Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ x^{2} + 2 x w + 4 y^{2} - 2 y z + 2 z^{2} $ |
| $=$ | $3 x^{2} + 4 x y - 8 x z - 2 x w + 3 y^{2} + 2 y z - 2 z^{2} + 2 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 3 x^{4} - 24 x^{3} y + 4 x^{3} z + 156 x^{2} y^{2} - 24 x^{2} y z - 2 x^{2} z^{2} - 432 x y^{3} + \cdots + 9 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:
$\displaystyle X$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{4}y$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{2}w$ |
Maps to other modular curves
$j$-invariant map
of degree 32 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle 2^2\,\frac{249639393336350976xz^{7}+301903394224153792xz^{6}w-921197455240748672xz^{5}w^{2}-1992817169472539200xz^{4}w^{3}-1267840576144981440xz^{3}w^{4}-170473427908869008xz^{2}w^{5}+89326187893017824xzw^{6}+19367452715264096xw^{7}+8599150715519504y^{2}z^{6}-900639052740617232y^{2}z^{5}w-1794166412613580460y^{2}z^{4}w^{2}-1038222679913278160y^{2}z^{3}w^{3}+13330144157016960y^{2}z^{2}w^{4}+156585866224179868y^{2}zw^{5}+24175845200161421y^{2}w^{6}-60780036327237696yz^{7}+422905285819233728yz^{6}w+1044490290170575136yz^{5}w^{2}+766413213110508880yz^{4}w^{3}+175959603446436880yz^{3}w^{4}-6574897655879392yz^{2}w^{5}-1965490212651464yzw^{6}-1589416018153228yw^{7}+116885370294948528z^{8}-401995434073576448z^{7}w-1236322315246296320z^{6}w^{2}-912328040411339776z^{5}w^{3}-3209721338588240z^{4}w^{4}+253584390852879296z^{3}w^{5}+91650270883541624z^{2}w^{6}+3787705669219680zw^{7}-2650047313156667w^{8}}{690499576142656xz^{7}+1306768933406412xz^{6}w-865571261290232xz^{5}w^{2}-3378738938981420xz^{4}w^{3}-2618169095836000xz^{3}w^{4}-764320754091328xz^{2}w^{5}-54849222766576xzw^{6}+5687117164056xw^{7}+23785774937974y^{2}z^{6}-1869897180624627y^{2}z^{5}w-3937644417391160y^{2}z^{4}w^{2}-2737440963970160y^{2}z^{3}w^{3}-658129842321140y^{2}z^{2}w^{4}+7101143182658y^{2}zw^{5}+13661722850256y^{2}w^{6}-168117359221476yz^{7}+725995872974008yz^{6}w+2035260281874316yz^{5}w^{2}+1664978978148640yz^{4}w^{3}+606785516210160yz^{3}w^{4}+129836538859888yz^{2}w^{5}+23192604213416yzw^{6}+2095698964992yw^{7}+323302992772193z^{8}-562136969312408z^{7}w-2355931521141660z^{6}w^{2}-2080928450169226z^{5}w^{3}-483170041754960z^{4}w^{4}+223089302810416z^{3}w^{5}+145680410322824z^{2}w^{6}+26298465935260zw^{7}+1041648124288w^{8}}$ |
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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.