Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ x^{2} + 2 x w + 2 y^{2} + 2 y z + 2 z^{2} $ |
| $=$ | $3 x^{2} - 4 x y - 8 x z - 2 x w + y^{2} - 2 y z - 2 z^{2} + 2 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 3 x^{4} + 24 x^{3} y + 4 x^{3} z + 108 x^{2} y^{2} + 24 x^{2} y z - 2 x^{2} z^{2} + 240 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
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle 2^2\cdot3\,\frac{10265572806912xz^{7}+178324783644096xz^{6}w+390783415641984xz^{5}w^{2}+172944842299200xz^{4}w^{3}-225542719880640xz^{3}w^{4}-245022988403280xz^{2}w^{5}-68629851690144xzw^{6}-694574489664xw^{7}+45798916647888y^{2}z^{6}+122893656854256y^{2}z^{5}w+218116301364y^{2}z^{4}w^{2}-194661963373968y^{2}z^{3}w^{3}-158500726412544y^{2}z^{2}w^{4}-32965479334212y^{2}zw^{5}+3761412930253y^{2}w^{6}+51490452403008yz^{7}+160172022715200yz^{6}w+57131006155488yz^{5}w^{2}-159829078253328yz^{4}w^{3}-156473762040144yz^{3}w^{4}-45778214341344yz^{2}w^{5}-5742427080440yzw^{6}-2576125624516yw^{7}+53797600129392z^{8}+186484486616064z^{7}w+84724194433536z^{6}w^{2}-196960274051328z^{5}w^{3}-225184767530832z^{4}w^{4}-64472579581248z^{3}w^{5}+14102581945144z^{2}w^{6}+10606425408928zw^{7}+2155107153753w^{8}}{85183405056xz^{7}+1281326316372xz^{6}w+2949215030424xz^{5}w^{2}+1909804498596xz^{4}w^{3}-348658321536xz^{3}w^{4}-695676124464xz^{2}w^{5}-182856194256xzw^{6}-8922901176xw^{7}+380038563558y^{2}z^{6}+1090063166679y^{2}z^{5}w+497439638100y^{2}z^{4}w^{2}-639109961316y^{2}z^{3}w^{3}-557271724284y^{2}z^{2}w^{4}-113992849242y^{2}zw^{5}-1684272424y^{2}w^{6}+427268670012yz^{7}+1359991344600yz^{6}w+922147071588yz^{5}w^{2}-318250150848yz^{4}w^{3}-419118827040yz^{3}w^{4}-101597486736yz^{2}w^{5}-15309318424yzw^{6}-3088128896yw^{7}+446413057839z^{8}+1533735675864z^{7}w+1103367099564z^{6}w^{2}-549469621494z^{5}w^{3}-857447867448z^{4}w^{4}-285957985704z^{3}w^{5}+1721662232z^{2}w^{6}+20430737060zw^{7}+3133923552w^{8}}$ |
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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.