Canonical model in $\mathbb{P}^{ 8 }$ defined by 21 equations
| $ 0 $ | $=$ | $ x^{2} + x y + y r $ |
| $=$ | $x z + x u - x s - y z - y t - y s$ |
| $=$ | $x z - x t - y z + y t + y u$ |
| $=$ | $x z + x w + 2 x v - x s + u r$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - x^{9} y + 3 x^{8} z^{2} + 3 x^{7} y^{3} + 3 x^{7} y z^{2} - 18 x^{6} y^{2} z^{2} - 3 x^{5} y^{5} + \cdots + 3 y^{8} z^{2} $ |
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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 |
|---|
| $(0:0:0:1/2:0:0:1:0:0)$, $(0:0:1:0:-2:0:0:0:1)$, $(0:0:0:0:-1:1:1:0:1)$, $(0:0:0:0:0:0:1/2:0:1)$, $(0:0:1:0:1:0:0:0:1)$, $(0:0:-1/2:-3/10:-1/2:0:9/10:0:1)$, $(0:0:0:-2:0:0:1:0:0)$, $(0:0:-1/5:-6/25:-4/5:6/5:18/25:0:1)$ |
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 \frac{2410871852091480982502400000xr^{9}-17426214713292751239581047500xr^{7}s^{2}+173929393657525295679303884400xr^{5}s^{4}+100414093997391705845329469544xr^{3}s^{6}+106318762497774517196497098447xrs^{8}-766587741849286886679930000yvr^{7}s-15379607967652732818275550000yvr^{5}s^{3}-96860101776098574648863736000yvr^{3}s^{5}-78001984760668131446463633996yvrs^{7}-715781831615443099468800000yr^{9}+2856443521979155436656818750yr^{7}s^{2}-36530485111799671896471879900yr^{5}s^{4}+63476642433467886616912581324yr^{3}s^{6}-72523949705009609428791581553yrs^{8}+119606439181731482243029879065zvs^{8}-670963745577847810908960000zr^{8}s+21216625033543153741762841250zr^{6}s^{3}-4486552431362593679951077860zr^{4}s^{5}+22698506535348131706967845768zr^{2}s^{7}+9069681879008898645442655601zs^{9}-292256812560000000000000000wv^{9}+1729995826080000000000000000wv^{8}s-4752936081667000000000000000wv^{7}s^{2}+14031854058552700000000000000wv^{6}s^{3}-17776629680800939940000000000wv^{5}s^{4}-9830344664009677414295000000wv^{4}s^{5}+50905523699522718767174860000wv^{3}s^{6}-51328601100838836566464784200wv^{2}s^{7}-162883457458460849810590249wvs^{8}-651180488955375418845705000wr^{8}s-10441823125337083694131785000wr^{6}s^{3}+25428614500501290968342628240wr^{4}s^{5}-1596270459246813275500967718wr^{2}s^{7}-161275078829509251843495425400ws^{9}-341154478811111631897600000tvr^{8}+12248299785044137895373858750tvr^{6}s^{2}-41630118134459935437372737700tvr^{4}s^{4}+81191153349311082632602236tvr^{2}s^{6}+19774547901724885426439280272tvs^{8}-281485991114123954485860000tr^{8}s-9740715671318808537047373750tr^{6}s^{3}-43112795263732889521033171140tr^{4}s^{5}-35966407886580041113974069396tr^{2}s^{7}-3414700474405920000000000000ts^{9}+323123884389621899136000000uvr^{8}+5127914840847584091398527500uvr^{6}s^{2}+753220955458558975205107800uvr^{4}s^{4}+19473259459767491538456196584uvr^{2}s^{6}+58944525711801032984482913897uvs^{8}+1117577850203467974975750000ur^{8}s-4044077052978186042154852500ur^{6}s^{3}+34988277665593825027623684360ur^{4}s^{5}+18663874703253090995977620204ur^{2}s^{7}-58952097923908392984482913897us^{9}+153953140480000000000000000v^{10}-1062657870880000000000000000v^{9}s+3257287959706000000000000000v^{8}s^{2}-9116871107304600000000000000v^{7}s^{3}+14943567218612756600000000000v^{6}s^{4}+4339170346504507030760000000v^{5}s^{5}-76764661757993380069710480000v^{4}s^{6}+124088412403265006382561253600v^{3}s^{7}+45909118549358284032000000v^{2}r^{8}-5547295672255486550488893750v^{2}r^{6}s^{2}+52472685415905484922292754500v^{2}r^{4}s^{4}+19218932251033280561612084100v^{2}r^{2}s^{6}-53387588640603349985030291058v^{2}s^{8}-3869167707150805350530955000vr^{8}s+24501812370562554363326801250vr^{6}s^{3}-200139726394294283097254488500vr^{4}s^{5}-63568531961465612471185231806vr^{2}s^{7}+15199091478056604113301453331vs^{9}+734931449695114555852800000r^{10}-4237495451499779502698182500r^{8}s^{2}+46362542307350689209541693500r^{6}s^{4}+78700847306011436914408875744r^{4}s^{6}+10909814482510210761563568027r^{2}s^{8}-5274606018790338645442655601s^{10}}{853044103354513121280000xr^{9}-12087490650519932983296000xr^{7}s^{2}+2783360070099900873235936800xr^{5}s^{4}+76021017287490504448419401118xr^{3}s^{6}+7901397994054934412108933246xrs^{8}+2891017316945119555584000yvr^{7}s+97437058959728386741968000yvr^{5}s^{3}+8114565243356303308672535640yvr^{3}s^{5}-57253432305764675667378789852yvrs^{7}-83714214836056144896000yr^{9}-575357078087748191232000yr^{7}s^{2}-683728128173638088471062800yr^{5}s^{4}-35832779968549545423144861507yr^{3}s^{6}+188153139679833743186140665459yrs^{8}-96427313607723876261264040006zvs^{8}-679260047838611208192000zr^{8}s-20260488121315957792512000zr^{6}s^{3}-2919036999598729808891923440zr^{4}s^{5}-15824546273090588639250529656zr^{2}s^{7}-25344713191841532078903889929zs^{9}+2363093649920000000000000wv^{6}s^{3}+464148269949075120000000000wv^{5}s^{4}-1547269081184147811536000000wv^{4}s^{5}+16185748528788611406417470000wv^{3}s^{6}-32755846798999362520105100850wv^{2}s^{7}+31862594643301127182509855180wvs^{8}-587353001916146973696000wr^{8}s-29591627165661379079736000wr^{6}s^{3}-2217611936957954843747288580wr^{4}s^{5}+24535923659228630366404350552wr^{2}s^{7}+151193523935416540802468997474ws^{9}-524532690814452166656000tvr^{8}+4047196145547557855232000tvr^{6}s^{2}-1064745683099087715292928400tvr^{4}s^{4}-28684780761694178596831827003tvr^{2}s^{6}-19369550703834548847810754329tvs^{8}-83285496156514111488000tr^{8}s+7211523833273007728928000tr^{6}s^{3}+319652447524778230988280000tr^{4}s^{5}-7756758427852852177153447395tr^{2}s^{7}-338085437557158899712000uvr^{8}+1509378833332585101312000uvr^{6}s^{2}-499812227824165532674058400uvr^{4}s^{4}-1565439730731070429806303462uvr^{2}s^{6}-52583940124084082558847635703uvs^{8}-1229385302804218152960000ur^{8}s-49191551616450203559216000ur^{6}s^{3}-3640987323616963358641502280ur^{4}s^{5}+33903818630360075795710046940ur^{2}s^{7}+52583940124084082558847635703us^{9}-1181546824960000000000000v^{7}s^{3}-231182754441392800000000000v^{6}s^{4}+1010268771112074651488000000v^{5}s^{5}-9573478342769307883127460000v^{4}s^{6}+6062654167729634573310536800v^{3}s^{7}+601688822708081664000000v^{2}r^{8}-6298545375370158243840000v^{2}r^{6}s^{2}+1254533108578151344063074000v^{2}r^{4}s^{4}+22154889109408870008909162975v^{2}r^{2}s^{6}+18165919635472501976153521000v^{2}s^{8}-40483802606806416384000vr^{8}s+58584322837320413755608000vr^{6}s^{3}-96559708111071859585598460vr^{4}s^{5}-90599377375207387269695296467vr^{2}s^{7}-60147263825954631444539242058vs^{9}+304055288780335506432000r^{10}-2473495780939207271424000r^{8}s^{2}+952662135361879317788604000r^{6}s^{4}+27891398932357813841012532258r^{4}s^{6}+48926017892784972895307866980r^{2}s^{8}+25344713191841532078903889929s^{10}}$ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
60.144.9.fv.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{3}w$ |
Equation of the image curve:
| $0$ |
$=$ |
$ -X^{9}Y+3X^{8}Z^{2}+3X^{7}Y^{3}+3X^{7}YZ^{2}-18X^{6}Y^{2}Z^{2}-3X^{5}Y^{5}+3X^{5}Y^{3}Z^{2}+78X^{4}Y^{4}Z^{2}+X^{3}Y^{7}-3X^{3}Y^{5}Z^{2}-225X^{3}Y^{3}Z^{4}-18X^{2}Y^{6}Z^{2}-3XY^{7}Z^{2}+225XY^{5}Z^{4}+3Y^{8}Z^{2} $ |
The following modular covers realize this modular curve as a fiber product over $X(1)$.
This modular curve minimally covers the modular curves listed below.
This modular curve is minimally covered by the modular curves in the database listed below.
