Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ 12 x^{2} + 10 x y + 21 x z + x t + 12 y^{2} - 21 y z + y t + 14 z^{2} - t^{2} $ |
| $=$ | $14 x^{2} + 7 x y + 14 x z - 12 y^{2} + 21 y z + y w - 14 z^{2} + w^{2}$ |
| $=$ | $2 x^{2} + 14 x y - 7 x z - x w - x t + 2 y^{2} + 7 y z + y w - 28 z^{2} + 2 w^{2} + 2 w t + t^{2}$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 70756 x^{4} y^{4} - 17052 x^{4} y^{3} z - 16856 x^{4} y^{2} z^{2} + 392 x^{4} y z^{3} + 196 x^{4} z^{4} + \cdots + 5 z^{8} $ |
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This modular curve has no $\Q_p$ points for $p=13,17$, 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
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x-y+3z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle -2x-3y-z$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle -3x-2y+z$ |
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
70.120.5.a.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x+z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{5}y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{5}w$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 70756X^{4}Y^{4}-17052X^{4}Y^{3}Z-16856X^{4}Y^{2}Z^{2}+392X^{4}YZ^{3}+196X^{4}Z^{4}-4033925X^{2}Y^{6}+742350X^{2}Y^{5}Z+726425X^{2}Y^{4}Z^{2}-31850X^{2}Y^{3}Z^{3}-15925X^{2}Y^{2}Z^{4}+55644480Y^{8}-14578320Y^{7}Z-13590115Y^{6}Z^{2}+1972040Y^{5}Z^{3}+975100Y^{4}Z^{4}-13090Y^{3}Z^{5}-4340Y^{2}Z^{6}+20YZ^{7}+5Z^{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.
