Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ - x t + y^{2} $ |
| $=$ | $y z + y v + w t$ |
| $=$ | $x z + x v + y w$ |
| $=$ | $x^{2} + x t - x v - z w - w u$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 5 x^{8} + 16 x^{7} y + 5 x^{7} z + 25 x^{6} y^{2} + 32 x^{6} y z - 5 x^{6} z^{2} + 25 x^{5} y^{2} z + \cdots + 5 y^{6} z^{2} $ |
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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 |
|---|
| $(0:0:0:-1:0:0:1)$, $(0:0:1:0:0:-1:1)$, $(0:0:-1:0:0:-1:1)$, $(0:0:0:0:0:1:0)$ |
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{1}{2\cdot5}\cdot\frac{4394552450849989923296545826749359255196714991393993147xtu^{10}-8439655486601005663482844777562806224251351154990809806xtu^{9}v+11165741010320668432814059632896834853847687994374836119xtu^{8}v^{2}+38247080181808559236870293623547163897777638262549838592xtu^{7}v^{3}-168010638759425329762999409825120875017264516354301541234xtu^{6}v^{4}+212605264507709507430393614651824607666772314075183877300xtu^{5}v^{5}-937171091664267776702300195447601963935828572147051139394xtu^{4}v^{6}+893530485559124914773196775529626437063442326553553500528xtu^{3}v^{7}-993367628915377564072897490903309031869578964371610267961xtu^{2}v^{8}+1525530384982745924359515122728319908502246558459205823082xtuv^{9}-629112706345904682765110404047077774945045946857677075797xtv^{10}-184870088269704225265729344946837876980847639529661672xu^{11}+934918325620594487464897567526200834717625973103827106xu^{10}v-16314912148521665526694382963524874314007586841919179894xu^{9}v^{2}+42318540377489973261877451796623033098714304934408760808xu^{8}v^{3}-65594880684182329678051125370073872016146648885856738216xu^{7}v^{4}+454561840016031573768966114129238205882363692714835267100xu^{6}v^{5}-125143010556068688107861310344451123842126698824203251556xu^{5}v^{6}+1020103143882378405394683115238941263013269679178914356472xu^{4}v^{7}-1199433676610894868051056172305555546592226129120025849664xu^{3}v^{8}+538299063463880697481553977679949712773004790057402504018xu^{2}v^{9}-1489273555914501698637137684473908457043503024675928421478xuv^{10}+809680214840865590007350687978886009951780330937500000000xv^{11}-6184518269356679499005997085364908444461977749326598380ytu^{10}+34733029547250710896323732582466119442160259991024701970ytu^{9}v-92818976594638588307315082458039946767890237296850921050ytu^{8}v^{2}+302471632751556500908352715757775619703887720952878630200ytu^{7}v^{3}-437543544202372988132478493130172083215458903683794338400ytu^{6}v^{4}+893325996402187899235829766960461401050299733482785054300ytu^{5}v^{5}-1334453051633026652347469378102758839565704794267581195100ytu^{4}v^{6}+987911311513070055593098525430510040245281452934175686600ytu^{3}v^{7}-1496951508179001389212014733494551519981423776940211206500ytu^{2}v^{8}+913592683813776909968772718735363161361832478961260462850ytuv^{9}-82729967484326372513434500565839693191391264259899428650ytv^{10}+2019562034227221104739958073031309666778123830542787932yu^{11}-16527010905096783013115170855441698807147064950400411969yu^{10}v+41434529332902146350053477073980009915932882630208479348yu^{9}v^{2}-179930121757002470093431655652220768068669402844664798759yu^{8}v^{3}+276530396096490924597559103637579935428610555042930154448yu^{7}v^{4}-640268787498841389158954675683685099154376239584062900994yu^{6}v^{5}+1128189611329955682223616639372152468465367600204956594576yu^{5}v^{6}-1135757618823939405070800149303425632975850566577367913046yu^{4}v^{7}+1909030969421230766284392683397075455621723529843102289652yu^{3}v^{8}-1333049366773266521911574950516049066071970541355201121949yu^{2}v^{9}+1282836330013097449146499300147650693714153063557075177420yuv^{10}-675496462929428676486686991087405654451724979437889941267yv^{11}-4159006488941128630631107697289469605802425183660270226ztu^{10}+12233589191374076455755977160416407001980018415601983978ztu^{9}v-30535016237689852009775872731102332450497212127323420332ztu^{8}v^{2}+63649887280515177926947149896095879120452687302410266424ztu^{7}v^{3}+11604724853706214498149435363137318159403526516443938452ztu^{6}v^{4}+155057311451200159459309939412640814195921109253256155020ztu^{5}v^{5}+277951810404163796190661386866859839210191409845607286432ztu^{4}v^{6}-338542493059255536186222733974431899118756109412755042984ztu^{3}v^{7}+184024364271827414227167290583123238543580980999486943758ztu^{2}v^{8}-849896837352236956466898697586087290286970091843827305046ztuv^{9}+458798814199545482084338100763629711724517756017697445836ztv^{10}-3002777903768416584702349351473924781495265546517640705zu^{11}+14678469400191327635024351556231548235072121392219731325zu^{10}v-40416182701079455534515664198637624899764093524634426495zu^{9}v^{2}+72030876338440426293958202060970948292871607950971526515zu^{8}v^{3}-228757848611010330360644913400055963429553379089215581930zu^{7}v^{4}+54653805463573690921644650525472887731790618195409246450zu^{6}v^{5}-444721370644370812116875406243873018298785232516490968350zu^{5}v^{6}+172760121813878874375343344735137318429100377815626760310zu^{4}v^{7}+304941229001670078095034943427779069963830586104253422555zu^{3}v^{8}+313073733542786951507236757035287754364535509280531909265zu^{2}v^{9}+538680543553234962317684644275604844577573791195129104285zuv^{10}-500427355004373328997603816938048424372123990520068362745zv^{11}-5513867611263527865968096959137323295045749369344968964wtu^{10}+30450208149883496702898652477086045542568752558939916642wtu^{9}v-88036258001396231531647247983984057848039807465859377858wtu^{8}v^{2}+199178739902900754797952156632303920635676530075212488856wtu^{7}v^{3}-5634307946749881751292254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|
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
40.144.7.fr.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle z$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 5X^{8}+16X^{7}Y+5X^{7}Z+25X^{6}Y^{2}+32X^{6}YZ-5X^{6}Z^{2}+25X^{5}Y^{2}Z-16X^{5}YZ^{2}-25X^{4}Y^{4}-25X^{4}Y^{2}Z^{2}-32X^{4}YZ^{3}-25X^{3}Y^{5}-25X^{3}Y^{4}Z+16X^{3}YZ^{4}-5X^{2}Y^{6}+25X^{2}Y^{4}Z^{2}-5XY^{6}Z+5Y^{6}Z^{2} $ |
This modular curve minimally covers the modular curves listed below.
