Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ x y - x w - 2 y^{2} + y z + y w - y t + y v + z^{2} - w v $ |
| $=$ | $2 x^{2} + x y - 2 x z + 2 x w + x t - x u - y w - y t - y u - 2 z^{2} + z w + z u - z v + w^{2} + \cdots - u^{2}$ |
| $=$ | $x^{2} - 3 x z + 2 x w - x t - x u + x v + y z + y u + y v + z^{2} - 2 z w - 3 z u - w t - w v - t^{2} + \cdots - u v$ |
| $=$ | $x^{2} - x y - x u + x v - 3 y z + y w - y t - y v - 2 z w - z t - 3 z u - w t - t^{2} - 2 t u + t v - u v$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 494800 x^{12} + 2328800 x^{11} y + 2594800 x^{11} z + 2706120 x^{10} y^{2} + 6084400 x^{10} y z + \cdots - 7200 z^{12} $ |
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This modular curve has 2 rational CM points but no rational cusps or other known rational points.
Maps to other modular curves
$j$-invariant map
of degree 120 from the canonical model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -2^4\cdot3^3\,\frac{183363779328804309802692697693679100044xtu^{8}+319525648542582618352211655132956824715xtu^{7}v+241087364498808418855021343908156133598xtu^{6}v^{2}+49765902174929244546001996893898090112xtu^{5}v^{3}+435900298127713760498272628040718960xtu^{4}v^{4}-3923561848545766314907157971474092240xtu^{3}v^{5}-447494799489511744751210235760168480xtu^{2}v^{6}-39206007208051925142485975006289536xtuv^{7}+1726006458227419938224579718955008xtv^{8}+144057734600205951568964254377245842605xu^{9}+89272385614525278286056030001348798877xu^{8}v+37796591169781720055347110738904384790xu^{7}v^{2}-95207322700176826183114722309063334740xu^{6}v^{3}+34331616718537979444462299011496089392xu^{5}v^{4}+14887616039175391703366690779825038736xu^{4}v^{5}+7434716988471171426654685683280931040xu^{3}v^{6}+646091323413504359222370505496617280xu^{2}v^{7}+53614034935720156882394802337017088xuv^{8}-1972206142904154524955156356126208xv^{9}-566624329373656673364557896858094080546ytu^{8}-1024827218890380614213940399131397008410ytu^{7}v-1040894828694184622830451064845083978620ytu^{6}v^{2}-398922290421509670001473301895385940096ytu^{5}v^{3}-106538954435290205985387960141961934720ytu^{4}v^{4}-11194960639486417240631403158113347744ytu^{3}v^{5}-786143412198813905353944898769763520ytu^{2}v^{6}-9505270051737511979202911176398080ytuv^{7}-261526441428254556558234344251392ytv^{8}+82353232488317701442266175869945960277yu^{9}+514430537224239796937062215082396978164yu^{8}v+710372271833208943226296395989363181022yu^{7}v^{2}+697075735691575968161276544826778881580yu^{6}v^{3}+242420931959293497386561240869530634256yu^{5}v^{4}+69450659407327090080954140083092291104yu^{4}v^{5}+6940914086046142017675368155767903584yu^{3}v^{6}+588228281262436134050557202642607296yu^{2}v^{7}+4756232409151108937779241804030720yuv^{8}+261526441428254556558234344251392yv^{9}-184359132645376724245006735153051883675ztu^{8}-463685752244379325541705013923677265547ztu^{7}v-297668289702168933015238229163841880494ztu^{6}v^{2}-132850105820785050269100854015404450688ztu^{5}v^{3}-15013054255838519666979453578221187840ztu^{4}v^{4}-914722333054433041256122541850214448ztu^{3}v^{5}+229319673357371841193901002493372320ztu^{2}v^{6}+13438612485545029520591862878574208ztuv^{7}-361141741440509134126375478763008ztv^{8}+92794002794027356594821053865708397812zu^{9}+221592838589966948178862744435875351379zu^{8}v+538747286792964939289955644198049588862zu^{7}v^{2}+244715423408348037841759792587473392424zu^{6}v^{3}+118966256163448116619135570837651198960zu^{5}v^{4}+7833276639966890114285180317951853488zu^{4}v^{5}+400898850486726003851145877790971360zu^{3}v^{6}-282779857721856983209316387926340096zu^{2}v^{7}-9555957406368822437048141581644800zuv^{8}-290532407397706473920671422479360zv^{9}+92794024405253659989087927273834768372wtu^{8}+22637432139756375715890825338084977393wtu^{7}v+31281358174577632819963363261062151306wtu^{6}v^{2}-77366144269982177255023096880336951808wtu^{5}v^{3}-26343680125171290572214974769696923120wtu^{4}v^{4}-8769708203061188582006924554732197232wtu^{3}v^{5}-718780889014696381836029660563985760wtu^{2}v^{6}-45982586216759700269669164427287424wtuv^{7}+738104646831370999136069288059904wtv^{8}-21611226303394266873408126370560wu^{9}-232785072027102041838374202188732215391wu^{8}v-252953069566379160022049160185338065604wu^{7}v^{2}-259541411015927851984619223070729211096wu^{6}v^{3}+4883181741604539684032140783471360192wu^{5}v^{4}+9234477744995577309542213568116248560wu^{4}v^{5}+7978323452856814226361471391969538240wu^{3}v^{6}+719412061332636118682345518096948480wu^{2}v^{7}+46573748697313218160360829229123072wuv^{8}+112226921461588570928388317083136wv^{9}-42923618575846200053402564718178206734t^{3}u^{7}-75499326196735988827109267238833916092t^{3}u^{6}v-71685559495762563608565135017037890784t^{3}u^{5}v^{2}-25490713182871447341308914687363340000t^{3}u^{4}v^{3}-6034383637356226541497094292977436640t^{3}u^{3}v^{4}-513845605519227658838859470271034560t^{3}u^{2}v^{5}-27082460646831871321322025327146240t^{3}uv^{6}+117526349639077651259082217352192t^{3}v^{7}-86419591368966846903528895408921275543t^{2}u^{8}-130965855008014901994481089857145887127t^{2}u^{7}v-16969482000917098716183275576759957478t^{2}u^{6}v^{2}+53478494419334564397816796937676744096t^{2}u^{5}v^{3}+32748950492980036978647958783789878720t^{2}u^{4}v^{4}+8748444959136154505890729782246342672t^{2}u^{3}v^{5}+890137557896060956231371950063281440t^{2}u^{2}v^{6}+43497238273846059945362848004769408t^{2}uv^{7}+29090987071263164511457702525440t^{2}v^{8}+46989782955458614966263423584016186960tu^{8}v+67928015598878085433214939616805576073tu^{7}v^{2}-11840765625323105755517904367480695958tu^{6}v^{3}-42474493670161925996076869826841608288tu^{5}v^{4}-25414144664027293540338295529446337200tu^{4}v^{5}-6120978573988061235083047534778310896tu^{3}v^{6}-664703345515571470171295068939630176tu^{2}v^{7}-32414766525549542271119585968620928tuv^{8}+586441520182562892986166833256448tv^{9}+13359667169371001339925023574528u^{10}-97939735777446607889879700417267943172u^{9}v-229980656760069521471893495543839955577u^{8}v^{2}-49368202473527004672330253027435812582u^{7}v^{3}+44198552828051755231196278145508940324u^{6}v^{4}+68073147343240221617804032197370869632u^{5}v^{5}+23880234950138910591773708721701839920u^{4}v^{6}+4861248968724969644873086317921802144u^{3}v^{7}+428095221431039641991417994579983552u^{2}v^{8}+20138189106794654722264985690138880uv^{9}-528114535240808687285213685198336v^{10}}{3075944479395641891997825644083396xtu^{8}-3096423491182668103243830385103351xtu^{7}v+942014795186605856738934992630998xtu^{6}v^{2}-92966854811944676574519407585216xtu^{5}v^{3}+3437310598717686183902201507520xtu^{4}v^{4}-1600657162770695963089423208192xtu^{3}v^{5}-454957547369219599851880292352xtu^{2}v^{6}+170628151865192471006274348032xtuv^{7}+80114943969973484165391894528xtv^{8}+3724813078294690798082329173986895xu^{9}-14847292503660471942122360091760493xu^{8}v+8949543762407741378399897923782238xu^{7}v^{2}-1853820574985239242835751091327212xu^{6}v^{3}+129413625095780287182662997023296xu^{5}v^{4}-1791836496103675153593343131744xu^{4}v^{5}-886708468230051409987311102464xu^{3}v^{6}+748665637056483638568965998848xu^{2}v^{7}+434240969242615218648469313024xuv^{8}-56459841929512638558369862656xv^{9}-24525163948523227881527876488885814ytu^{8}+12133292867901354419152010070251674ytu^{7}v-1839737916710140936814301808769132ytu^{6}v^{2}+69528133737281500142067250870240ytu^{5}v^{3}-1002391508891105925657469847040ytu^{4}v^{4}+3045526651200807045621599678464ytu^{3}v^{5}-1447214060157552252417205309440ytu^{2}v^{6}-603631160406955579804237772800ytuv^{7}-8269616609682285039465345024ytv^{8}+1160021895075191872768065444208519yu^{9}+22245701848521744852741840913763856yu^{8}v-11647827020833576839713494429827630yu^{7}v^{2}+1839070546256093008338650733863196yu^{6}v^{3}-74769340582111139170876000386880yu^{5}v^{4}-765722612946382467570889927424yu^{4}v^{5}+179946492683140162788749404928yu^{3}v^{6}+464486910641471662078339688960yu^{2}v^{7}+263003466539505468746616828928yuv^{8}+8269616609682285039465345024yv^{9}+9707971987019747889810630237291767ztu^{8}-6034698148181492062011755568985605ztu^{7}v+1152109464989937271559955731070354ztu^{6}v^{2}-69574923710543377649625899154720ztu^{5}v^{3}-721989590812824972469849617440ztu^{4}v^{4}+1210049084758014848558103627008ztu^{3}v^{5}-863730201301172090968379037440ztu^{2}v^{6}-239503235112438099679488839168ztuv^{7}+51748454101637181332199099392ztv^{8}+12381892993008563969936380893330108zu^{9}-20123848002207282409041333503236927zu^{8}v+8809506916530029684348586600248214zu^{7}v^{2}-1392433831738616063356321499895624zu^{6}v^{3}+67900473355781786952293228276448zu^{5}v^{4}+372571949847680230471185629632zu^{4}v^{5}-1230086877641466942393571349248zu^{3}v^{6}+1037339101275798981823144775680zu^{2}v^{7}+292473839883483845131774879744zuv^{8}-48019988359846763682966851584zv^{9}+12381892993008563969936380893330108wtu^{8}-8718313420547364182658011444529205wtu^{7}v+1991088839385200016050722530410642wtu^{6}v^{2}-159722089102174242070841896274560wtu^{5}v^{3}+1723238256443677707492535343360wtu^{4}v^{4}-819608523503482009507929228288wtu^{3}v^{5}+510381964213712038587141905408wtu^{2}v^{6}-40444554476454327362011923456wtuv^{7}-71268747837256884419488458752wtv^{8}-19491173737955618096991816075045301wu^{8}v+12370569241838215226218927689125848wu^{7}v^{2}-2562065027204855435869606515139712wu^{6}v^{3}+183251610025167135125062044182752wu^{5}v^{4}-1757116283913281487583611337120wu^{4}v^{5}+1078483940539215081363733156352wu^{3}v^{6}-809835399264353664345203570944wu^{2}v^{7}+86632544274551474119848139264wuv^{8}+75809898705148751809721555968wv^{9}-1723259463450561247273387080442714t^{3}u^{7}+760796713151032957018172173598132t^{3}u^{6}v-91211692977290174999048725524832t^{3}u^{5}v^{2}+1678088835148153029919633518080t^{3}u^{4}v^{3}-222762394815119226997470405120t^{3}u^{3}v^{4}+394188318904202353527522818048t^{3}u^{2}v^{5}-199167899068324526627376205824t^{3}uv^{6}-63137790586742480189097267200t^{3}v^{7}+2224445771577171456205040563519779t^{2}u^{8}+476726055121948299650204428255239t^{2}u^{7}v-536820928221017316791477692081062t^{2}u^{6}v^{2}+78109092772438184763252983386560t^{2}u^{5}v^{3}-1634622246946119343229446399200t^{2}u^{4}v^{4}+691490088657141498285239384064t^{2}u^{3}v^{5}-912153879156907848620201958144t^{2}u^{2}v^{6}+78828498658729569171631256064t^{2}uv^{7}+91642072554270772089977760768t^{2}v^{8}-1661208203201802081699718927285584tu^{8}v+620560964636180674612121757370819tu^{7}v^{2}-46418110279268498568256796928206tu^{6}v^{3}-7038491053129534119332261678432tu^{5}v^{4}+528577857458489289479548662400tu^{4}v^{5}-411306709916936929854751719168tu^{3}v^{6}-208985649208279618040021652992tu^{2}v^{7}+128838164449600025424919056384tuv^{8}-17119532825724224349185327104tv^{9}+7483526215442576433605589673771988u^{9}v-5037931951231840746140787068646307u^{8}v^{2}+1461143457190865356495255522550234u^{7}v^{3}-227792659347532477047653990473636u^{6}v^{4}+19389609935793917326708670500704u^{5}v^{5}+333325653278036523194308825184u^{4}v^{6}-200322749717919233288199419136u^{3}v^{7}+87387699684154989911734183168u^{2}v^{8}+111434043205925313735348420096uv^{9}-3115132532121782512229821440v^{10}}$ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
30.120.7.h.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle 5u$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle v$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 494800X^{12}+2328800X^{11}Y+2706120X^{10}Y^{2}+772580X^{9}Y^{3}+74092X^{8}Y^{4}+468X^{7}Y^{5}-180X^{6}Y^{6}+X^{5}Y^{7}+2594800X^{11}Z+6084400X^{10}YZ+6117800X^{9}Y^{2}Z+2335300X^{8}Y^{3}Z+320138X^{7}Y^{4}Z+10334X^{6}Y^{5}Z-409X^{5}Y^{6}Z+2X^{4}Y^{7}Z+3665200X^{10}Z^{2}+7136000X^{9}YZ^{2}+2757700X^{8}Y^{2}Z^{2}+1701530X^{7}Y^{3}Z^{2}+400249X^{6}Y^{4}Z^{2}+29765X^{5}Y^{5}Z^{2}+7X^{4}Y^{6}Z^{2}-X^{3}Y^{7}Z^{2}-1558000X^{9}Z^{3}+7217500X^{8}YZ^{3}+2270400X^{7}Y^{2}Z^{3}-24980X^{6}Y^{3}Z^{3}+176229X^{5}Y^{4}Z^{3}+23465X^{4}Y^{5}Z^{3}+184X^{3}Y^{6}Z^{3}-2X^{2}Y^{7}Z^{3}-5614000X^{8}Z^{4}-1492000X^{7}YZ^{4}+2741325X^{6}Y^{2}Z^{4}-41440X^{5}Y^{3}Z^{4}-6910X^{4}Y^{4}Z^{4}+13510X^{3}Y^{5}Z^{4}-282X^{2}Y^{6}Z^{4}+XY^{7}Z^{4}+528800X^{7}Z^{5}-4772200X^{6}YZ^{5}-1074330X^{5}Y^{2}Z^{5}+284115X^{4}Y^{3}Z^{5}-49239X^{3}Y^{4}Z^{5}+5949X^{2}Y^{5}Z^{5}-24XY^{6}Z^{5}+3652800X^{6}Z^{6}+2143400X^{5}YZ^{6}-1742725X^{4}Y^{2}Z^{6}+102000X^{3}Y^{3}Z^{6}-9156X^{2}Y^{4}Z^{6}-413XY^{5}Z^{6}-1066800X^{5}Z^{7}+2095500X^{4}YZ^{7}-109150X^{3}Y^{2}Z^{7}-44135X^{2}Y^{3}Z^{7}+7952XY^{4}Z^{7}-1224000X^{4}Z^{8}-845000X^{3}YZ^{8}+120450X^{2}Y^{2}Z^{8}-19115XY^{3}Z^{8}-288Y^{4}Z^{8}+638000X^{3}Z^{9}-361000X^{2}YZ^{9}-31150XY^{2}Z^{9}+2880Y^{3}Z^{9}+183200X^{2}Z^{10}+64200XYZ^{10}-4320Y^{2}Z^{10}-104800XZ^{11}-14400YZ^{11}-7200Z^{12} $ |
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.
