Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 3 x^{2} + x y - 2 x z - 2 y^{2} - 2 y z + 2 z^{2} $ |
| $=$ | $2 x^{2} + 7 x y + x z + x w + y^{2} + y z - z^{2} - w^{2}$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 944 x^{4} - 16 x^{3} y - 86 x^{3} z + x^{2} y^{2} + 2 x^{2} y z + 39 x^{2} z^{2} - x y z^{2} + 4 x z^{3} - z^{4} $ |
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This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle 30z$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle 2w$ |
Maps to other modular curves
$j$-invariant map
of degree 72 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -\frac{1}{2^{12}}\cdot\frac{82768833494691407790995435110571293591797760xz^{17}+20064032257277951977030590043608328035434496xz^{16}w+73241275135023049632076626148033863105576960xz^{15}w^{2}+18610042494650076762878485968424138020634624xz^{14}w^{3}+25598472574216310654590495580073699042729984xz^{13}w^{4}+6662512420740630368215287348674421256310784xz^{12}w^{5}+4556060747839479294396655538728803157428224xz^{11}w^{6}+1098152416622808118112647826238285585805824xz^{10}w^{7}+405677854122505394884177196975945211144960xz^{9}w^{8}+73826929785652902031319601611714885184000xz^{8}w^{9}+11249305968687116986216236819442995133056xz^{7}w^{10}-25478322877201794603935194244179967904xz^{6}w^{11}-544434483194323105464672270732235041744xz^{5}w^{12}-131428989426830494510407386553664262904xz^{4}w^{13}-13015124305634247846703609734629406684xz^{3}w^{14}+1706673633407160519819138309832779045xz^{2}w^{15}+944437744020484054090915229999086299xzw^{16}-75676536978939616398841078539776460xw^{17}-148902623844060919070239403570002848764592128y^{2}z^{16}-12271806477406694142520316867873522818482176y^{2}z^{15}w-85963720277855191909777535613332473495879680y^{2}z^{14}w^{2}-12490651816299156030814012188056130061205504y^{2}z^{13}w^{3}-20421172143857998878051490178589644589424640y^{2}z^{12}w^{4}-3701486646389803528109320639663098148356096y^{2}z^{11}w^{5}-2412779501629565391083027455440649540986880y^{2}z^{10}w^{6}-428757106566804941806903782279257519042560y^{2}z^{9}w^{7}-122234558663461101322128319018049309210880y^{2}z^{8}w^{8}-13202314466605183352232732557276638371840y^{2}z^{7}w^{9}+734341455361451585934639869818591420800y^{2}z^{6}w^{10}+707213731230799954218047094319594987776y^{2}z^{5}w^{11}+186193241518427226449320887083308881840y^{2}z^{4}w^{12}+14589242464411824130261949007118556544y^{2}z^{3}w^{13}-3089367109851944315317078287508691220y^{2}z^{2}w^{14}-1111937415876887519615736302645770584y^{2}zw^{15}+113966012494442490552119692381407093y^{2}w^{16}+142047222239289061024169722178437766101139456yz^{17}-12271806477406694142520316867873522818482176yz^{16}w+98927048142818350468319035141658928651501568yz^{15}w^{2}+1187976203634318769808241760000659351535616yz^{14}w^{3}+28096287352910735909896138448498298642178048yz^{13}w^{4}+2513921815006724184071906983707118339588096yz^{12}w^{5}+4136006840577925072371444152389923744716800yz^{11}w^{6}+571453957743780764950724734889720860651520yz^{10}w^{7}+312159713491038317414388693602728135304960yz^{9}w^{8}+43382284937675260977770061795376603345920yz^{8}w^{9}+7304997749069324882711028801889546924416yz^{7}w^{10}+11366018445301216329342024236323090176yz^{6}w^{11}-381951348080332629930477183614452715088yz^{5}w^{12}-82525946947679076524559061937910388416yz^{4}w^{13}-8282568064568317205303920744855969428yz^{3}w^{14}+818426029903197808569519516939188568yz^{2}w^{15}+666453390051262174186981154337643653yzw^{16}-60861469939318181692307691262558494yw^{17}-30914289544139513673867165245887113286713344z^{18}+12271806477406694142520316867873522818482176z^{17}w-48961828606330515214440275908575297953595392z^{16}w^{2}+2335297811969938123858077826638967375527936z^{15}w^{3}-22595310312074211765631022786459146321272832z^{14}w^{4}-2055201437426021567634197708946446438137856z^{13}w^{5}-5003336852032214458210829564030973175592960z^{12}w^{6}-814492781126071599261657666345607133425664z^{11}w^{7}-583522059822131117148376185697653003849472z^{10}w^{8}-102942452503921099703161121781919024675840z^{9}w^{9}-30227530979051575973491696370004581962368z^{8}w^{10}-3429386043870006090521930550715417940736z^{7}w^{11}+144630784228323720758101854541402149840z^{6}w^{12}+170900049671930421290917013975907928896z^{5}w^{13}+46965661049556293386518275433276813252z^{4}w^{14}+4020581223553934713722964189816944264z^{3}w^{15}-760516651912389122494874639968692633z^{2}w^{16}-277984353969221879903934075661442646zw^{17}+25685268965801578026706911129703289w^{18}}{207594528567685944854400000xz^{17}+252677478574997904146400000xz^{16}w-53931808411807942843295904000xz^{15}w^{2}+138371725929628091573457811200xz^{14}w^{3}+1278573109303799484712040366400xz^{13}w^{4}-3492550027018647967360004803152xz^{12}w^{5}-6061973139081054898622710071168xz^{11}w^{6}+16291811681221803571911959827968xz^{10}w^{7}+8133636427271322723713998831212xz^{9}w^{8}-19173216639895920117947970148633xz^{8}w^{9}-8043888878287078376412806462223xz^{7}w^{10}+7808990443063676360485096090524xz^{6}w^{11}-911637355011350757930637983744xz^{5}w^{12}-5690758956456945850750102142976xz^{4}w^{13}-4878183743571919740350188486656xz^{3}w^{14}-1470494061492078441458971967488xz^{2}w^{15}-235111013047284995502070824960xzw^{16}-14324456674899559632705945600xw^{17}+36550612640723075452800000y^{2}z^{16}+3301923215217303690393600000y^{2}z^{15}w-25508034578780209606059360000y^{2}z^{14}w^{2}-156181275511024949944063027200y^{2}z^{13}w^{3}+1018568362435090748464560475200y^{2}z^{12}w^{4}+809982725680435937290979348736y^{2}z^{11}w^{5}-7759366117624150418910661508544y^{2}z^{10}w^{6}+1167898777257984476026795228800y^{2}z^{9}w^{7}+15161151688838490838603935893868y^{2}z^{8}w^{8}-2305232868657035903965603063840y^{2}z^{7}w^{9}-9968203206096150945420083894113y^{2}z^{6}w^{10}-1449527209733511018736640851968y^{2}z^{5}w^{11}+13258751964045099347961760972800y^{2}z^{4}w^{12}+6672352517216969289268583202816y^{2}z^{3}w^{13}+2134457806499335229805774766080y^{2}z^{2}w^{14}+298307695714562715626913136640y^{2}zw^{15}+18407438135159400205962444800y^{2}w^{16}+36550612640723075452800000yz^{17}+3301923215217303690393600000yz^{16}w-25530840434237137988646240000yz^{15}w^{2}-155771343737743801183103347200yz^{14}w^{3}+1022332980973540666243212686400yz^{13}w^{4}+769889950040575089625989870336yz^{12}w^{5}-7789622617590883473038311333824yz^{11}w^{6}+1759516713095249400212258328672yz^{10}w^{7}+14839980924088678832417524945068yz^{9}w^{8}-4378511482883178012514307614976yz^{8}w^{9}-8551147504605900340073652701857yz^{7}w^{10}+703684908670192791365115565862yz^{6}w^{11}-6960778548615285822531154477056yz^{5}w^{12}-3991686916736776074712773033984yz^{4}w^{13}-3474645214078631290734045560832yz^{3}w^{14}-952435765293020169337472811008yz^{2}w^{15}-160534089118644316595342540800yzw^{16}-9722597141109709581215334400yw^{17}-36550612640723075452800000z^{18}-3301923215217303690393600000z^{17}w+25482792015813899685108960000z^{16}w^{2}+156374119850685868910008051200z^{15}w^{3}-1013160310643195298107130120000z^{14}w^{4}-840210824291480232890679150336z^{13}w^{5}+7667410054961008905218109569024z^{12}w^{6}-625601364724394760599735857056z^{11}w^{7}-15070264224282590498432525897004z^{10}w^{8}+224199697573133965750099241024z^{9}w^{9}+10817195297238379511686327366477z^{8}w^{10}+3349433306799899020171531935878z^{7}w^{11}-1997233530150047671181761982037z^{6}w^{12}-1339621292471434447214459486208z^{5}w^{13}+2918583690305966189014708912128z^{4}w^{14}+1593257806736085932492527239168z^{3}w^{15}+528805040861631953608659435520z^{2}w^{16}+74576923928640678906728284160zw^{17}+4601859533789850051490611200w^{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.
