Canonical model in $\mathbb{P}^{ 2 }$
| $ 0 $ | $=$ | $ x^{4} + 3 x^{3} y - 2 x^{3} z + x^{2} y^{2} - x^{2} y z - 2 x^{2} z^{2} - x y^{3} + 2 x y^{2} z + \cdots + z^{4} $ |
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This modular curve has 2 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}{3^4}\cdot\frac{43302601023592x^{3}y^{21}-609763750207824x^{3}y^{20}z+4323687737815488x^{3}y^{19}z^{2}-20423129094341120x^{3}y^{18}z^{3}+71829664079798592x^{3}y^{17}z^{4}-199526717997924480x^{3}y^{16}z^{5}+453599832933801984x^{3}y^{15}z^{6}-863692209216258048x^{3}y^{14}z^{7}+1398892625687236608x^{3}y^{13}z^{8}-1947430708550017024x^{3}y^{12}z^{9}+2345673351571046400x^{3}y^{11}z^{10}-2453431871927746560x^{3}y^{10}z^{11}+2230424648070594560x^{3}y^{9}z^{12}-1759491863602790400x^{3}y^{8}z^{13}+1198947516442214400x^{3}y^{7}z^{14}-700120381972807680x^{3}y^{6}z^{15}+345955927051468800x^{3}y^{5}z^{16}-141894692335779840x^{3}y^{4}z^{17}+46849071775744000x^{3}y^{3}z^{18}-11835554974924800x^{3}y^{2}z^{19}+2071222120611840x^{3}yz^{20}-197259249582080x^{3}z^{21}+15530374133968x^{2}y^{22}-238624495396808x^{2}y^{21}z+1802475877016784x^{2}y^{20}z^{2}-8933389678762688x^{2}y^{19}z^{3}+32579290348333888x^{2}y^{18}z^{4}-92939409989848512x^{2}y^{17}z^{5}+215107772160642816x^{2}y^{16}z^{6}-413551027964534784x^{2}y^{15}z^{7}+670527260685066240x^{2}y^{14}z^{8}-925705005231861760x^{2}y^{13}z^{9}+1093536405931163648x^{2}y^{12}z^{10}-1106261421658079232x^{2}y^{11}z^{11}+954673301443051520x^{2}y^{10}z^{12}-695818581596569600x^{2}y^{9}z^{13}+419629689264144384x^{2}y^{8}z^{14}-200696946587860992x^{2}y^{7}z^{15}+68168098451030016x^{2}y^{6}z^{16}-9439725768671232x^{2}y^{5}z^{17}-6301592937562112x^{2}y^{4}z^{18}+5582862124318720x^{2}y^{3}z^{19}-2372524050481152x^{2}y^{2}z^{20}+602893346406400x^{2}yz^{21}-81707474616320x^{2}z^{22}+2201317345320xy^{23}+1217453218544xy^{22}z-292810150682728xy^{21}z^{2}+2995037070048336xy^{20}z^{3}-17290146072722176xy^{19}z^{4}+70281765000525440xy^{18}z^{5}-219799504985879232xy^{17}z^{6}+554456854528983168xy^{16}z^{7}-1161376678460780544xy^{15}z^{8}+2058516507840135168xy^{14}z^{9}-3127089229277818880xy^{13}z^{10}+4105551739615682560xy^{12}z^{11}-4682646490630324224xy^{11}z^{12}+4651076308085506048xy^{10}z^{13}-4022992194382462976xy^{9}z^{14}+3022460398826422272xy^{8}z^{15}-1961820676982046720xy^{7}z^{16}+1090359210154131456xy^{6}z^{17}-511893032129789952xy^{5}z^{18}+198808517554995200xy^{4}z^{19}-61841481782001664xy^{3}z^{20}+14584368630595584xy^{2}z^{21}-2350188844810240xyz^{22}+197259249582080xz^{23}-49219651922329y^{24}+696692129140800y^{23}z-4987129533416360y^{22}z^{2}+23843477000965224y^{21}z^{3}-85047658163525952y^{20}z^{4}+239856025435673920y^{19}z^{5}-553766930863084544y^{18}z^{6}+1069842442351440960y^{17}z^{7}-1753808806693397760y^{16}z^{8}+2459300840026341376y^{15}z^{9}-2958864513258737664y^{14}z^{10}+3047028622545047552y^{13}z^{11}-2659378733802160128y^{12}z^{12}+1921377754770702336y^{11}z^{13}-1085167378744606720y^{10}z^{14}+395687072966770688y^{9}z^{15}+20154495948226560y^{8}z^{16}-176638628475174912y^{7}z^{17}+171762268452159488y^{6}z^{18}-109212918014803968y^{5}z^{19}+51813836482674688y^{4}z^{20}-18668660277116928y^{3}z^{21}+4948008584085504y^{2}z^{22}-881859869278208yz^{23}+81707457839104z^{24}}{y^{8}(y^{2}-yz+z^{2})(5636x^{3}y^{11}+5620x^{3}y^{10}z-88564x^{3}y^{9}z^{2}+204792x^{3}y^{8}z^{3}-291840x^{3}y^{7}z^{4}+282624x^{3}y^{6}z^{5}-215040x^{3}y^{5}z^{6}+141312x^{3}y^{4}z^{7}-76800x^{3}y^{3}z^{8}+35840x^{3}y^{2}z^{9}-11264x^{3}yz^{10}+2048x^{3}z^{11}-101179x^{2}y^{12}+316440x^{2}y^{11}z-822312x^{2}y^{10}z^{2}+1355540x^{2}y^{9}z^{3}-1805328x^{2}y^{8}z^{4}+1831680x^{2}y^{7}z^{5}-1552128x^{2}y^{6}z^{6}+1078272x^{2}y^{5}z^{7}-628992x^{2}y^{4}z^{8}+296960x^{2}y^{3}z^{9}-110592x^{2}y^{2}z^{10}+27648x^{2}yz^{11}-4096x^{2}z^{12}-131907xy^{13}+470068xy^{12}z-1198416xy^{11}z^{2}+1993788xy^{10}z^{3}-2602532xy^{9}z^{4}+2662920xy^{8}z^{5}-2262528xy^{7}z^{6}+1586688xy^{6}z^{7}-887040xy^{5}z^{8}+365568xy^{4}z^{9}-88064xy^{3}z^{10}-3072xy^{2}z^{11}+9216xyz^{12}-2048xz^{13}-133531y^{14}+377644y^{13}z-806708y^{12}z^{2}+1107556y^{11}z^{3}-1246972y^{10}z^{4}+1047540y^{9}z^{5}-676944y^{8}z^{6}+251136y^{7}z^{7}+44544y^{6}z^{8}-150016y^{5}z^{9}+119296y^{4}z^{10}-54272y^{3}z^{11}+13312y^{2}z^{12}-1024yz^{13})}$ |
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.
