Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 3 y^{2} - 2 y z - 2 y w - 3 z^{2} - z w + 2 w^{2} $ |
| $=$ | $15 x^{2} + y^{2} - y z - z^{2} - z w + w^{2}$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 3 x^{4} + 6 x^{3} z - 10 x^{2} y^{2} - 18 x^{2} z^{2} - 85 x y^{2} z - 21 x z^{3} - 75 y^{4} + \cdots + 3 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:
| $\displaystyle X$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle 3x$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Maps to other modular curves
$j$-invariant map
of degree 40 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle 3^3\cdot5^2\,\frac{649976yz^{9}-1347068yz^{8}w-158712yz^{7}w^{2}+1026872yz^{6}w^{3}-1108396yz^{5}w^{4}-89448yz^{4}w^{5}+501088yz^{3}w^{6}-97856yz^{2}w^{7}-33984yzw^{8}+7552yw^{9}-920787z^{10}+1319194z^{9}w+681875z^{8}w^{2}-2597430z^{7}w^{3}+569653z^{6}w^{4}+1256728z^{5}w^{5}-598620z^{4}w^{6}-44704z^{3}w^{7}+97520z^{2}w^{8}-41856zw^{9}+7616w^{10}}{205yz^{9}-360yz^{8}w+225yz^{7}w^{2}-8250yz^{6}w^{3}+34650yz^{5}w^{4}-53550yz^{4}w^{5}+27300yz^{3}w^{6}+9900yz^{2}w^{7}-13275yzw^{8}+2950yw^{9}+132z^{10}+110z^{9}w-3705z^{8}w^{2}+14100z^{7}w^{3}-30825z^{6}w^{4}+39300z^{5}w^{5}-12750z^{4}w^{6}-27150z^{3}w^{7}+34500z^{2}w^{8}-16350zw^{9}+2975w^{10}}$ |
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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.
