Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ x z + x w - x t + x v + y w + y t - y u + y v - z w + z t + z u - u v $ |
| $=$ | $x y - x w - x v + y z - z w + z u - t u - t v + u^{2} - v^{2}$ |
| $=$ | $x y - x z - x w + x t - x v + y t - z t - w t + t^{2} - t v - v^{2}$ |
| $=$ | $x^{2} + 2 x y - x z + x w + 2 x v + y^{2} + y w + y t - y u + 2 y v - z u - z v + t v$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 11 x^{9} y^{3} + 33 x^{9} y^{2} z - 33 x^{9} y z^{2} + 11 x^{9} z^{3} - 106 x^{8} y^{4} + \cdots + 11 y^{3} z^{9} $ |
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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. The following are the known rational points on this modular curve (one row per $j$-invariant).
Maps to other modular curves
$j$-invariant map
of degree 126 from the canonical model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -\frac{1}{2}\cdot\frac{257576432830305xtu^{9}+848318821285173xtu^{8}v+206356468601014304xtu^{7}v^{2}+3712974688741235784xtu^{6}v^{3}+11785626931410790122xtu^{5}v^{4}+6258688253115592762xtu^{4}v^{5}-8303255976949091576xtu^{3}v^{6}+3521689854803552032xtu^{2}v^{7}+12048355926917373197xtuv^{8}+4055207095180774313xtv^{9}-242192172588460xu^{10}-746378696739074xu^{9}v+5707129645750224426xu^{8}v^{2}+10511179739619488880xu^{7}v^{3}-5138998702317764120xu^{6}v^{4}+3397388299191573148xu^{5}v^{5}+25867361101258363364xu^{4}v^{6}+4774115947072941824xu^{3}v^{7}+6444044081659693748xu^{2}v^{8}+18208235819494886902xuv^{9}+6780163301215293442xv^{10}-348599642896128ytu^{9}-1014296072445152ytu^{8}v+6861094726954391904ytu^{7}v^{2}+12582210316778231648ytu^{6}v^{3}-12422310486674423776ytu^{5}v^{4}-16349685414552477728ytu^{4}v^{5}+19060736345312237728ytu^{3}v^{6}+9789933805749304992ytu^{2}v^{7}-12994366460813155104ytuv^{8}-6131787647217125632ytv^{9}-56764694187030yu^{10}-158356209272940yu^{9}v+4115017790471144962yu^{8}v^{2}+11372558110085041072yu^{7}v^{3}+6405990888671913492yu^{6}v^{4}+4689716003035854968yu^{5}v^{5}+12078893372875896852yu^{4}v^{6}+4412698579416754096yu^{3}v^{7}+7465620734211558146yu^{2}v^{8}+11201046712880561812yuv^{9}+3676610278922756842yv^{10}-715126726135767ztu^{9}-3268203171535603ztu^{8}v+4543926598718059680ztu^{7}v^{2}+12170228426550237576ztu^{6}v^{3}+1835936120914239162ztu^{5}v^{4}-8029259414698156438ztu^{4}v^{5}+8095621220027434952ztu^{3}v^{6}+11655261256606866592ztu^{2}v^{7}+3356780844586651957ztuv^{8}+226520307983314145ztv^{9}-21747097447170zu^{10}-893516522961910zu^{9}v-7600392594016752052zu^{8}v^{2}-18255027812410816512zu^{7}v^{3}-4492767736577601252zu^{6}v^{4}-6326672422238197148zu^{5}v^{5}-25323944242454659304zu^{4}v^{6}-830431502337951088zu^{3}v^{7}+3481772696186945494zu^{2}v^{8}-10838942556358509566zuv^{9}-5085897994619992660zv^{10}+955455629062128wtu^{9}+2140115532716336wtu^{8}v-25081043760889367648wtu^{7}v^{2}-54997864615191989216wtu^{6}v^{3}+24818131171989400000wtu^{5}v^{4}+53831277869124887616wtu^{4}v^{5}-60164959121261091360wtu^{3}v^{6}-15640896453390266784wtu^{2}v^{7}+55489549209369766608wtuv^{8}+22363038480430690832wtv^{9}-69772520626425wu^{10}-469013918749798wu^{9}v+3697937524198093371wu^{8}v^{2}+15249542611284197496wu^{7}v^{3}+23220104994306470334wu^{6}v^{4}+13125067939721695324wu^{5}v^{5}+1026312165698839230wu^{4}v^{6}+15088481086250549880wu^{3}v^{7}+23829135307055611323wu^{2}v^{8}+14530895550074576922wuv^{9}+3311334305826064263wv^{10}-4141465505759232t^{9}v^{2}-591637929394176t^{8}v^{3}+4035815875510272t^{7}v^{4}+1071589106810880t^{6}v^{5}-841998804885504t^{5}v^{6}-258394970308608t^{4}v^{7}+6162141353984t^{3}v^{8}-306939159165432t^{2}u^{9}-436036849486280t^{2}u^{8}v+11162295994211411808t^{2}u^{7}v^{2}+21248944848058173984t^{2}u^{6}v^{3}-19192629941996762384t^{2}u^{5}v^{4}-27301287321717543152t^{2}u^{4}v^{5}+31657534121843989472t^{2}u^{3}v^{6}+14991000569905839776t^{2}u^{2}v^{7}-21852508077333944888t^{2}uv^{8}-9978963127841363976t^{2}v^{9}-634694824446460tu^{10}-1433755303028304tu^{9}v+17538933674754145556tu^{8}v^{2}+38786357852374787200tu^{7}v^{3}-2007078192059235096tu^{6}v^{4}+5339725765660941216tu^{5}v^{5}+60442233844836124392tu^{4}v^{6}+888922618857589888tu^{3}v^{7}+3110996929788817684tu^{2}v^{8}+39093921420151109040tuv^{9}+15633634563085959172tv^{10}+477386280323325u^{11}+1770270477080525u^{10}v-14833630686811189185u^{9}v^{2}-41774602081445772273u^{8}v^{3}-18238010243630159134u^{7}v^{4}+22574431887843451330u^{6}v^{5}+12615370885126397758u^{5}v^{6}-16117956654292446690u^{4}v^{7}-21759536416146582415u^{3}v^{8}+22825028148388469441u^{2}v^{9}+41051093752616075827uv^{10}+13243661285637708675v^{11}}{5311931174xtu^{9}+63322653030xtu^{8}v-59166604749662xtu^{7}v^{2}-1074131382079536xtu^{6}v^{3}-3415317157186620xtu^{5}v^{4}-1701900391060972xtu^{4}v^{5}+2554998194516398xtu^{3}v^{6}-1119884450804641xtu^{2}v^{7}-3278965167394318xtuv^{8}-1024644398696301xtv^{9}+15606946014xu^{10}+63905571576xu^{9}v-1651910201601786xu^{8}v^{2}-3042883683569972xu^{7}v^{3}+1683266940000476xu^{6}v^{4}-913876402114704xu^{5}v^{5}-7611137633737124xu^{4}v^{6}-741212369951684xu^{3}v^{7}-1881030038275356xu^{2}v^{8}-4985753341566024xuv^{9}-1715210044288000xv^{10}+21233990018ytu^{9}+101903636758ytu^{8}v-1985675746143064ytu^{7}v^{2}-3641696903207688ytu^{6}v^{3}+3827302105935532ytu^{5}v^{4}+4811061460962852ytu^{4}v^{5}-5878589686672648ytu^{3}v^{6}-2423324633986840ytu^{2}v^{7}+3638334025829954ytuv^{8}+1552361980216542ytv^{9}+12227605500yu^{10}+69949652836yu^{9}v-1190792865896260yu^{8}v^{2}-3291487301295564yu^{7}v^{3}-1714854183942808yu^{6}v^{4}-1179493957110552yu^{5}v^{5}-3445403558394092yu^{4}v^{6}-955927626735406yu^{3}v^{7}-2110221979088158yu^{2}v^{8}-3041033463827442yuv^{9}-930718323350710yv^{10}+13396903264ztu^{9}+75181890140ztu^{8}v-1316829233552878ztu^{7}v^{2}-3524669462189704ztu^{6}v^{3}-376496761954544ztu^{5}v^{4}+2504306368455672ztu^{4}v^{5}-2442750306758322ztu^{3}v^{6}-3187737681496189ztu^{2}v^{7}-859863885249604ztuv^{8}-65849320166611ztv^{9}-21758414610zu^{10}-109363896044zu^{9}v+2198286343470358zu^{8}v^{2}+5281406931722176zu^{7}v^{3}+1038866365917708zu^{6}v^{4}+1588631099099128zu^{5}v^{5}+7327056356173488zu^{4}v^{6}-380120930181766zu^{3}v^{7}-837442388145714zu^{2}v^{8}+3028784493088154zuv^{9}+1284190790155386zv^{10}-71174241580wtu^{9}-340063076892wtu^{8}v+7257684611651944wtu^{7}v^{2}+15915845563921696wtu^{6}v^{3}-8046699461518648wtu^{5}v^{4}-16192287159425320wtu^{4}v^{5}+18417538236115464wtu^{3}v^{6}+3448596556552204wtu^{2}v^{7}-15305273638466372wtuv^{8}-5653765257540600wtv^{9}+12038838290wu^{10}+86130330418wu^{9}v-1070296054740294wu^{8}v^{2}-4413674976789162wu^{7}v^{3}-6597280557999628wu^{6}v^{4}-3466492852508548wu^{5}v^{5}+29350322381090wu^{4}v^{6}-4119598370089293wu^{3}v^{7}-6540206643874113wu^{2}v^{8}-3855049417166659wuv^{9}-839174125166725wv^{10}+1198340713472t^{9}v^{2}+171191530496t^{8}v^{3}-2066526332416t^{7}v^{4}-566853588224t^{6}v^{5}+1160221899648t^{5}v^{6}+440077561280t^{4}v^{7}-205901626784t^{3}v^{8}+34601199208t^{2}u^{9}+173054565824t^{2}u^{8}v-3229726530824264t^{2}u^{7}v^{2}-6149163255215472t^{2}u^{6}v^{3}+5931849956938752t^{2}u^{5}v^{4}+8054390576137568t^{2}u^{4}v^{5}-9725270557352472t^{2}u^{3}v^{6}-3699538120927868t^{2}u^{2}v^{7}+6105089990413472t^{2}uv^{8}+2528147081854332t^{2}v^{9}+49321637312tu^{10}+234359444464tu^{9}v-5075438693361120tu^{8}v^{2}-11225180985496968tu^{7}v^{3}+1184907929238416tu^{6}v^{4}-1113081860390464tu^{5}v^{5}-17758735516775208tu^{4}v^{6}+1122480874746216tu^{3}v^{7}-1216740516620276tu^{2}v^{8}-10769946059943964tuv^{9}-3950594927726648tv^{10}-43836895402u^{11}-243596435040u^{10}v+4293057058346048u^{9}v^{2}+12090428115783780u^{8}v^{3}+4772510587403378u^{7}v^{4}-7211363963742984u^{6}v^{5}-3645916143343982u^{5}v^{6}+4286595163706407u^{4}v^{7}+6624937393633224u^{3}v^{8}-6625426200178406u^{2}v^{9}-11129177553224210uv^{10}-3350472212157773v^{11}}$ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
14.126.7.a.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle z$ |
Equation of the image curve:
| $0$ |
$=$ |
$ -11X^{9}Y^{3}+33X^{9}Y^{2}Z-33X^{9}YZ^{2}+11X^{9}Z^{3}-106X^{8}Y^{4}+314X^{8}Y^{3}Z-416X^{8}Y^{2}Z^{2}+314X^{8}YZ^{3}-106X^{8}Z^{4}-925X^{7}Y^{5}+1742X^{7}Y^{4}Z-865X^{7}Y^{3}Z^{2}+865X^{7}Y^{2}Z^{3}-1742X^{7}YZ^{4}+925X^{7}Z^{5}-1804X^{6}Y^{6}-793X^{6}Y^{5}Z-3104X^{6}Y^{4}Z^{2}+11314X^{6}Y^{3}Z^{3}-3104X^{6}Y^{2}Z^{4}-793X^{6}YZ^{5}-1804X^{6}Z^{6}-925X^{5}Y^{7}-793X^{5}Y^{6}Z-14276X^{5}Y^{5}Z^{2}-660X^{5}Y^{4}Z^{3}+660X^{5}Y^{3}Z^{4}+14276X^{5}Y^{2}Z^{5}+793X^{5}YZ^{6}+925X^{5}Z^{7}-106X^{4}Y^{8}+1742X^{4}Y^{7}Z-3104X^{4}Y^{6}Z^{2}-660X^{4}Y^{5}Z^{3}-13760X^{4}Y^{4}Z^{4}-660X^{4}Y^{3}Z^{5}-3104X^{4}Y^{2}Z^{6}+1742X^{4}YZ^{7}-106X^{4}Z^{8}-11X^{3}Y^{9}+314X^{3}Y^{8}Z-865X^{3}Y^{7}Z^{2}+11314X^{3}Y^{6}Z^{3}+660X^{3}Y^{5}Z^{4}-660X^{3}Y^{4}Z^{5}-11314X^{3}Y^{3}Z^{6}+865X^{3}Y^{2}Z^{7}-314X^{3}YZ^{8}+11X^{3}Z^{9}+33X^{2}Y^{9}Z-416X^{2}Y^{8}Z^{2}+865X^{2}Y^{7}Z^{3}-3104X^{2}Y^{6}Z^{4}+14276X^{2}Y^{5}Z^{5}-3104X^{2}Y^{4}Z^{6}+865X^{2}Y^{3}Z^{7}-416X^{2}Y^{2}Z^{8}+33X^{2}YZ^{9}-33XY^{9}Z^{2}+314XY^{8}Z^{3}-1742XY^{7}Z^{4}-793XY^{6}Z^{5}+793XY^{5}Z^{6}+1742XY^{4}Z^{7}-314XY^{3}Z^{8}+33XY^{2}Z^{9}+11Y^{9}Z^{3}-106Y^{8}Z^{4}+925Y^{7}Z^{5}-1804Y^{6}Z^{6}+925Y^{5}Z^{7}-106Y^{4}Z^{8}+11Y^{3}Z^{9} $ |
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.
