Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 10 x^{2} + 15 x y + y^{2} - y z - z^{2} - w^{2} $ |
| $=$ | $20 x^{2} - 10 x y + 3 y^{2} + 2 y z + 2 z^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} + 4 x^{3} y + 26 x^{2} y^{2} + 4 x^{2} z^{2} + 44 x y^{3} + 8 x y z^{2} + 221 y^{4} - 36 y^{2} z^{2} + 4 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:
$\displaystyle X$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{2}y$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{2}w$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{2^8}{5^2}\cdot\frac{198857498398817850000000xz^{11}+7345936402972136155800000xz^{9}w^{2}-16894905198340428178920000xz^{7}w^{4}-8334818814472939784008000xz^{5}w^{6}+1177099709498380550480000xz^{3}w^{8}+107484680214076246952000xzw^{10}+758145691225714995562500y^{2}z^{10}-1648751038484850735654375y^{2}z^{8}w^{2}-12885010914658163176233000y^{2}z^{6}w^{4}+11204906750633157935243000y^{2}z^{4}w^{6}-421780752892522348107900y^{2}z^{2}w^{8}+2047137852448967472688y^{2}w^{10}+433051678341418252500000yz^{11}+1865824107132581540257500yz^{9}w^{2}-16375991862895258217832000yz^{7}w^{4}+8900089881824269824072000yz^{5}w^{6}+772868137178418895073200yz^{3}w^{8}-56680033045629608918368yzw^{10}+166766162159468469375000z^{12}+789830921135373127132500z^{10}w^{2}-2858800794360311339340750z^{8}w^{4}-1353775765800946910994000z^{6}w^{6}+173747721749069911991600z^{4}w^{8}+91071330890254214274632z^{2}w^{10}-1111925904585833629656w^{12}}{1178414805326328000000xz^{11}-4274333236331840928000xz^{9}w^{2}-11293548758993466700800xz^{7}w^{4}-9035818392179283269760xz^{5}w^{6}-3072538755534642912960xz^{3}w^{8}-385037084659496168640xzw^{10}+4492715207263496270000y^{2}z^{10}+12564054546856638207200y^{2}z^{8}w^{2}+13544707768205481480080y^{2}z^{6}w^{4}+6958895592478324646710y^{2}z^{4}w^{6}+1689711307166345296600y^{2}z^{2}w^{8}+154273410583993134421y^{2}w^{10}+2566232167949145200000yz^{11}+8033994817278071968800yz^{9}w^{2}+10014072125940779060320yz^{7}w^{4}+6072855040022798695720yz^{5}w^{6}+1765260794972467117104yz^{3}w^{8}+195049110665196494252yzw^{10}+988243923907961300000z^{12}+2724755234691976068800z^{10}w^{2}+2767287800065853173720z^{8}w^{4}+1182014020616182137800z^{6}w^{6}+100317024962980441876z^{4}w^{8}-68666831186226267124z^{2}w^{10}-14909116987888943326w^{12}}$ |
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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.