Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ y^{2} - 2 y z - z^{2} - w^{2} $ |
| $=$ | $3 x^{2} - y^{2} - 2 y z + z^{2}$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} - 6 x^{2} y^{2} - 2 x^{2} z^{2} - 9 y^{4} + 6 y^{2} z^{2} - z^{4} $ |
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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:
| $\displaystyle X$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{2}x$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{2}w$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle 2^2\,\frac{1532805120yz^{23}+1040670720yz^{21}w^{2}-468725760yz^{19}w^{4}-436582400yz^{17}w^{6}+7165952yz^{15}w^{8}+57977856yz^{13}w^{10}+8712704yz^{11}w^{12}-1975040yz^{9}w^{14}-645696yz^{7}w^{16}-61984yz^{5}w^{18}-2288yz^{3}w^{20}-24yzw^{22}+634908672z^{24}+972988416z^{22}w^{2}+106039296z^{20}w^{4}-375613440z^{18}w^{6}-124464896z^{16}w^{8}+38925312z^{14}w^{10}+21149440z^{12}w^{12}+521984z^{10}w^{14}-975600z^{8}w^{16}-176608z^{6}w^{18}-11512z^{4}w^{20}-264z^{2}w^{22}-w^{24}}{z^{4}(2z^{2}+w^{2})^{4}(221760yz^{11}+258176yz^{9}w^{2}+105920yz^{7}w^{4}+18192yz^{5}w^{6}+1196yz^{3}w^{8}+20yzw^{10}+91856z^{12}+185344z^{10}w^{2}+125352z^{8}w^{4}+36024z^{6}w^{6}+4333z^{4}w^{8}+174z^{2}w^{10}+w^{12})}$ |
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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.
