Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ x w + x u + y^{2} $ |
| $=$ | $x z - x w + y z + y t$ |
| $=$ | $y z - y w - z w - z u - w t - t u$ |
| $=$ | $x y - x w - y w + y t - y u + z^{2} + z w + z v + w t + w v + t u$ |
| $=$ | $\cdots$ |
![Copy content]()
![Toggle raw display]()
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - x^{7} z^{2} - x^{6} y^{2} z - 4 x^{6} y z^{2} + x^{5} y^{4} - 4 x^{5} y^{3} z - 9 x^{5} y^{2} z^{2} + \cdots + 5 y^{8} z $ |
![Copy content]()
![Toggle raw display]()
This modular curve has 6 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:0:1:-1:-1:1:1)$, $(0:0:-1:-1:-3:1:1)$, $(0:0:0:0:0:0:1)$, $(0:0:-1:0:2:0:1)$, $(0:0:0:0:0:1:0)$, $(0:0:0:1:0:0:0)$ |
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}{2^6}\cdot\frac{11378480213131774152818203307066778175660544xu^{11}-47035106206752499274612596990396227121923072xu^{10}v+91724539766043885345845851066792166158471296xu^{9}v^{2}-114292126410056121489582851666751321583448064xu^{8}v^{3}+96214181473079478862501097996467924932062880xu^{7}v^{4}-43775628072027573807997800553424183244989600xu^{6}v^{5}-10555181779801289435485553044900986699604680xu^{5}v^{6}+34845176077802593916910133130884548781985264xu^{4}v^{7}-28360472673177456383370425953284665684286184xu^{3}v^{8}+12697130330271548676039866767372872786511342xu^{2}v^{9}-3156384686207241053516574555082386402347745xuv^{10}+340308081688609986640215789591244800000000xv^{11}+58307319803133815418809324436880321027241728ytu^{10}-306736447744613486322402078493137625673595264ytu^{9}v+769626811173571598047196650424011461974789952ytu^{8}v^{2}-1223475787052941579267942732296710485673897568ytu^{7}v^{3}+1363294380885603256886214848369787630500704560ytu^{6}v^{4}-1105600986524922012801442598339878274264096200ytu^{5}v^{5}+656720132280100122385805804455421846794676940ytu^{4}v^{6}-280949441092864408870053764640553941848812682ytu^{3}v^{7}+82597817005863456911115852933454737106473742ytu^{2}v^{8}-15010745301000742303254184678159074368975596ytuv^{9}+1271356865516063132118115726579428882681135ytv^{10}+12721770708327087516385155575374477130187008yu^{11}-67217388397367719417266294221652923526648832yu^{10}v+174283536022472768711206298839808169693327936yu^{9}v^{2}-298076281213926126188183047950590949471624000yu^{8}v^{3}+369077050404261717530225381767221021993307728yu^{7}v^{4}-335428362206236981085127525420361240247984960yu^{6}v^{5}+219182817437402883502757376638838143528412940yu^{5}v^{6}-98505238421012364595835018503233033261726292yu^{4}v^{7}+27696318017681959062128319194409222718858294yu^{3}v^{8}-3609429703026566982601069492518551188403523yu^{2}v^{9}-204569024629887579507027204881562994194194yuv^{10}+89875461238376814442747431785550317318865yv^{11}+15480102068064674529033705939053931301132032ztu^{10}-84863107076949049186408072211295617932972416ztu^{9}v+215562935161479030279028579491954479786743488ztu^{8}v^{2}-337956227911896729709554427769611916305168992ztu^{7}v^{3}+364651357911486843847898700926572027577109840ztu^{6}v^{4}-283255248405756569101389841785469833601737000ztu^{5}v^{5}+159773486278444143145569502426127277705377460ztu^{4}v^{6}-64081996867657046574838908864553793376723258ztu^{3}v^{7}+17251891124294292941738535202944806519491698ztu^{2}v^{8}-2750896631038024800128367495036599159020299ztuv^{9}+189435574786364092360538389699366885688040ztv^{10}+16468438724132097517412210641252203435631616zu^{11}-92732239146034031810971371628661872888775040zu^{10}v+214066935745837012445581171690393630899980160zu^{9}v^{2}-253133451340991588806390392546891863822657184zu^{8}v^{3}+126440005545750296855377539693998442454317312zu^{7}v^{4}+70825833785770842309443807855155884413650360zu^{6}v^{5}-183375967730647829540452598546818217613644520zu^{5}v^{6}+168018344104669578926233290198321626986823186zu^{4}v^{7}-93478674286641917301451335544646011780985368zu^{3}v^{8}+33173993773441758362523487939630084353466915zu^{2}v^{9}-6946121875088586079905430963556977938064656zuv^{10}+651881627166032819055550645678988314311960zv^{11}-151248036306048882951207017596108800w^{12}-1209984290448391063609656140768870400w^{11}v-4310569034722393164109400001489100800w^{10}v^{2}-8734574096674322990432205266175283200w^{9}v^{3}-10658260058441882220467869521225792000w^{8}v^{4}-7621483079484494492463166121053920000w^{7}v^{5}-2759685849943571610410499918716500800w^{6}v^{6}-117867122043190438081116406290717600w^{5}v^{7}-256541926815496732007693055358634250w^{4}v^{8}-964400091849158981508157099685540175w^{3}v^{9}-33196451292043169594918182754213925209673600w^{2}u^{10}+185504843799399509542184296151110733179977600w^{2}u^{9}v-483262408420901714624174609410925214084592800w^{2}u^{8}v^{2}+779817112097875041410986582833950235157296000w^{2}u^{7}v^{3}-867768314547720181991117182928201478108251400w^{2}u^{6}v^{4}+696910070022764100074348131034932975079391000w^{2}u^{5}v^{5}-408593211846427601346557977660560236709871950w^{2}u^{4}v^{6}+172164387027022682489627221448722815455862900w^{2}u^{3}v^{7}-49658217946498220455001870985475605863000775w^{2}u^{2}v^{8}+8790984171524023482726243745311519449348175w^{2}uv^{9}-718442595478600507837888604579738558117100w^{2}v^{10}-416483550324866077841676639672276589930368wtu^{10}+16720803541659611964691108481964228508646784wtu^{9}v-79414669612043022854128767155034292066190112wtu^{8}v^{2}+186233109854908738282474515772598227201070208wtu^{7}v^{3}-278239807140125400082624689670320171191923560wtu^{6}v^{4}+292591108039614117141455247107185508169651000wtu^{5}v^{5}-224291330858989505218499284434720883040906790wtu^{4}v^{6}+124473452636531817591811360046931207070379892wtu^{3}v^{7}-47689089046491434351047299700346941656151677wtu^{2}v^{8}+11273276615311315277122930512053607704156601wtuv^{9}-1226126784327593085719865886922629900388910wtv^{10}-111423959372655041862702487557170219518908160wu^{11}+640162973891221342769546208692616451677272576wu^{10}v-1736568785628041389725030410894577560283684288wu^{9}v^{2}+2961754996792509486261896448271952520770178624wu^{8}v^{3}-3543777195474906424448530068965424463144691376wu^{7}v^{4}+3125341120800564289283296638497336779860197920wu^{6}v^{5}-2070146202183897658519160059085003722946496500wu^{5}v^{6}+1026971331947158459123688297587670666865276020wu^{4}v^{7}-371965781570943609923363903641451712693417464wu^{3}v^{8}+92733259710492593442427799806056063215896314wu^{2}v^{9}-14067528907569528680259014370865967095499372wuv^{10}+957446526637071396717135515213830455197145wv^{11}+15387008846739517494699636073705935762921216t^{2}u^{10}-91234142401752876951233236028721872644823808t^{2}u^{9}v+249191413400293711495730271495963381888951744t^{2}u^{8}v^{2}-419042024666989931087741026868636760877840896t^{2}u^{7}v^{3}+486472887109049069205263850376822487733713520t^{2}u^{6}v^{4}-410979786974030884018091488493437627815056400t^{2}u^{5}v^{5}+257165398608262784810657581255762770982315380t^{2}u^{4}v^{6}-117902547331325325953312097900368179227373704t^{2}u^{3}v^{7}+37848005804279389916639878436710725620157324t^{2}u^{2}v^{8}-7635728625024132457166832132272344038903162t^{2}uv^{9}+725705679404477460614280786933011243087220t^{2}v^{10}+82934655910692108022831241733938730140430592tu^{11}-483871097873920819377240169090707603999338112tu^{10}v+1329184297607181605082384006627592591401155136tu^{9}v^{2}-2283777238966491728130469424326709637549076896tu^{8}v^{3}+2738239012758778084207477415969138966336676336tu^{7}v^{4}-2412269876361014710762266121992368069363773720tu^{6}v^{5}+1596890812352594792579246789924138137044806460tu^{5}v^{6}-795793539457861660437393329780630330436719078tu^{4}v^{7}+292730561655165339799398292549914741024464992tu^{3}v^{8}-75557559344927470270386001893359838945488668tu^{2}v^{9}+12249491704743205187858353650853516492986627tuv^{10}-936034970439574419340247861327161443330420tv^{11}-151248036306048882951207017596108800u^{12}-16117984528899681472058086834898467073090688u^{11}v+104389437282597309067822150201348525317740544u^{10}v^{2}-296832601434713684849153662176669531971690592u^{9}v^{3}+502502414675164764545912784231079692505751328u^{8}v^{4}-575314178379157402441094407152572450237915160u^{7}v^{5}+475128592398536621403065772596402531508027600u^{6}v^{6}-290465295940254263897511328242310689725254290u^{5}v^{7}+130406844597600809241375577186809327028502922u^{4}v^{8}-41007741955034096161942785323841060097900707u^{3}v^{9}+8068339184623345785687336099443160391633041u^{2}v^{10}-740734742446173882143558228655877815934935uv^{11}-9453002269128055184450438599756800v^{12}}{18192840441311984454798831227721600xu^{11}-47147034063874193568037920095806240xu^{10}v+83112561280674990124252265791470304xu^{9}v^{2}-96248326946171915425759990113428192xu^{8}v^{3}+65503931005456709512429756392067184xu^{7}v^{4}-24548289951319314819247800069608544xu^{6}v^{5}+4202927387986615120454375347630732xu^{5}v^{6}-600989503416158155683254372402484xu^{4}v^{7}+758181962333503981218370016190846xu^{3}v^{8}-25440281127260942153270235494735xu^{2}v^{9}-14080705010871227019232096544052xuv^{10}+178186883993269356693189464660116800ytu^{10}-361458061134782430174919284978390080ytu^{9}v+415946789368629567838859445584020448ytu^{8}v^{2}-321201631858425426583631012092195504ytu^{7}v^{3}+170715210352442559199659015513922408ytu^{6}v^{4}-65483950911383096263912748942183628ytu^{5}v^{5}+18758673149456147777182867802216234ytu^{4}v^{6}-4718808470524050286615363330428483ytu^{3}v^{7}+1848908907666748955947389962451177ytu^{2}v^{8}-84989130427799160569214115810495ytuv^{9}-26340966471763189029251862096174ytv^{10}+426729604770939767375769570867019200yu^{11}-1052001163863688164991087494791992480yu^{10}v+1345380657604965415584321942843460608yu^{9}v^{2}-1159686672381344222907407281513780192yu^{8}v^{3}+717264802250701875472142218830172392yu^{7}v^{4}-325435063359188856899002843588193416yu^{6}v^{5}+110605292497536712556572658507629702yu^{5}v^{6}-30234600430564007771844532805129072yu^{4}v^{7}+8053113912982925912944121921498880yu^{3}v^{8}-1797468291552794053288838576943122yu^{2}v^{9}+27291833825483765171860508277481yuv^{10}+26340966471763189029251862096174yv^{11}+10075085022773064101754340988256000ztu^{10}-3541164477734912393020334282154720ztu^{9}v-27055761365726329015902488596034688ztu^{8}v^{2}+49498304476963299911847549758074224ztu^{7}v^{3}-38752666566690674454087866710090248ztu^{6}v^{4}+14558670625567455593088519189053268ztu^{5}v^{5}-1972095186730385226739330817810754ztu^{4}v^{6}-160542537448896053780542333982127ztu^{3}v^{7}+183511331126130465643192195854813ztu^{2}v^{8}-127284357675164593812207941707380ztuv^{9}+3343575772526011559233255560444ztv^{10}-295881355915294216206098698543797600zu^{11}+642575525782477716296935718959480640zu^{10}v-739547307058331186896553575506902624zu^{9}v^{2}+562087851653369805140903559108588400zu^{8}v^{3}-307499690977342797247324783358156848zu^{7}v^{4}+127504675176813143352732975062229132zu^{6}v^{5}-38538860028746744224383721188378320zu^{5}v^{6}+7909052126432907301906718550266853zu^{4}v^{7}-682157604184954998432499863778929zu^{3}v^{8}-693610935385671362732179747609328zu^{2}v^{9}+168671415123294834858009453825552zuv^{10}-3343575772526011559233255560444zv^{11}-9231447528445366391064881445075w^{4}v^{8}-18462895056890732782129762890150w^{3}v^{9}-183662681074041050864887231348896000w^{2}u^{10}+451474987015192892076186959975412000w^{2}u^{9}v-435114315844316225293056832608742800w^{2}u^{8}v^{2}+202473025682539646514723852191475600w^{2}u^{7}v^{3}-18676398277418246843914092408630900w^{2}u^{6}v^{4}-31682971387761223814665604933602800w^{2}u^{5}v^{5}+19610391358865641597314917527594725w^{2}u^{4}v^{6}-5732538862397693317772437801086375w^{2}u^{3}v^{7}+579282227976079575413075851144425w^{2}u^{2}v^{8}+155880991920174174701200755282000w^{2}uv^{9}-27592204390626677323078513828050w^{2}v^{10}-157506402754569667599104767762677600wtu^{10}+290274485319835558685744828493106080wtu^{9}v-379375879352354424796687218225102288wtu^{8}v^{2}+383734633162295089606464447459046224wtu^{7}v^{3}-277153483535929339007195529042249348wtu^{6}v^{4}+130533683485040622575922951368939568wtu^{5}v^{5}-37089980244191368724092094143403079wtu^{4}v^{6}+5739413081031156775795480814930373wtu^{3}v^{7}+153902702185321353240142717668138wtu^{2}v^{8}-196951493799264102986884823518305wtuv^{9}+17177155016789934305780692324869wtv^{10}-1039005935905365893614985992815888000wu^{11}+2386394133847802141009205973265604800wu^{10}v-2936524506829348697022482598682451520wu^{9}v^{2}+2479416737437089893202639523618066016wu^{8}v^{3}-1490135992282568011673621839754883288wu^{7}v^{4}+637730174941661153343726650873867816wu^{6}v^{5}-191558109683829858326402693923928426wu^{5}v^{6}+40265449057904338411613800710670948wu^{4}v^{7}-6415439982096115976502307015812621wu^{3}v^{8}+409604715355278409276156598825164wu^{2}v^{9}+265272470380141452861819005566315wuv^{10}-32929647913924167104629416649143wv^{11}+2878831435135509103297766349986400t^{2}u^{10}-1683637081276372767527563302536160t^{2}u^{9}v+32888564140033299327013774014262656t^{2}u^{8}v^{2}-70575118540035501660972527644831488t^{2}u^{7}v^{3}+62854358043628107212426656583751576t^{2}u^{6}v^{4}-26248556442999105113203150053165216t^{2}u^{5}v^{5}+3989704077926889255633042736211298t^{2}u^{4}v^{6}+528362253783798779005218069117474t^{2}u^{3}v^{7}-250990543849701639460964089573431t^{2}u^{2}v^{8}-56732043796955884943243904915240t^{2}uv^{9}+1671787886263005779616627780222t^{2}v^{10}+156652052081522190917999365832095200tu^{11}-338626681174741502398873056736449920tu^{10}v+461494390377732897375930695640686432tu^{9}v^{2}-471835333395745897246368836343957712tu^{8}v^{3}+330626579750140842686089116847731400tu^{7}v^{4}-146225430600346667303945817198108468tu^{6}v^{5}+34171427345529305526311924681020442tu^{5}v^{6}-959768157788846608270299509362185tu^{4}v^{7}-1159997283489140216711185104819546tu^{3}v^{8}+265199666262064491090066821701526tu^{2}v^{9}+55010070349867664977120530491229tuv^{10}-1671787886263005779616627780222tv^{11}-79871461568804859557559178786488000u^{11}v+148945148236674667866364913897033280u^{10}v^{2}-140431099582820294032502255636216208u^{9}v^{3}+87543993109531501686745542449841984u^{8}v^{4}-36721144098124119166632405296206668u^{7}v^{5}+6641993790819639663444526225561188u^{6}v^{6}+4091492486730293508027781392370311u^{5}v^{7}-4245872781181761877630849275279282u^{4}v^{8}+1886376124864775229023245125264483u^{3}v^{9}-103828242049327605585036076668255u^{2}v^{10}-24767648688391228292724190883871uv^{11}}$ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
40.144.7.fp.2
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Equation of the image curve:
| $0$ |
$=$ |
$ -X^{7}Z^{2}-X^{6}Y^{2}Z-4X^{6}YZ^{2}+X^{5}Y^{4}-4X^{5}Y^{3}Z-9X^{5}Y^{2}Z^{2}+4X^{4}Y^{5}-9X^{4}Y^{4}Z-16X^{4}Y^{3}Z^{2}+16X^{4}YZ^{4}+4X^{3}Y^{6}-16X^{3}Y^{5}Z-11X^{3}Y^{4}Z^{2}+32X^{3}Y^{3}Z^{3}-4X^{2}Y^{7}-11X^{2}Y^{6}Z+20X^{2}Y^{5}Z^{2}-9XY^{8}+4XY^{7}Z+5XY^{6}Z^{2}-5Y^{9}+5Y^{8}Z $ |
This modular curve minimally covers the modular curves listed below.
