Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ x^{2} + x y + x z + 2 x w - y^{2} - 2 y z - y w - z^{2} - z w - w^{2} $ |
| $=$ | $5 x^{2} + y^{2} + 3 y z - y t + z^{2} + z t$ |
| $=$ | $2 x^{2} + 2 x y + 2 x z + 4 x w + y^{2} + 3 y w - y t + z^{2} + 3 z w + z t + 3 w^{2} + 3 t^{2}$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 245025 x^{8} + 133650 x^{7} y + 33075 x^{6} y^{2} + 9315 x^{6} z^{2} + 4050 x^{5} y^{3} + 5940 x^{5} y z^{2} + \cdots + z^{8} $ |
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This modular curve has no real points, and therefore no rational points.
Maps to other modular curves
Map
of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve
20.48.3.i.2
:
| $\displaystyle X$ |
$=$ |
$\displaystyle -2y-3z-3t$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle -y-4z+t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle 2y-2z-2t$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 6X^{3}Y-22X^{2}Y^{2}+6XY^{3}+12X^{2}YZ+14XY^{2}Z-6Y^{3}Z-3X^{2}Z^{2}-2XYZ^{2}+5Y^{2}Z^{2}-10YZ^{3}+2Z^{4} $ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
60.96.5.x.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle 3w$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle 3t$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 245025X^{8}+133650X^{7}Y+33075X^{6}Y^{2}+4050X^{5}Y^{3}+225X^{4}Y^{4}+9315X^{6}Z^{2}+5940X^{5}YZ^{2}+1755X^{4}Y^{2}Z^{2}+360X^{3}Y^{3}Z^{2}+30X^{2}Y^{4}Z^{2}+1296X^{4}Z^{4}+414X^{3}YZ^{4}+33X^{2}Y^{2}Z^{4}+6XY^{3}Z^{4}+Y^{4}Z^{4}+27X^{2}Z^{6}+12XYZ^{6}+Y^{2}Z^{6}+Z^{8} $ |
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.
