Canonical model in $\mathbb{P}^{ 2 }$
| $ 0 $ | $=$ | $ x^{3} y - 2 x^{2} y^{2} - 2 x^{2} y z + x y^{3} - 2 x y^{2} z - 2 x y z^{2} + x z^{3} + y z^{3} + z^{4} $ |
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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 |
|---|
| $(0:-1:1)$, $(1:0:0)$, $(0:1:0)$, $(-1:0:1)$, $(1:1:0)$, $(-1/2:-1/2:1)$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the canonical model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -\frac{x^{24}-744x^{22}z^{2}-1464x^{21}z^{3}+193200x^{20}z^{4}+762288x^{19}z^{5}-18811556x^{18}z^{6}-116544768x^{17}z^{7}+328568076x^{16}z^{8}+4429896112x^{15}z^{9}+7792911792x^{14}z^{10}-51228097032x^{13}z^{11}-285457755626x^{12}z^{12}-337843534848x^{11}z^{13}+1909658462112x^{10}z^{14}+9998199392160x^{9}z^{15}+19664560837002x^{8}z^{16}-7232092094400x^{7}z^{17}-179601630593832x^{6}z^{18}-675400242994488x^{5}z^{19}-1528808426231700x^{4}z^{20}-1673457456956144x^{3}z^{21}+184549354x^{2}y^{22}+15032377948x^{2}y^{21}z+405370417188x^{2}y^{20}z^{2}+6611525128098x^{2}y^{19}z^{3}+78147121413632x^{2}y^{18}z^{4}+706527238562328x^{2}y^{17}z^{5}+5088139297699672x^{2}y^{16}z^{6}+29611233583715472x^{2}y^{15}z^{7}+140178925003903520x^{2}y^{14}z^{8}+539719005202719612x^{2}y^{13}z^{9}+1676496980663455640x^{2}y^{12}z^{10}+4162622801117770884x^{2}y^{11}z^{11}+8212335960761008482x^{2}y^{10}z^{12}+12825501383482139892x^{2}y^{9}z^{13}+15776913518768708368x^{2}y^{8}z^{14}+15146610289714390114x^{2}y^{7}z^{15}+11181744932714125133x^{2}y^{6}z^{16}+6185479604981041328x^{2}y^{5}z^{17}+2358345953529735514x^{2}y^{4}z^{18}+358991675463285497x^{2}y^{3}z^{19}-209705527240413651x^{2}y^{2}z^{20}-103590634628115513x^{2}yz^{21}+4181076538412340x^{2}z^{22}-167772140xy^{23}-11140064852xy^{22}z-248402939168xy^{21}z^{2}-3529625931026xy^{20}z^{3}-36356707695602xy^{19}z^{4}-283549808690344xy^{18}z^{5}-1729228902136728xy^{17}z^{6}-8165964249771904xy^{16}z^{7}-29149236788767640xy^{15}z^{8}-71954831692778020xy^{14}z^{9}-81642002341397820xy^{13}z^{10}+193801983871218788xy^{12}z^{11}+1241288467526007680xy^{11}z^{12}+3375048689666085804xy^{10}z^{13}+5992904775462384296xy^{9}z^{14}+7492261745705942510xy^{8}z^{15}+6578732809753417264xy^{7}z^{16}+3781301243902950408xy^{6}z^{17}+1012476443211800786xy^{5}z^{18}-471866075149393129xy^{4}z^{19}-787440422537238133xy^{3}z^{20}-469975973595976981xy^{2}z^{21}-71535093045552117xyz^{22}+33706179176519448xz^{23}+y^{24}-744y^{22}z^{2}-167773604y^{21}z^{3}-11643188072y^{20}z^{4}-283164548556y^{19}z^{5}-4367346298916y^{18}z^{6}-49166934022232y^{17}z^{7}-426495673749604y^{16}z^{8}-2956870579503568y^{15}z^{9}-16582066995662752y^{14}z^{10}-75710913292156280y^{13}z^{11}-281026475336627868y^{12}z^{12}-841263070222277856y^{11}z^{13}-2016072892310028212y^{10}z^{14}-3853145447576566460y^{9}z^{15}-5861158910783955576y^{8}z^{16}-7072858284950638964y^{7}z^{17}-6728075460911769300y^{6}z^{18}-4996240384730630606y^{5}z^{19}-2845851025947959534y^{4}z^{20}-1170433533788893870y^{3}z^{21}-260270446355563798y^{2}z^{22}+32055541582563420yz^{23}+28865739480355800z^{24}}{z^{2}(x^{22}+2x^{21}z+5x^{20}z^{2}+34x^{19}z^{3}+100x^{18}z^{4}+246x^{17}z^{5}+653x^{16}z^{6}+1162x^{15}z^{7}-101x^{14}z^{8}-12202x^{13}z^{9}-72650x^{12}z^{10}-308712x^{11}z^{11}-1082512x^{10}z^{12}-3212246x^{9}z^{13}-7726568x^{8}z^{14}-12017772x^{7}z^{15}+10513409x^{6}z^{16}+190636480x^{5}z^{17}+1022171892x^{4}z^{18}+4038773040x^{3}z^{19}-41943060x^{2}y^{20}-2868909712x^{2}y^{19}z-60892296180x^{2}y^{18}z^{2}-625521774653x^{2}y^{17}z^{3}-3713578464512x^{2}y^{16}z^{4}-13842775453208x^{2}y^{15}z^{5}-33366193751595x^{2}y^{14}z^{6}-50691726383586x^{2}y^{13}z^{7}-41362358927175x^{2}y^{12}z^{8}-810072919327x^{2}y^{11}z^{9}+32948233058945x^{2}y^{10}z^{10}+23686532257796x^{2}y^{9}z^{11}-8216725697454x^{2}y^{8}z^{12}-18551982280360x^{2}y^{7}z^{13}-5008147606918x^{2}y^{6}z^{14}+4935364105296x^{2}y^{5}z^{15}+3051405817008x^{2}y^{4}z^{16}-268545823038x^{2}y^{3}z^{17}-493529753073x^{2}y^{2}z^{18}-37192246698x^{2}yz^{19}+13040136525x^{2}z^{20}+37748754xy^{21}+2063602496xy^{20}z+33990970532xy^{19}z^{2}+254253612525xy^{18}z^{3}+946539184781xy^{17}z^{4}+1298580875656xy^{16}z^{5}-3041911051346xy^{15}z^{6}-16857854600684xy^{14}z^{7}-31679225083776xy^{13}z^{8}-24061212143673xy^{12}z^{9}+9903184408455xy^{11}z^{10}+32314521093558xy^{10}z^{11}+14176416613692xy^{9}z^{12}-15054558665612xy^{8}z^{13}-15784453873608xy^{7}z^{14}+1174564788668xy^{6}z^{15}+6807064257380xy^{5}z^{16}+1480781811982xy^{4}z^{17}-1368490968118xy^{3}z^{18}-476001201918xy^{2}z^{19}+116015372738xyz^{20}+34028406486xz^{21}+y^{22}+2y^{21}z+5y^{20}z^{2}+37748788y^{19}z^{3}+2176848858y^{18}z^{4}+40483768298y^{17}z^{5}+373494514128y^{16}z^{6}+2024953507168y^{15}z^{7}+6975148841178y^{14}z^{8}+15664335396516y^{13}z^{9}+22266852412378y^{12}z^{10}+16878851782610y^{11}z^{11}-399327502764y^{10}z^{12}-13191956249028y^{9}z^{13}-8730700276164y^{8}z^{14}+3441578175792y^{7}z^{15}+6967998779372y^{6}z^{16}+1925716810422y^{5}z^{17}-1676909877634y^{4}z^{18}-1116767943726y^{3}z^{19}+17528551156y^{2}z^{20}+153207619436yz^{21}+24178325764z^{22})}$ |
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.
