Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x^{2} - y z $ |
| $=$ | $2 x^{2} - x y - x z - 2 x w + 3 y^{2} + 2 y z + 2 y w + 3 z^{2} + 2 z w + 2 w^{2}$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 3 x^{4} - x^{3} z + 2 x^{2} y z + 4 x^{2} z^{2} - 2 x y z^{2} - x z^{3} + 2 y^{2} z^{2} + 2 y z^{3} + 3 z^{4} $ |
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This modular curve has no real points, and therefore no rational points.
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 3^3\,\frac{1146140513066xz^{11}+2264648509012xz^{10}w+182811735946xz^{9}w^{2}+3633348801600xz^{8}w^{3}-12814799504184xz^{7}w^{4}-13757875083216xz^{6}w^{5}-21454544311344xz^{5}w^{6}-22172744003328xz^{4}w^{7}-5772135103200xz^{3}w^{8}+4340747102400xz^{2}w^{9}+1691077069728xzw^{10}-110413756800xw^{11}-3310922061373y^{2}z^{10}+6198936933810y^{2}z^{9}w+4824585059796y^{2}z^{8}w^{2}+36463362727464y^{2}z^{7}w^{3}+59359694533380y^{2}z^{6}w^{4}+72065652653232y^{2}z^{5}w^{5}+82975729426272y^{2}z^{4}w^{6}+54787453090560y^{2}z^{3}w^{7}+15188890834800y^{2}z^{2}w^{8}+2995566792480y^{2}zw^{9}+682889875200y^{2}w^{10}-4457062574439yz^{11}+3934288424798yz^{10}w+1033244203750yz^{9}w^{2}+35746584048000yz^{8}w^{3}+67524740834436yz^{7}w^{4}+90126008202384yz^{6}w^{5}+109257040007856yz^{5}w^{6}+81041037648192yz^{4}w^{7}+30008701974960yz^{3}w^{8}+5450776973280yz^{2}w^{9}+1377958016160yzw^{10}+406761004800yw^{11}-1915196746110z^{12}+3934288424798z^{11}w+6060021000078z^{10}w^{2}+39232575120952z^{9}w^{3}+74574771723648z^{8}w^{4}+118866957592608z^{7}w^{5}+146645400522096z^{6}w^{6}+132572012078016z^{5}w^{7}+84490819286400z^{4}w^{8}+36188462210400z^{3}w^{9}+7653717228960z^{2}w^{10}+1686640993920zw^{11}+451638244800w^{12}}{2171699200xz^{11}+15507660800xz^{10}w+77431709696xz^{9}w^{2}+238898672640xz^{8}w^{3}+450244505088xz^{7}w^{4}+477799354368xz^{6}w^{5}+157823576334xz^{5}w^{6}-213073211556xz^{4}w^{7}-139816029744xz^{3}w^{8}+10435532940xz^{2}w^{9}+5021015202xzw^{10}-423105768xw^{11}+21221785600y^{2}z^{10}+64434094080y^{2}z^{9}w+6927897600y^{2}z^{8}w^{2}-310478635008y^{2}z^{7}w^{3}-690484642560y^{2}z^{6}w^{4}-597464418816y^{2}z^{5}w^{5}+30590269569y^{2}z^{4}w^{6}+235379024280y^{2}z^{3}w^{7}+17531648100y^{2}z^{2}w^{8}-9236247750y^{2}zw^{9}+449008596y^{2}w^{10}+19050086400yz^{11}+48926433280yz^{10}w-69635132416yz^{9}w^{2}-536730743808yz^{8}w^{3}-1095709208832yz^{7}w^{4}-1009858392576yz^{6}w^{5}-114576467085yz^{5}w^{6}+346123041894yz^{4}w^{7}+135435326589yz^{3}w^{8}-2838879090yz^{2}w^{9}-4622001426yzw^{10}+275089608yw^{11}+19050086400z^{12}+48926433280z^{11}w-53526383616z^{10}w^{2}-498177504256z^{9}w^{3}-1141063540992z^{8}w^{4}-1345245977088z^{7}w^{5}-729073881450z^{6}w^{6}-23265553860z^{5}w^{7}+253813107069z^{4}w^{8}+152886910806z^{3}w^{9}-552423078z^{2}w^{10}-5733736632zw^{11}+297528228w^{12}}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
60.48.1.be.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle w$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle z$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 3X^{4}-X^{3}Z+2X^{2}YZ+4X^{2}Z^{2}-2XYZ^{2}+2Y^{2}Z^{2}-XZ^{3}+2YZ^{3}+3Z^{4} $ |
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.
