Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 4 x^{2} - x y + y^{2} + w^{2} $ |
| $=$ | $x w - 2 y w + 3 z^{2} - w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 4 x^{4} - 15 x^{2} y z - 24 x^{2} z^{2} + 15 y^{2} z^{2} + 45 y z^{3} + 45 z^{4} $ |
This modular curve has no real points, and therefore no rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
$\displaystyle X$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{3}w$ |
Maps to other modular curves
$j$-invariant map
of degree 72 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{3^3}{2^2}\cdot\frac{190153669921875xy^{17}+64435585556250000xy^{16}w+6908354275387500000xy^{15}w^{2}+243599175478125000000xy^{14}w^{3}+1330470514205268750000xy^{13}w^{4}+2443747148801532000000xy^{12}w^{5}+2571149604507084000000xy^{11}w^{6}+5483665617448147200000xy^{10}w^{7}+10627118767735524000000xy^{9}w^{8}+16102738315610918400000xy^{8}w^{9}+16179546468938757120000xy^{7}w^{10}+11650559372883267584000xy^{6}w^{11}+5501881036696200192000xy^{5}w^{12}+1574586681932660736000xy^{4}w^{13}+135628810672717824000xy^{3}w^{14}-83985339416696586240xy^{2}w^{15}-7155095468769607680xyw^{16}+149134468753391616xw^{17}-39017162109375y^{18}-21866523581250000y^{17}w-3510300720779296875y^{16}w^{2}-185927805026831250000y^{15}w^{3}-1688650306617473437500y^{14}w^{4}-6296639441760234000000y^{13}w^{5}-13218138029403110250000y^{12}w^{6}-20743160012200382400000y^{11}w^{7}-26004260392040077200000y^{10}w^{8}-24416424925383436800000y^{9}w^{9}-18637659240494729760000y^{8}w^{10}-9275576205802943488000y^{7}w^{11}-3218193944186100480000y^{6}w^{12}+186792662153748480000y^{5}w^{13}+491356820073636864000y^{4}w^{14}+291383597509612666880y^{3}w^{15}+32520211290972487680y^{2}w^{16}-2908146245484675072yw^{17}-134467123384877056w^{18}}{w(614322319921875xy^{16}-9521599992187500xy^{15}w+57977447228437500xy^{14}w^{2}-175059645187500000xy^{13}w^{3}+294326568803250000xy^{12}w^{4}-497455470522000000xy^{11}w^{5}+1448675929911600000xy^{10}w^{6}-2764626408144000000xy^{9}w^{7}+1311026976727200000xy^{8}w^{8}+2982185665056000000xy^{7}w^{9}-3565212598861056000xy^{6}w^{10}-422161442887680000xy^{5}w^{11}+1696805664692428800xy^{4}w^{12}-64794880696320000xy^{3}w^{13}-266703241855303680xy^{2}w^{14}-13147084176752640xyw^{15}+3738339953278976xw^{16}-208473212109375y^{17}+4838481210937500y^{16}w-43303298643984375y^{15}w^{2}+199028157117187500y^{14}w^{3}-508270927188375000y^{13}w^{4}+700312839788250000y^{12}w^{5}-400831745483700000y^{11}w^{6}-217495217566800000y^{10}w^{7}+950410250085600000y^{9}w^{8}-1813109124060000000y^{8}w^{9}+1370877604656192000y^{7}w^{10}+829334545862400000y^{6}w^{11}-1478161087719321600y^{5}w^{12}+12443261230080000y^{4}w^{13}+417477542203637760y^{3}w^{14}+4718559305072640y^{2}w^{15}-25492207040069632yw^{16}-1011706805026816w^{17})}$ |
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Cover information
Click on a modular curve in the diagram to see information about it.
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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.