Canonical model in $\mathbb{P}^{ 2 }$
| $ 0 $ | $=$ | $ 6 x^{4} - 8 x^{3} y - 4 x^{3} z - 3 x^{2} y^{2} - 6 x^{2} y z + 6 x^{2} z^{2} + 2 x y^{3} + \cdots + 2 y z^{3} $ |
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This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
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{1}{3}\cdot\frac{280482888913061733164328400x^{3}y^{21}+8501705111563716468538558248x^{3}y^{20}z+111491080925352158972466748704x^{3}y^{19}z^{2}+826792022886655247285776969088x^{3}y^{18}z^{3}+3792096459480649786720664494464x^{3}y^{17}z^{4}+10963387852179206560456332979296x^{3}y^{16}z^{5}+18975894284363461614252262354944x^{3}y^{15}z^{6}+14902335201681288028431184458240x^{3}y^{14}z^{7}-7987701894221758132894342169088x^{3}y^{13}z^{8}-27272707655627930237657415160064x^{3}y^{12}z^{9}-15439357149673229917262498030592x^{3}y^{11}z^{10}+10639543649542419885449440364544x^{3}y^{10}z^{11}+14155259245842405688727098019840x^{3}y^{9}z^{12}+90637532051879482806758114304x^{3}y^{8}z^{13}-4937258356577918453321158164480x^{3}y^{7}z^{14}-836011287487049922355754311680x^{3}y^{6}z^{15}+874174010817350917014639611904x^{3}y^{5}z^{16}+153618741465798714696241072128x^{3}y^{4}z^{17}-77628576840446932058690969600x^{3}y^{3}z^{18}-6540231506629482513009278976x^{3}y^{2}z^{19}+2235477558083354689719336960x^{3}yz^{20}-3302836342593910807601152x^{3}z^{21}+51811841458566575593152474x^{2}y^{22}+1851229712384878835035512156x^{2}y^{21}z+28514434357262279091581097756x^{2}y^{20}z^{2}+247640683395489036370347553824x^{2}y^{19}z^{3}+1326520416997536992001187685400x^{2}y^{18}z^{4}+4464957601726536883858908902928x^{2}y^{17}z^{5}+8954844309996133554576995172048x^{2}y^{16}z^{6}+8036545819504472807905076670720x^{2}y^{15}z^{7}-5398358111759186208515933234496x^{2}y^{14}z^{8}-20307666028241774620690862340480x^{2}y^{13}z^{9}-12957063751384573398781917040512x^{2}y^{12}z^{10}+10870799505218647433160950851584x^{2}y^{11}z^{11}+16281592585158260723908460217600x^{2}y^{10}z^{12}-22239738445842019206578084352x^{2}y^{9}z^{13}-7824604728831076435293240482304x^{2}y^{8}z^{14}-1584328151008056033355356868608x^{2}y^{7}z^{15}+1974280404075973390460035109376x^{2}y^{6}z^{16}+460228257177241435236590570496x^{2}y^{5}z^{17}-274176616635268264555409052672x^{2}y^{4}z^{18}-40050273180389610763470151680x^{2}y^{3}z^{19}+16890588456337295914861049856x^{2}y^{2}z^{20}+381418838557375402272313344x^{2}yz^{21}-159717004002820416940634112x^{2}z^{22}-61588126260963715532419204xy^{23}-1634658247528756954810237252xy^{22}z-17702128863626261973808862272xy^{21}z^{2}-96574576411869623154597511304xy^{20}z^{3}-237333728671084021113692276144xy^{19}z^{4}+123546743713113257809282268464xy^{18}z^{5}+2362604023655151575849750068800xy^{17}z^{6}+5780053989385173349289421737568xy^{16}z^{7}+3812103968484011918325999052416xy^{15}z^{8}-7460891751822894305070201473920xy^{14}z^{9}-14512914929400364725324265980928xy^{13}z^{10}-2733393777858644113121174484736xy^{12}z^{11}+12268710189665472471431261195776xy^{11}z^{12}+7953221391777342892348170463744xy^{10}z^{13}-4240522562327610492184485926912xy^{9}z^{14}-4679934173837631466609861929984xy^{8}z^{15}+582212273463571767392018795520xy^{7}z^{16}+1277233163351219377507844140032xy^{6}z^{17}-18369769175292848835126231040xy^{5}z^{18}-170481970583865212165935802368xy^{4}z^{19}+1503485221894035960827285504xy^{3}z^{20}+9009523393916439208483876864xy^{2}z^{21}-312607279800064200437645312xyz^{22}-70529504394230197031149568xz^{23}+7625597484987y^{24}-30794063130664872105849290y^{23}z-807186536779122693450258014y^{22}z^{2}-8657932989112723747220364516y^{21}z^{3}-47438937373436347929520692076y^{20}z^{4}-126349268098391898954389634376y^{19}z^{5}-44524726698422424721366781784y^{18}z^{6}+649904750009671311317018472240y^{17}z^{7}+1562328087241499953796974729440y^{16}z^{8}+582911327190171048083077886784y^{15}z^{9}-2561048459826136026081041508416y^{14}z^{10}-3110286922011405497473487232896y^{13}z^{11}+988903678889421650664241869696y^{12}z^{12}+3230452489810196606576152908032y^{11}z^{13}+407769793671203433083437617920y^{10}z^{14}-1598410536366751721540505220608y^{9}z^{15}-395639143461719288949799218432y^{8}z^{16}+445393293398044537326627540480y^{7}z^{17}+97006494544163231216155937280y^{6}z^{18}-69363480192305091798022501376y^{5}z^{19}-6440419235373076657903483904y^{4}z^{20}+4453079783373139083184674816y^{3}z^{21}-89922674254387503590815744y^{2}z^{22}-35265021758364567478669312yz^{23}+11231718727873462272z^{24}}{(y-z)(15787499231985688337344x^{3}y^{20}+288485056755433105134400x^{3}y^{19}z+1955500937163928570042900x^{3}y^{18}z^{2}+5193991220602855675859256x^{3}y^{17}z^{3}-1757830099309023189974712x^{3}y^{16}z^{4}-34164362983215109198128864x^{3}y^{15}z^{5}-33089324733738965321932512x^{3}y^{14}z^{6}+88018467614161646794783008x^{3}y^{13}z^{7}+126133669665524188195523232x^{3}y^{12}z^{8}-136144396571336103270873344x^{3}y^{11}z^{9}-204018953224095205182649856x^{3}y^{10}z^{10}+148414293049992336974635648x^{3}y^{9}z^{11}+166289953974674889201346176x^{3}y^{8}z^{12}-108264934856226267393388032x^{3}y^{7}z^{13}-63801442160571413676274176x^{3}y^{6}z^{14}+43022823127858603500252672x^{3}y^{5}z^{15}+8772576472226418885189120x^{3}y^{4}z^{16}-7102603502551979881562112x^{3}y^{3}z^{17}-50584648061350014510080x^{3}y^{2}z^{18}+300704467810336100024320x^{3}yz^{19}-11820306896912671944704x^{3}z^{20}+2916324868026910030464x^{2}y^{21}+69093317789739086556768x^{2}y^{20}z+616745627487697002564288x^{2}y^{19}z^{2}+2377809115642010847649254x^{2}y^{18}z^{3}+1801951519355356378055328x^{2}y^{17}z^{4}-13594354735428214166051244x^{2}y^{16}z^{5}-29434934137035064237000200x^{2}y^{15}z^{6}+29175841234742467651482168x^{2}y^{14}z^{7}+109604569353142029362340480x^{2}y^{13}z^{8}-37579164131377293374392944x^{2}y^{12}z^{9}-211291373892999155739905568x^{2}y^{11}z^{10}+53913562155591525107911200x^{2}y^{10}z^{11}+230193355847518448210212224x^{2}y^{9}z^{12}-76742637237254453413442112x^{2}y^{8}z^{13}-131930270588223629321788800x^{2}y^{7}z^{14}+59133247962518041685817984x^{2}y^{6}z^{15}+32734218906297569455423488x^{2}y^{5}z^{16}-18828336301629886663107840x^{2}y^{4}z^{17}-1993434963198020774094336x^{2}y^{3}z^{18}+1877056970608928477091840x^{2}y^{2}z^{19}-81302985848092064372736x^{2}yz^{20}-24559362568609227190272x^{2}z^{21}-3466602178909435415776xy^{22}-50278973876946644483168xy^{21}z-205111374021005206709802xy^{20}z^{2}+241291020802587489195160xy^{19}z^{3}+3304127421709692747893156xy^{18}z^{4}+3297208428330110678633112xy^{17}z^{5}-17358527255757863715862824xy^{16}z^{6}-26615658218046865293593472xy^{15}z^{7}+51463825176757714559402544xy^{14}z^{8}+81365355395766194573389280xy^{13}z^{9}-103823447358013075629072800xy^{12}z^{10}-126829174458330770805420672xy^{11}z^{11}+142795954980742802600138560xy^{10}z^{12}+101443441704027504801229952xy^{9}z^{13}-120406950895826551399353216xy^{8}z^{14}-35902907323268489599921152xy^{7}z^{15}+54690812960017073477519616xy^{6}z^{16}+2319900995739004848585216xy^{5}z^{17}-11476110419603127029029888xy^{4}z^{18}+1020289947487926043107328xy^{3}z^{19}+840008527934005648730112xy^{2}z^{20}-101985679687668346413056xyz^{21}-8598000703504754630656xz^{22}-1733301089454717707888y^{22}z-24568592338389011468284y^{21}z^{2}-98557329933119600149771y^{20}z^{3}+89628300820332852715924y^{19}z^{4}+1294490058796859277002982y^{18}z^{5}+854387864119145095636200y^{17}z^{6}-6951250180023711027117732y^{16}z^{7}-6202822334307391742497872y^{15}z^{8}+22415938081764661087897224y^{14}z^{9}+14830631310644595332352064y^{13}z^{10}-46313705243126835390842800y^{12}z^{11}-11810740166763562707790336y^{11}z^{12}+56413632915324148187355424y^{10}z^{13}-6084692844855579447282816y^{9}z^{14}-34708746773515549832090304y^{8}z^{15}+12647485594802547295808256y^{7}z^{16}+8345536202564737630687104y^{6}z^{17}-4839491110356906678875136y^{5}z^{18}-275769919091561914359296y^{4}z^{19}+466275215854651784680448y^{3}z^{20}-41628748937867583422464y^{2}z^{21}-4299000351752377315328yz^{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.
