Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ 2 x y + x t - 2 y^{2} + 3 y z - y w + y t - 2 w t + t^{2} $ |
| $=$ | $2 x^{2} - 2 x y - 3 x z + x w - x t + y w - w^{2} + 2 w t$ |
| $=$ | $2 x^{2} - 3 x y - 2 x w + 3 y z + 3 y t - 6 z^{2} + 2 w^{2} + 3 t^{2}$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 8 x^{8} + 20 x^{7} y + 18 x^{7} z + 40 x^{6} y^{2} - 165 x^{6} y z + 69 x^{6} z^{2} - 120 x^{5} y^{3} + \cdots + 27 z^{8} $ |
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This modular curve has no $\Q_p$ points for $p=13$, 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
10.60.3.a.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x+y-3z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle 3x-2y+z$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle 2x+2y-z$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 2X^{4}-3X^{3}Y-5X^{2}Y^{2}-4XY^{3}-2Y^{4}+3X^{3}Z-18X^{2}YZ-17XY^{2}Z+4Y^{3}Z-5X^{2}Z^{2}+17XYZ^{2}-6Y^{2}Z^{2}+4XZ^{3}+4YZ^{3}-2Z^{4} $ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
30.120.5.b.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Equation of the image curve:
| $0$ |
$=$ |
$ -8X^{8}+20X^{7}Y+18X^{7}Z+40X^{6}Y^{2}-165X^{6}YZ+69X^{6}Z^{2}-120X^{5}Y^{3}+585X^{5}Y^{2}Z-150X^{5}YZ^{2}-136X^{5}Z^{3}+60X^{4}Y^{4}-840X^{4}Y^{3}Z-210X^{4}Y^{2}Z^{2}+550X^{4}YZ^{3}+420X^{3}Y^{4}Z+720X^{3}Y^{3}Z^{2}-670X^{3}Y^{2}Z^{3}-75X^{3}YZ^{4}+6X^{3}Z^{5}-360X^{2}Y^{4}Z^{2}+240X^{2}Y^{3}Z^{3}+135X^{2}Y^{2}Z^{4}-240X^{2}YZ^{5}+124X^{2}Z^{6}-120XY^{4}Z^{3}-120XY^{3}Z^{4}+240XY^{2}Z^{5}+80XYZ^{6}-108XZ^{7}+60Y^{4}Z^{4}-80Y^{2}Z^{6}+27Z^{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.
