Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ - 2 x w + 2 y^{2} - z^{2} $ |
| $=$ | $x^{2} - 4 x w + 4 y z + 4 z^{2} + 2 w^{2}$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 4 x^{4} - 4 x^{2} y^{2} + 24 x^{2} z^{2} + 8 x y z^{2} + y^{4} - 4 y^{2} z^{2} + 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:
| $\displaystyle X$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle 2y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle 2w$ |
Maps to other modular curves
$j$-invariant map
of degree 64 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{356896xyz^{13}w-3088000xyz^{11}w^{3}-22881216xyz^{9}w^{5}+174947584xyz^{7}w^{7}-213786240xyz^{5}w^{9}-683611648xyz^{3}w^{11}-52733184xyzw^{13}+190408xz^{14}w+1855072xz^{12}w^{3}-42234768xz^{10}w^{5}+179132224xz^{8}w^{7}-700022816xz^{6}w^{9}+1842040448xz^{4}w^{11}+534199872xz^{2}w^{13}+7533312xw^{15}-87488yz^{15}+2771696yz^{13}w^{2}+18018048yz^{11}w^{4}-212477536yz^{9}w^{6}+638201600yz^{7}w^{8}-821742016yz^{5}w^{10}-174444544yz^{3}w^{12}+20236672yzw^{14}-61791z^{16}+1633552z^{14}w^{2}+17906488z^{12}w^{4}-168929440z^{10}w^{6}+566970904z^{8}w^{8}-866638912z^{6}w^{10}-1262238496z^{4}w^{12}-283015552z^{2}w^{14}-4412912w^{16}}{z^{8}(160xyz^{5}w-480xyz^{3}w^{3}+944xyzw^{5}+48xz^{6}w+688xz^{4}w^{3}-1960xz^{2}w^{5}-408xw^{7}-96yz^{7}+272yz^{5}w^{2}+592yz^{3}w^{4}+24yzw^{6}-68z^{8}+48z^{6}w^{2}+732z^{4}w^{4}+1112z^{2}w^{6}+239w^{8})}$ |
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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.
