Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ 8 x z - 4 y z + 6 z^{2} - 2 w t - t^{2} $ |
| $=$ | $8 x^{2} - 2 y^{2} + 6 y z - 2 z^{2} - w^{2} + w t - t^{2}$ |
| $=$ | $8 x y + 6 y^{2} - 4 y z - w^{2} - 2 w t$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 100 x^{4} y^{4} - 100 x^{4} y^{2} z^{2} + 5 x^{4} z^{4} - 3600 x^{2} y^{6} + 2800 x^{2} y^{4} z^{2} + \cdots + z^{8} $ |
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This modular curve has no $\Q_p$ points for $p=17$, 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
:
| $\displaystyle X$ |
$=$ |
$\displaystyle -3x-y-z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle x-3y+2z$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle -x-2y+3z$ |
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
40.120.5.c.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x+z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{4}y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{4}w$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 100X^{4}Y^{4}-100X^{4}Y^{2}Z^{2}+5X^{4}Z^{4}-3600X^{2}Y^{6}+2800X^{2}Y^{4}Z^{2}-260X^{2}Y^{2}Z^{4}+32400Y^{8}-36000Y^{6}Z^{2}+10360Y^{4}Z^{4}-200Y^{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.
