941956xtu^{9}v-334873103218088285769341985xtu^{8}v^{2}+89279519893761859072571520xtu^{7}v^{3}-12445108303314394715701950xtu^{6}v^{4}-194047554384526715933400xtu^{5}v^{5}+547708600174055577558750xtu^{4}v^{6}-143188438402570249980000xtu^{3}v^{7}+19846987633194370700625xtu^{2}v^{8}-1873936486269350727500xtuv^{9}+86511972404165301875xtv^{10}-128519607963095608552582482xu^{11}+1152604785734811710375587425xu^{10}v-855030413355827266004032242xu^{9}v^{2}-110883731757130378648771965xu^{8}v^{3}+131889951850249674940645260xu^{7}v^{4}-45440964935797397609928150xu^{6}v^{5}+10606283006416247279634300xu^{5}v^{6}-1741854404575830133316250xu^{4}v^{7}+211975406526808619943750xu^{3}v^{8}-17632239271103320626875xu^{2}v^{9}+1029036920369180533750xuv^{10}-7568428112910365625xv^{11}+720749793294913367502376146ytu^{10}-1036474087183118951751644832ytu^{9}v-87183299438957866132195650ytu^{8}v^{2}+220942651710379006117114560ytu^{7}v^{3}-80985205431766823480282700ytu^{6}v^{4}+19615225597071005404828800ytu^{5}v^{5}-3349332596002846995202500ytu^{4}v^{6}+400135035635573744040000ytu^{3}v^{7}-33490988748739468643750ytu^{2}v^{8}+1308892086801899060000ytuv^{9}-23423451011276876250ytv^{10}+124814230076099238014722221yu^{11}+307902902519367684422529525yu^{10}v-807596192276716920456218517yu^{9}v^{2}+92189248359677530756602435yu^{8}v^{3}+59598625584371771219111010yu^{7}v^{4}-30521263571555470279347150yu^{6}v^{5}+8810257140642530674070550yu^{5}v^{6}-1741624909774518141176250yu^{4}v^{7}+249360731398312661310625yu^{3}v^{8}-26780184908225902064375yu^{2}v^{9}+1858669885238871079375yuv^{10}-81036399236192285625yv^{11}-973336814121078738476437374ztu^{10}+318208269097025317612036440ztu^{9}v+371700876403211491798654950ztu^{8}v^{2}-113819138499523524036868800ztu^{7}v^{3}+20434283029634009954836500ztu^{6}v^{4}-1759744686769005871890000ztu^{5}v^{5}-211855533853170592552500ztu^{4}v^{6}+89440213514977650960000ztu^{3}v^{7}-13339833584484403413750ztu^{2}v^{8}+1241220388139742195000ztuv^{9}-54066164085936231250ztv^{10}+128519632450329605270662482zu^{11}-965830693762009766016394344zu^{10}v+306093166101838901193254550zu^{9}v^{2}+210650093880372099405789120zu^{8}v^{3}-121469513528263668827949900zu^{7}v^{4}+36637704372677562664743600zu^{6}v^{5}-7640142579317997460732500zu^{5}v^{6}+1103682201817464420480000zu^{4}v^{7}-112803793728052615803750zu^{3}v^{8}+4592182997714480235000zu^{2}v^{9}+58397394375385278750zuv^{10}-73467971123281920000zv^{11}+440730013959695560855895118wtu^{10}+725955738810820063726941648wtu^{9}v-467167735833854693263830990wtu^{8}v^{2}+31943092741832377294364160wtu^{7}v^{3}+16063112190092072095745100wtu^{6}v^{4}-8162569702586685264415200wtu^{5}v^{5}+2072004497811348648502500wtu^{4}v^{6}-345015781069687306320000wtu^{3}v^{7}+40559955149697320863750wtu^{2}v^{8}-2980249606325520070000wtuv^{9}+145823915033215346250wtv^{10}+195233602229708929561221534wu^{11}-138034317098457679255899552wu^{10}v+167660617897145135236161330wu^{9}v^{2}-96790140800118139604394240wu^{8}v^{3}+11729926631380677528494700wu^{7}v^{4}+3413615136404803198084800wu^{6}v^{5}-2187146371880937296857500wu^{5}v^{6}+639855380303963291040000wu^{4}v^{7}-121326035575463614006250wu^{3}v^{8}+17283879115799868300000wu^{2}v^{9}-1494927338010066573750wuv^{10}+97955205120000000000wv^{11}-1530550080000000000t^{12}+6122200320000000000t^{11}v-15305500800000000000t^{10}v^{2}+27890023680000000000t^{9}v^{3}-40039945920000000000t^{8}v^{4}+46777305600000000000t^{7}v^{5}-45710749440000000000t^{6}v^{6}+38205043200000000000t^{5}v^{7}-27832178880000000000t^{4}v^{8}+17876465920000000000t^{3}v^{9}-425321174081094705100501974t^{2}u^{10}-29323724939759017041540408t^{2}u^{9}v+192516618723592835207448270t^{2}u^{8}v^{2}-18550063953496213623682560t^{2}u^{7}v^{3}-4572759160681207998860700t^{2}u^{6}v^{4}+2734288681148358795925200t^{2}u^{5}v^{5}-717562588907839995742500t^{2}u^{4}v^{6}+113998788648382540200000t^{2}u^{3}v^{7}-11552013539121612128750t^{2}u^{2}v^{8}+628426124711231785000t^{2}uv^{9}-23375206783369586250t^{2}v^{10}-308910902043267228159440712tu^{11}-262262610802071323889610680tu^{10}v-146077270615423139925250800tu^{9}v^{2}+202839861807911655392673600tu^{8}v^{3}-43860779241081475724442000tu^{7}v^{4}+4033912538547705680010000tu^{6}v^{5}+1061175356400824459460000tu^{5}v^{6}-536054270815546332960000tu^{4}v^{7}+116809973591290117615000tu^{3}v^{8}-17733990505211085295000tu^{2}v^{9}+1507564161719972090000tuv^{10}-97961474253127680000tv^{11}-1567283281920000u^{12}+128519632475406137781382482u^{11}v-656919767403909702150073632u^{10}v^{2}+215518212026762005391943870u^{9}v^{3}+77728763502406292059604160u^{8}v^{4}-46519461082067941816289100u^{7}v^{5}+13606804486123288722748800u^{6}v^{6}-2770782431470032163522500u^{5}v^{7}+392806572252109188840000u^{4}v^{8}-42560933578862417043750u^{3}v^{9}+2767068281798219300000u^{2}v^{10}-137945137167994956250uv^{11}-1567283281920000v^{12}}{2999590521363137xtu^{10}+3570676584641022xtu^{9}v+2055424553520965xtu^{8}v^{2}+748817328204840xtu^{7}v^{3}+187506006409650xtu^{6}v^{4}+33170942723700xtu^{5}v^{5}+4551602201250xtu^{4}v^{6}+132159865000xtu^{3}v^{7}+64377458125xtu^{2}v^{8}-13061181250xtuv^{9}+5402545625xtv^{10}+1400923501926086xu^{11}+1762315572723505xu^{10}v+580499131171716xu^{9}v^{2}-42641879544575xu^{8}v^{3}-89917898651980xu^{7}v^{4}-29573533853150xu^{6}v^{5}-5716556792400xu^{5}v^{6}-491644813750xu^{4}v^{7}-121947366250xu^{3}v^{8}+36197338125xu^{2}v^{9}-3924402500xuv^{10}+1128183125xv^{11}-392895451308358ytu^{10}-2023507050818204ytu^{9}v-1947159293392470ytu^{8}v^{2}-973494023364880ytu^{7}v^{3}-313473716175500ytu^{6}v^{4}-70244419495400ytu^{5}v^{5}-10560850417500ytu^{4}v^{6}-1327827450000ytu^{3}v^{7}-27029088750ytu^{2}v^{8}-12759937500ytuv^{9}+4352571250ytv^{10}+94779750994417yu^{11}-1496907545899165yu^{10}v-2241960285559309yu^{9}v^{2}-1411500726711975yu^{8}v^{3}-525270244196230yu^{7}v^{4}-128648644817650yu^{6}v^{5}-20785348808650yu^{5}v^{6}-2255366758750yu^{4}v^{7}-166295926875yu^{3}v^{8}+16132429375yu^{2}v^{9}-621805625yuv^{10}+1128183125yv^{11}-2438035403837398ztu^{10}-3145851485330660ztu^{9}v-2045993291273150ztu^{8}v^{2}-862745348526000ztu^{7}v^{3}-254665885855500ztu^{6}v^{4}-54028738903000ztu^{5}v^{5}-8860421927500ztu^{4}v^{6}-761012590000ztu^{3}v^{7}-79058398750ztu^{2}v^{8}-304342500ztuv^{9}-236693750ztv^{10}-1400923501926086zu^{11}-3564040880068868zu^{10}v-2856660047247790zu^{9}v^{2}-1290639005203760zu^{8}v^{3}-381461977035500zu^{7}v^{4}-80667663003800zu^{6}v^{5}-12101111107500zu^{5}v^{6}-1304081630000zu^{4}v^{7}-4894268750zu^{3}v^{8}-26048682500zu^{2}v^{9}+5204956250zuv^{10}+4662944292141286wtu^{10}+6311081246268876wtu^{9}v+4063903712963750wtu^{8}v^{2}+1642071592199120wtu^{7}v^{3}+458229549193100wtu^{6}v^{4}+92426883934600wtu^{5}v^{5}+13260947037500wtu^{4}v^{6}+1344236370000wtu^{3}v^{7}+89781068750wtu^{2}v^{8}-1799792500wtuv^{9}+1089078750wtv^{10}-4115181798282wu^{11}+1023164419499836wu^{10}v+1245560183881190wu^{9}v^{2}+710443655391120wu^{8}v^{3}+247400193832300wu^{7}v^{4}+60388100434600wu^{6}v^{5}+9674159557500wu^{5}v^{6}+1166235210000wu^{4}v^{7}+78866518750wu^{3}v^{8}-1566842500wu^{2}v^{9}+1089078750wuv^{10}-1569356469091598t^{2}u^{10}-2357759889053156t^{2}u^{9}v-1663223760155590t^{2}u^{8}v^{2}-726265572700720t^{2}u^{7}v^{3}-216948525978300t^{2}u^{6}v^{4}-46038423424600t^{2}u^{5}v^{5}-7380465057500t^{2}u^{4}v^{6}-744562590000t^{2}u^{3}v^{7}-41336013750t^{2}u^{2}v^{8}-63462500t^{2}uv^{9}-1089078750t^{2}v^{10}-1240170189819624tu^{11}-4125053435235880tu^{10}v-4164211245814000tu^{9}v^{2}-2291061365156000tu^{8}v^{3}-817186729302000tu^{7}v^{4}-203128809614000tu^{6}v^{5}-36050539840000tu^{5}v^{6}-4732476820000tu^{4}v^{7}-203554945000tu^{3}v^{8}-28609105000tu^{2}v^{9}+2948590000tuv^{10}-1400923501926086u^{11}v-2323870690249244u^{10}v^{2}-1636417382380630u^{9}v^{3}-690078273412880u^{8}v^{4}-195768581308300u^{7}v^{5}-40637325895400u^{6}v^{6}-5794313137500u^{5}v^{7}-581168570000u^{4}v^{8}+22314671250u^{3}v^{9}-19280817500u^{2}v^{10}+4352571250uv^{11}}$

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

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

Equation of the image curve:

$0$

$=$

$ -X^{5}Y^{2}Z-X^{4}Y^{4}-5X^{4}Y^{2}Z^{2}-2X^{3}Y^{5}+2X^{3}Y^{4}Z-5X^{3}Y^{3}Z^{2}-10X^{3}Y^{2}Z^{3}-4X^{2}Y^{5}Z+10X^{2}Y^{4}Z^{2}-20X^{2}Y^{3}Z^{3}-5X^{2}Y^{2}Z^{4}+5X^{2}YZ^{5}+X^{2}Z^{6}-10XY^{3}Z^{4}+6XY^{2}Z^{5}+3XYZ^{6}+2Y^{2}Z^{6} $

Modular covers

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

Factor curve	Level	Index	Degree	Genus	Rank
$X_0(5)$	$5$	$48$	$24$	$0$	$0$
16.48.0-8.bb.1.6	$16$	$6$	$6$	$0$	$0$

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
16.48.0-8.bb.1.6	$16$	$6$	$6$	$0$	$0$
80.144.3-40.bx.1.7	$80$	$2$	$2$	$3$	$?$
80.144.3-40.bx.1.18	$80$	$2$	$2$	$3$	$?$