Canonical model in $\mathbb{P}^{ 8 }$ defined by 21 equations
| $ 0 $ | $=$ | $ w^{2} - r s $ |
| $=$ | $z r - w u - t r$ |
| $=$ | $z w - w t - u s$ |
| $=$ | $y t - z w + w v$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 2401 x^{14} - 4459 x^{12} z^{2} - 1029 x^{10} y^{2} z^{2} + 882 x^{10} z^{4} - 931 x^{8} y^{2} z^{4} + \cdots + 64 z^{14} $ |
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This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:0:0:-1:1:1:0:1:1)$, $(0:0:0:1:-1:1:0:1:1)$, $(0:0:0:-1:-1:-1:0:1:1)$, $(0:0:0:1:1:-1:0:1:1)$ |
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
28.96.4.c.3
:
| $\displaystyle X$ |
$=$ |
$\displaystyle -x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle -y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle -w$ |
| $\displaystyle W$ |
$=$ |
$\displaystyle -u$ |
Equation of the image curve:
| $0$ |
$=$ |
$ X^{2}+Y^{2}-4YZ+4Z^{2}-4W^{2} $ |
|
$=$ |
$ X^{2}Z-Y^{2}Z+YZ^{2}-2Z^{3}+2XYW-4XZW-3YW^{2}+2ZW^{2} $ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
56.192.9.s.3
:
| $\displaystyle X$ |
$=$ |
$\displaystyle w$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle v$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle s$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 2401X^{14}-4459X^{12}Z^{2}-1029X^{10}Y^{2}Z^{2}+882X^{10}Z^{4}-931X^{8}Y^{2}Z^{4}+147X^{6}Y^{4}Z^{4}+2898X^{8}Z^{6}-1239X^{6}Y^{2}Z^{6}+161X^{4}Y^{4}Z^{6}-7X^{2}Y^{6}Z^{6}-2883X^{6}Z^{8}+939X^{4}Y^{2}Z^{8}-94X^{2}Y^{4}Z^{8}+Y^{6}Z^{8}+1497X^{4}Z^{10}-380X^{2}Y^{2}Z^{10}+12Y^{4}Z^{10}-400X^{2}Z^{12}+48Y^{2}Z^{12}+64Z^{14} $ |
This modular curve minimally covers the modular curves listed below.
