Canonical model in $\mathbb{P}^{ 2 }$
| $ 0 $ | $=$ | $ x^{3} y - x^{3} z - x^{2} y^{2} + x^{2} y z + 2 x^{2} z^{2} - 3 x y z^{2} - x z^{3} - y^{3} z - y^{2} z^{2} $ |
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This modular curve has 8 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:1:0)$, $(1:0:1)$, $(1:-1:1)$, $(1:0:0)$, $(-1:-1:1)$, $(1:1:0)$, $(0:0:1)$, $(0:-1:1)$ |
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{2985984x^{24}-71663616x^{23}z+859963392x^{22}z^{2}-6903595008x^{21}z^{3}+41690308608x^{20}z^{4}-201016442880x^{19}z^{5}+797998252032x^{18}z^{6}-2637579386880x^{17}z^{7}+7198870007808x^{16}z^{8}-15578284621824x^{15}z^{9}+23254556737536x^{14}z^{10}-7212656295936x^{13}z^{11}-88242814402560x^{12}z^{12}+322678760472576x^{11}z^{13}-574498257371136x^{10}z^{14}+92811310202880x^{9}z^{15}+2865144091078656x^{8}z^{16}-9501898696261632x^{7}z^{17}+13348487740391424x^{6}z^{18}+13120056787304448x^{5}z^{19}-117189055923904512x^{4}z^{20}+279949783612686336x^{3}z^{21}-2985244x^{2}y^{22}+11942616x^{2}y^{21}z-129794548x^{2}y^{20}z^{2}+529068912x^{2}y^{19}z^{3}-2129011524x^{2}y^{18}z^{4}+10735691000x^{2}y^{17}z^{5}-56397913996x^{2}y^{16}z^{6}+229448322880x^{2}y^{15}z^{7}-865319024024x^{2}y^{14}z^{8}+2520040341552x^{2}y^{13}z^{9}-6888217770120x^{2}y^{12}z^{10}+15246542191264x^{2}y^{11}z^{11}-24759988926600x^{2}y^{10}z^{12}-11245298122704x^{2}y^{9}z^{13}-17179015673240x^{2}y^{8}z^{14}-813962654904512x^{2}y^{7}z^{15}-210838776176524x^{2}y^{6}z^{16}+591083306394872x^{2}y^{5}z^{17}-13272851837035332x^{2}y^{4}z^{18}+59698440985998192x^{2}y^{3}z^{19}+274084107447844364x^{2}y^{2}z^{20}-189040787118343464x^{2}yz^{21}-288501866003238172x^{2}z^{22}-12xy^{23}-2977356xy^{22}z+1236684xy^{21}z^{2}+44976860xy^{20}z^{3}-1661643236xy^{19}z^{4}+6314705244xy^{18}z^{5}-10764188956xy^{17}z^{6}+17637657108xy^{16}z^{7}+24088093384xy^{15}z^{8}+153425003848xy^{14}z^{9}-225935611208xy^{13}z^{10}+2198595804504xy^{12}z^{11}-9808733970824xy^{11}z^{12}+41584237331960xy^{10}z^{13}+26369142990216xy^{9}z^{14}+605146542172648xy^{8}z^{15}+1659155311203972xy^{7}z^{16}+1566837956080196xy^{6}z^{17}+13752415963069244xy^{5}z^{18}-4587484317202036xy^{4}z^{19}-225560272735202772xy^{3}z^{20}-358573704266010772xy^{2}z^{21}+239295457838731988xyz^{22}+106150941926791748xz^{23}+y^{24}-2986012y^{23}z+9162796y^{22}z^{2}-139336892y^{21}z^{3}+1030714490y^{20}z^{4}-6062211460y^{19}z^{5}+16464115612y^{18}z^{6}-55663741988y^{17}z^{7}+182421629159y^{16}z^{8}-572424004632y^{15}z^{9}+1437048816088y^{14}z^{10}-3593590950232y^{13}z^{11}+6604818317708y^{12}z^{12}+10407532939384y^{11}z^{13}+16411308835864y^{10}z^{14}+230473939888184y^{9}z^{15}-357591575388385y^{8}z^{16}-1787625985837580y^{7}z^{17}-3268835007610116y^{6}z^{18}-46952419376322604y^{5}z^{19}-124058986232654054y^{4}z^{20}+26993573985335116y^{3}z^{21}+106150941926844236y^{2}z^{22}+8748yz^{23}+729z^{24}}{z(2985984x^{12}z^{11}-71663616x^{11}z^{12}+859963392x^{10}z^{13}-6831931392x^{9}z^{14}+39934550016x^{8}z^{15}-179517358080x^{7}z^{16}+623282356224x^{6}z^{17}-1586489131008x^{5}z^{18}+2278786535424x^{4}z^{19}+2622506139648x^{3}z^{20}-10x^{2}y^{21}-283x^{2}y^{20}z-4262x^{2}y^{19}z^{2}-46147x^{2}y^{18}z^{3}-395698x^{2}y^{17}z^{4}-2724764x^{2}y^{16}z^{5}-14761960x^{2}y^{15}z^{6}-60223628x^{2}y^{14}z^{7}-167243204x^{2}y^{13}z^{8}-190813818x^{2}y^{12}z^{9}+872427548x^{2}y^{11}z^{10}+6303701382x^{2}y^{10}z^{11}+22908441148x^{2}y^{9}z^{12}+57754398580x^{2}y^{8}z^{13}+112598638616x^{2}y^{7}z^{14}+113858817124x^{2}y^{6}z^{15}+118913431118x^{2}y^{5}z^{16}-918091588675x^{2}y^{4}z^{17}-415362322598x^{2}y^{3}z^{18}-7167137956123x^{2}y^{2}z^{19}-12811113529354x^{2}yz^{20}-8006665273344x^{2}z^{21}+11xy^{22}+289xy^{21}z+4020xy^{20}z^{2}+41262xy^{19}z^{3}+354237xy^{18}z^{4}+2590837xy^{17}z^{5}+15677568xy^{16}z^{6}+75627976xy^{15}z^{7}+278706526xy^{14}z^{8}+720121074xy^{13}z^{9}+897268744xy^{12}z^{10}-2260496204xy^{11}z^{11}-18018448502xy^{10}z^{12}-64329985214xy^{9}z^{13}-161831681248xy^{8}z^{14}-301869658488xy^{7}z^{15}-410972257753xy^{6}z^{16}-146966617043xy^{5}z^{17}+814916896932xy^{4}z^{18}+3368727434190xy^{3}z^{19}+14531350855977xy^{2}z^{20}+16966337200137xyz^{21}+4214202826752xz^{22}-y^{23}+3y^{22}z+538y^{21}z^{2}+9951y^{20}z^{3}+101277y^{19}z^{4}+668764y^{18}z^{5}+2775496y^{17}z^{6}+4017404y^{16}z^{7}-37797946y^{15}z^{8}-364052406y^{14}z^{9}-1847112884y^{13}z^{10}-6626252350y^{12}z^{11}-17920607830y^{11}z^{12}-36167748596y^{10}z^{13}-48090118104y^{9}z^{14}+1582436172y^{8}z^{15}+205271906795y^{7}z^{16}+925338577227y^{6}z^{17}+1884812825002y^{5}z^{18}+5462153036055y^{4}z^{19}+8537931546633y^{3}z^{20}+4214202826752y^{2}z^{21})}$ |
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.
