Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ y z - w u + t v $ |
| $=$ | $x z - w v - t u$ |
| $=$ | $x w - y t - z v$ |
| $=$ | $x t + y w - z u$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 16 x^{8} y^{2} - 8 x^{6} y^{4} + x^{4} y^{6} + 4 x^{4} y^{4} z^{2} - 24 x^{4} y^{2} z^{4} + \cdots + y^{2} z^{8} $ |
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This modular curve has 8 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(-1:-1:1:0:1:-1:1)$, $(-1:-1:-1:0:-1:-1:1)$, $(-1:1:-1:0:1:1:1)$, $(-1:1:1:0:-1:1:1)$, $(1:-1:-1:0:-1:1:1)$, $(1:-1:1:0:1:1:1)$, $(1:1:1:0:-1:-1:1)$, $(1:1:-1:0:1:-1:1)$ |
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
24.96.3.bq.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle t+u$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle y+t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle y-t$ |
Equation of the image curve:
| $0$ |
$=$ |
$ X^{3}Y-2X^{2}Y^{2}+XY^{3}+X^{3}Z+X^{2}YZ-3XY^{2}Z+X^{2}Z^{2}+Y^{2}Z^{2}-YZ^{3} $ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
24.192.7.i.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x+y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle 4z$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle 2w$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 16X^{8}Y^{2}-8X^{6}Y^{4}+X^{4}Y^{6}+4X^{4}Y^{4}Z^{2}-24X^{4}Y^{2}Z^{4}+64X^{4}Z^{6}-X^{2}Y^{6}Z^{2}+6X^{2}Y^{4}Z^{4}-16X^{2}Y^{2}Z^{6}+Y^{2}Z^{8} $ |
This modular curve minimally covers the modular curves listed below.
