Canonical model in $\mathbb{P}^{ 3 }$
| $ 0 $ | $=$ | $ 3 x^{2} + 3 x y + 2 x z + 2 x w + 2 y^{2} + y z + y w + 2 z^{2} - z w + 2 w^{2} $ |
| $=$ | $x^{2} z - x^{2} w + 2 x y z + x z^{2} - x w^{2} + y^{2} z + y z^{2} + z^{2} w - z w^{2}$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 6 x^{6} + 9 x^{5} y - x^{5} z + 13 x^{4} y^{2} - 20 x^{4} y z + 8 x^{4} z^{2} + 8 x^{3} y^{3} + \cdots + 15 y^{2} z^{4} $ |
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This modular curve has no real points, and therefore no rational points.
Maps between models of this curve
Birational map from canonical model to plane model:
| $\displaystyle X$ |
$=$ |
$\displaystyle x+w$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle z$ |
Maps to other modular curves
$j$-invariant map
of degree 100 from the canonical model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -2\cdot3^3\cdot5^4\,\frac{8756450474065920xy^{14}w^{3}+61987893962342400xy^{13}w^{4}+131887457925857280xy^{12}w^{5}+545703364185292800xy^{11}w^{6}-2699105529338265600xy^{10}w^{7}-41239928466953994240xy^{9}w^{8}-52022927941313495040xy^{8}w^{9}-504810626542821457920xy^{7}w^{10}-71619876835088609280xy^{6}w^{11}+1392566249013008428800xy^{5}w^{12}+5275935342373860787200xy^{4}w^{13}+57741836426664411815340xy^{3}w^{14}+101842107927021276227715xy^{2}w^{15}+161698798414408522356285xyw^{16}+212726268561457152xz^{17}-6158600230422970368xz^{16}w+65606254604448694272xz^{15}w^{2}-303819236604515450880xz^{14}w^{3}+199702234460195389440xz^{13}w^{4}+4472928440282089783296xz^{12}w^{5}-21601998884460291784704xz^{11}w^{6}+40138080292210504237056xz^{10}w^{7}+7056813490006990417920xz^{9}w^{8}-201280871404894537251840xz^{8}w^{9}+462326644387810101054336xz^{7}w^{10}-471645277156675161323904xz^{6}w^{11}-2387269065574178256584xz^{5}w^{12}+663580193269273320654420xz^{4}w^{13}-911525870254100151713865xz^{3}w^{14}+602818057300656861130677xz^{2}w^{15}-134110166684700453468263xzw^{16}-37796789705606913543313xw^{17}+10248908659752960y^{15}w^{3}+85352779119329280y^{14}w^{4}+207842523123548160y^{13}w^{5}+2895929535029575680y^{12}w^{6}+756418737497702400y^{11}w^{7}+8002272839671480320y^{10}w^{8}-45345784521212559360y^{9}w^{9}-376925705921295974400y^{8}w^{10}-364486946106011197440y^{7}w^{11}-2660342940135625075200y^{6}w^{12}+820399769841646629120y^{5}w^{13}+37994693045951317092600y^{4}w^{14}+80898846380591322399330y^{3}w^{15}+233768265619341312y^{2}z^{16}-3947003122701828096y^{2}z^{15}w+16088492120733646848y^{2}z^{14}w^{2}+75511056464433119232y^{2}z^{13}w^{3}-811429193215468634112y^{2}z^{12}w^{4}+2202894932981666807808y^{2}z^{11}w^{5}+1534535339066414530560y^{2}z^{10}w^{6}-25388480683858172903424y^{2}z^{9}w^{7}+70307088474067672449024y^{2}z^{8}w^{8}-82233431429554826287104y^{2}z^{7}w^{9}-17834939319970663783680y^{2}z^{6}w^{10}+196709130562028699236608y^{2}z^{5}w^{11}-255757594101184696500272y^{2}z^{4}w^{12}+60971394738922190377112y^{2}z^{3}w^{13}+219788416447113697882618y^{2}z^{2}w^{14}-235514088403030322602771y^{2}zw^{15}+212919700156984057714727y^{2}w^{16}+95842135751786496yz^{17}-3677820106102013952yz^{16}w+49930070267120320512yz^{15}w^{2}-322213741997385056256yz^{14}w^{3}+870553146038250110976yz^{13}w^{4}+1422273700128149471232yz^{12}w^{5}-18023008508759565434880yz^{11}w^{6}+59010305841465198428160yz^{10}w^{7}-89186759771833821376512yz^{9}w^{8}+658253774652369595392yz^{8}w^{9}+281183601055327422312576yz^{7}w^{10}-605014893140987118458304yz^{6}w^{11}+632190153775761161319464yz^{5}w^{12}-222794883501489329331696yz^{4}w^{13}-338494682482220828651709yz^{3}w^{14}+589591327127249499773430yz^{2}w^{15}-335350791769429749271771yzw^{16}+131286824625983540704996yw^{17}+233768265619341312z^{18}-3930645050412761088z^{17}w+12066546201730744320z^{16}w^{2}+157461013616828350464z^{15}w^{3}-1485309005508579950592z^{14}w^{4}+4566195789476630888448z^{13}w^{5}+1843511377791618514944z^{12}w^{6}-58618827476630123839488z^{11}w^{7}+202707912342698593320960z^{10}w^{8}-330619540622149606416384z^{9}w^{9}+122181189817945678794240z^{8}w^{10}+660302300652777993499392z^{7}w^{11}-1650138085783196977132256z^{6}w^{12}+1952477146340285515236528z^{5}w^{13}-1128955157306206706853512z^{4}w^{14}-150405949810945742795971z^{3}w^{15}+807730949493663306720860z^{2}w^{16}-575866995944895768755933zw^{17}+144114847886959549153532w^{18}}{183165322788864xy^{17}+2925948617883648xy^{16}w+17961943000154112xy^{15}w^{2}+50780302699659264xy^{14}w^{3}+109286200556126208xy^{13}w^{4}+547331169482440704xy^{12}w^{5}-478722045775970304xy^{11}w^{6}-24281999222592503808xy^{10}w^{7}-17409160676746506240xy^{9}w^{8}+869305274902652884992xy^{8}w^{9}+1277432819382064013568xy^{7}w^{10}-32820844603503573086592xy^{6}w^{11}-85757165621110410526242xy^{5}w^{12}+1205619435411322173101133xy^{4}w^{13}+5049925664107409026816911xy^{3}w^{14}-41554376268560164199236380xy^{2}w^{15}-206467364930993170734013770xyw^{16}-4133451792728457216xz^{17}+120303251693563281408xz^{16}w-1472335584377051086848xz^{15}w^{2}+10999134089332752384000xz^{14}w^{3}-58092875946193568624640xz^{13}w^{4}+233391988116358790860800xz^{12}w^{5}-740319231824473648527360xz^{11}w^{6}+1872456152357300969325312xz^{10}w^{7}-3664449291993473562047568xz^{9}w^{8}+4749648703667773814953536xz^{8}w^{9}+48371319129078576571866xz^{7}w^{10}-22834646080154919189688458xz^{6}w^{11}+82470636081562138595899828xz^{5}w^{12}-190139281637159669299335291xz^{4}w^{13}+310224595618249645557486513xz^{3}w^{14}-323499174308686482584149623xz^{2}w^{15}+122911173605063213338174849xzw^{16}+18416046649417844753938892xw^{17}+97061965922304y^{18}+1610915531194368y^{17}w+8251832618975232y^{16}w^{2}-2725432164089856y^{15}w^{3}-133496451895394304y^{14}w^{4}+58448969893675008y^{13}w^{5}+2181215047101972480y^{12}w^{6}-11597416859292401664y^{11}w^{7}-100361627356986335232y^{10}w^{8}+360016675879857463296y^{9}w^{9}+4044303623401660568064y^{8}w^{10}-10308832324165647906816y^{7}w^{11}-170641629692531851053492y^{6}w^{12}+201430617500449697395044y^{5}w^{13}+7127482655538523238572155y^{4}w^{14}+2018222329730363484667896y^{3}w^{15}-4582374747719860224y^{2}z^{16}+77056068032815693824y^{2}z^{15}w-588100877374352523264y^{2}z^{14}w^{2}+2586676445842890424320y^{2}z^{13}w^{3}-5957360358118899425280y^{2}z^{12}w^{4}-4544049003542819241984y^{2}z^{11}w^{5}+109205498406980827726848y^{2}z^{10}w^{6}-573056452956631280795136y^{2}z^{9}w^{7}+2043567553143438261343584y^{2}z^{8}w^{8}-5711299478956198736260896y^{2}z^{7}w^{9}+13020614475462252636461880y^{2}z^{6}w^{10}-24047807758683079788835776y^{2}z^{5}w^{11}+33835670938844343682633969y^{2}z^{4}w^{12}-28339512770628275947197253y^{2}z^{3}w^{13}-6151312131302356202132825y^{2}z^{2}w^{14}+52364730273665018085469391y^{2}zw^{15}-154726924385681242087144387y^{2}w^{16}-1866656595951747072yz^{17}+71905876415135612928yz^{16}w-1027570255927698456576yz^{15}w^{2}+8700210604646317817856yz^{14}w^{3}-51738877122838412857344yz^{13}w^{4}+235225074169658003060736yz^{12}w^{5}-858213343101670611388416yz^{11}w^{6}+2584301015348606699887104yz^{10}w^{7}-6500095712607172096704240yz^{9}w^{8}+13555794394456129431845400yz^{8}w^{9}-22409988975045726837220554yz^{7}w^{10}+24760682026555493720119500yz^{6}w^{11}+479721178009818769679000yz^{5}w^{12}-83323644233596841619066921yz^{4}w^{13}+231560916906465385423256175yz^{3}w^{14}-323748798597551666543137026yz^{2}w^{15}+171897689004578154182616722yzw^{16}-82326738218763101172982448yw^{17}-4574373410588590080z^{18}+77078327181035175936z^{17}w-511549539982164099072z^{16}w^{2}+1099955198958349123584z^{15}w^{3}+8489159411518955372544z^{14}w^{4}-98313933546673766277120z^{13}w^{5}+565264246517721174988800z^{12}w^{6}-2337955063217154229501440z^{11}w^{7}+7667999868192600038655168z^{10}w^{8}-20735300756879454650453472z^{9}w^{9}+46714335560434622125692816z^{8}w^{10}-86069938880595424516186830z^{7}w^{11}+120243847283254436229414292z^{6}w^{12}-91062363783254230674697494z^{5}w^{13}-94884026531384293850109824z^{4}w^{14}+479349650733449928499162424z^{3}w^{15}-750825408857472641623072336z^{2}w^{16}+552409917545775554632821916zw^{17}-160946857860359785173899812w^{18}}$ 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Cover information
Click on a modular curve in the diagram to see information about it.
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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.
