Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ x^{2} - x y - y w $ |
| $=$ | $x^{2} + 2 x w - y^{2} + z^{2} - t^{2}$ |
| $=$ | $x^{2} + 2 x y + y^{2} + 2 z^{2} - 2 z t + w^{2} + t^{2}$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{8} + 2 x^{6} y^{2} + 4 x^{5} y^{3} + x^{4} y^{4} + 6 x^{4} y^{2} z^{2} + 4 x^{3} y^{5} + \cdots + 5 y^{4} z^{4} $ |
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This modular curve has no real points, and therefore no rational points.
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{2}{3^2}\cdot\frac{19804183776668672xzw^{15}t-42102396565410816xzw^{13}t^{3}-110348293639802880xzw^{11}t^{5}-45319284326466304xzw^{9}t^{7}+41476519113428160xzw^{7}t^{9}+20774088197111040xzw^{5}t^{11}-1627304505374576xzw^{3}t^{13}-39946668333264xzwt^{15}-18342876511557120xw^{17}-33895350433680384xw^{15}t^{2}+71500648827383808xw^{13}t^{4}+169067685415497984xw^{11}t^{6}+70846234409523168xw^{9}t^{8}-46625897707879104xw^{7}t^{10}-23637089442700128xw^{5}t^{12}+1625081211919248xw^{3}t^{14}+29058516826566xwt^{16}+15852663452922880yzw^{15}t+22971058769636352yzw^{13}t^{3}-20317236237960192yzw^{11}t^{5}-30109631098057472yzw^{9}t^{7}-4997899035218112yzw^{7}t^{9}+3134181940728768yzw^{5}t^{11}+11672205522512yzw^{3}t^{13}+29151928211136yzwt^{15}+24007992757263872yw^{17}-28319753579013120yw^{15}t^{2}-91557307320098304yw^{13}t^{4}-27161117987388160yw^{11}t^{6}+52471266057837216yw^{9}t^{8}+17180731304706432yw^{7}t^{10}-4055719476844592yw^{5}t^{12}+7339296806016yw^{3}t^{14}-21055614832014ywt^{16}-39094192895394816z^{3}w^{14}t-70757460177865728z^{3}w^{12}t^{3}+31688770588114176z^{3}w^{10}t^{5}+62716360815427968z^{3}w^{8}t^{7}+11610080396853216z^{3}w^{6}t^{9}-7023535360111728z^{3}w^{4}t^{11}+98237435487552z^{3}w^{2}t^{13}-3058039057440z^{3}t^{15}-70645795278953216z^{2}w^{16}+96842424334181376z^{2}w^{14}t^{2}+313222263588604800z^{2}w^{12}t^{4}+86390972145827584z^{2}w^{10}t^{6}-163789465627028880z^{2}w^{8}t^{8}-53557546660149600z^{2}w^{6}t^{10}+14651774380690556z^{2}w^{4}t^{12}-43602884637720z^{2}w^{2}t^{14}+1990006865109z^{2}t^{16}+52556365944934912zw^{16}t-103342737718496256zw^{14}t^{3}-261000866160375552zw^{12}t^{5}-87514875892143488zw^{10}t^{7}+122851105792916928zw^{8}t^{9}+45353601011944752zw^{6}t^{11}-8944282125817504zw^{4}t^{13}-135924323167200zw^{2}t^{15}+3045038947698zt^{17}-34981932989305088w^{18}-6676364529587200w^{16}t^{2}+139813715528695680w^{14}t^{4}+171726352837800064w^{12}t^{6}+16711293665214416w^{10}t^{8}-58372164531452928w^{8}t^{10}-17272577960040004w^{6}t^{12}+2149742050470940w^{4}t^{14}+98299210599387w^{2}t^{16}-2008387814976t^{18}}{w(2672291047936xzw^{14}t+66138552012912xzw^{12}t^{3}-242080176594920xzw^{10}t^{5}+206388960393732xzw^{8}t^{7}-55532587894164xzw^{6}t^{9}+4694024665852xzw^{4}t^{11}-100490182850xzw^{2}t^{13}+239148450xzt^{15}+3479289197952xw^{16}-51726682793328xw^{14}t^{2}+407651836764xw^{12}t^{4}+258509497667004xw^{10}t^{6}-237065333116824xw^{8}t^{8}+61996513453518xw^{6}t^{10}-5027790081570xw^{4}t^{12}+103646937270xw^{2}t^{14}-239148450xt^{16}-11446132040128yzw^{14}t+70462475742096yzw^{12}t^{3}-101032979729224yzw^{10}t^{5}+46871210612028yzw^{8}t^{7}-7340959959888yzw^{6}t^{9}+342805033208yzw^{4}t^{11}-3156759220yzw^{2}t^{13}-4551866600864yw^{16}+91461176494320yw^{14}t^{2}-269124325838732yw^{12}t^{4}+245483987110260yw^{10}t^{6}-82568109510540yw^{8}t^{8}+10224099978562yw^{6}t^{10}-403299897824yw^{4}t^{12}+3300249570yw^{2}t^{14}+32217785243472z^{3}w^{13}t-181232009020104z^{3}w^{11}t^{3}+236309911651296z^{3}w^{9}t^{5}-99186580289448z^{3}w^{7}t^{7}+14015690648400z^{3}w^{5}t^{9}-591390245781z^{3}w^{3}t^{11}+4950371955z^{3}wt^{13}+13392711074216z^{2}w^{15}-286720013398272z^{2}w^{13}t^{2}+870897750891170z^{2}w^{11}t^{4}-798517403912364z^{2}w^{9}t^{6}+262755416868894z^{2}w^{7}t^{8}-31007394804508z^{2}w^{5}t^{10}+1141407724400z^{2}w^{3}t^{12}-8609344200z^{2}wt^{14}+2324920806512zw^{15}t+182180285093352zw^{13}t^{3}-643352034517540zw^{11}t^{5}+582339584501652zw^{9}t^{7}-177693245277396zw^{7}t^{9}+18678978077651zw^{5}t^{11}-594020787979zw^{3}t^{13}+3778546470zwt^{15}+6631274342696w^{17}-119075930945768w^{15}t^{2}+217451243883950w^{13}t^{4}+37150983604312w^{11}t^{6}-152849855257194w^{9}t^{8}+50188244918930w^{7}t^{10}-4504829893848w^{5}t^{12}+98911797640w^{3}t^{14}-239148450wt^{16})}$ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
60.144.5.oc.2
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle z$ |
Equation of the image curve:
| $0$ |
$=$ |
$ X^{8}+2X^{6}Y^{2}+4X^{5}Y^{3}+X^{4}Y^{4}+6X^{4}Y^{2}Z^{2}+4X^{3}Y^{5}-8X^{3}Y^{3}Z^{2}+4X^{2}Y^{6}+10X^{2}Y^{4}Z^{2}+12XY^{5}Z^{2}+4Y^{6}Z^{2}+5Y^{4}Z^{4} $ |
This modular curve minimally covers the modular curves listed below.
