Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 3 x^{2} - 6 y^{2} + 2 z^{2} - 2 z w $ |
| $=$ | $9 x^{2} + 12 x y + 6 y^{2} - 3 z^{2} + 4 z w + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 81 x^{4} - 180 x^{3} z - 84 x^{2} y^{2} + 82 x^{2} z^{2} + 96 x y^{2} z + 20 x z^{3} + 18 y^{4} + \cdots + z^{4} $ |
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
$\displaystyle X$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle 2y$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Maps to other modular curves
$j$-invariant map
of degree 64 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{1}{2^6}\cdot\frac{1848000y^{2}z^{14}-67824000y^{2}z^{13}w+884041920y^{2}z^{12}w^{2}-4449938688y^{2}z^{11}w^{3}+5570276544y^{2}z^{10}w^{4}+2411876736y^{2}z^{9}w^{5}-4614420288y^{2}z^{8}w^{6}-2164962816y^{2}z^{7}w^{7}+1012544064y^{2}z^{6}w^{8}+1062322560y^{2}z^{5}w^{9}+368986176y^{2}z^{4}w^{10}+68408064y^{2}z^{3}w^{11}+7228992y^{2}z^{2}w^{12}+411264y^{2}zw^{13}+9792y^{2}w^{14}-640625z^{16}+23693200z^{15}w-324863240z^{14}w^{2}+1825350288z^{13}w^{3}-3455417980z^{12}w^{4}+1094489232z^{11}w^{5}+2501798408z^{10}w^{6}-868802800z^{9}w^{7}-1188393286z^{8}w^{8}-36821712z^{7}w^{9}+271768648z^{6}w^{10}+128693040z^{5}w^{11}+29396164z^{4}w^{12}+3892912z^{3}w^{13}+304696z^{2}w^{14}+13104zw^{15}+239w^{16}}{z^{8}(10080y^{2}z^{6}-39936y^{2}z^{5}w+42912y^{2}z^{4}w^{2}+2304y^{2}z^{3}w^{3}-13536y^{2}z^{2}w^{4}-3840y^{2}zw^{5}-288y^{2}w^{6}-3431z^{8}+17144z^{7}w-28588z^{6}w^{2}+13880z^{5}w^{3}+6038z^{4}w^{4}-3448z^{3}w^{5}-1516z^{2}w^{6}-184zw^{7}-7w^{8})}$ |
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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.