Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
$ 0 $ | $=$ | $ 2 x y + 3 x t - y^{2} - 2 y t + t^{2} $ |
| $=$ | $x^{2} + 2 x y - 3 x t + 3 y^{2} - 3 y w - y t + 3 w^{2} + 2 t^{2}$ |
| $=$ | $3 x^{2} + x y + x t - 3 y^{2} + 2 y t - 7 z^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 139 x^{8} - 492 x^{7} z - 322 x^{6} y^{2} + 843 x^{6} z^{2} + 861 x^{5} y^{2} z - 729 x^{5} z^{3} + \cdots + 2268 y^{4} z^{4} $ |
This modular curve has no real points, and therefore no rational points.
Maps between models of this curve
Birational map from canonical model to plane model:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Maps to other modular curves
$j$-invariant map
of degree 84 from the canonical model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle 3^3\cdot7^2\,\frac{256664520000xw^{9}t+418552890000xw^{8}t^{2}-2009437096800xw^{7}t^{3}+675879800579158xw^{6}t^{4}+3026669797981326xw^{5}t^{5}-3467866796870238xw^{4}t^{6}-24704567846407827xw^{3}t^{7}-46845316061173875xw^{2}t^{8}-41288518324152183xwt^{9}-13334101877494769xt^{10}-7260264000yw^{10}+839807996400yw^{9}t+1618637688960yw^{8}t^{2}-3248664439941yw^{7}t^{3}-343786307644262yw^{6}t^{4}-2571495020227602yw^{5}t^{5}-34679276668323yw^{4}t^{6}+15807971337872241yw^{3}t^{7}+33320037818095866yw^{2}t^{8}+28293436507039230ywt^{9}+8626821736882647yt^{10}-102665808000z^{2}w^{9}-151174254000z^{2}w^{8}t+778535735040z^{2}w^{7}t^{2}-949950022007z^{2}w^{6}t^{3}+3027223751244z^{2}w^{5}t^{4}-6038880175096902z^{2}w^{4}t^{5}-19022423868844270z^{2}w^{3}t^{6}-8199859427001375z^{2}w^{2}t^{7}+13244489774292126z^{2}wt^{8}+8309406817110728z^{2}t^{9}-131998896000w^{11}-403255638000w^{10}t+1439307262080w^{9}t^{2}+1663874704401w^{8}t^{3}+1633771370163w^{7}t^{4}+908461545210751w^{6}t^{5}+3145744240270296w^{5}t^{6}+2586367883907051w^{4}t^{7}-3801984888932610w^{3}t^{8}-11821180181346636w^{2}t^{9}-11463013912645617wt^{10}-3775246865405927t^{11}}{42445894995xw^{9}t-350921723260473xw^{8}t^{2}-2173467567372120xw^{7}t^{3}-3059676898961886xw^{6}t^{4}+4857403226968971xw^{5}t^{5}+19117518718828446xw^{4}t^{6}+23150631226075353xw^{3}t^{7}+12115300289875314xw^{2}t^{8}+1414294568545272xwt^{9}-455594850948505xt^{10}-1200666159yw^{10}-7382752005528yw^{9}t+148954763866779yw^{8}t^{2}+1438014265514334yw^{7}t^{3}+2929382516291553yw^{6}t^{4}-2165699705369949yw^{5}t^{5}-13742655681642048yw^{4}t^{6}-16791927107461821yw^{3}t^{7}-7645361367805185yw^{2}t^{8}-367168801586499ywt^{9}+377098426340576yt^{10}-16978357998z^{2}w^{9}-40694523093837z^{2}w^{8}t-244854955070442z^{2}w^{7}t^{2}+1438967030394600z^{2}w^{6}t^{3}+8977203143183079z^{2}w^{5}t^{4}+11423551800119445z^{2}w^{4}t^{5}-4254389923618296z^{2}w^{3}t^{6}-14469512650901112z^{2}w^{2}t^{7}-6391933795723626z^{2}wt^{8}-452397303055345z^{2}t^{9}-21829317426w^{11}+7335980023572w^{10}t+43232853519621w^{9}t^{2}-188096405100588w^{8}t^{3}-1441086744750678w^{7}t^{4}-2548871252135271w^{6}t^{5}-179126472342069w^{5}t^{6}+4813180147486683w^{4}t^{7}+6880526478041121w^{3}t^{8}+4036574380577505w^{2}t^{9}+688820012123808wt^{10}-90996511582546t^{11}}$ |
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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.