Canonical model in $\mathbb{P}^{ 10 }$ defined by 36 equations
| $ 0 $ | $=$ | $ y r - t s $ |
| $=$ | $x w - y u$ |
| $=$ | $x y + x w + y r + z a$ |
| $=$ | $x t + x b + z u$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 64 x^{6} y^{6} z^{4} + 10976 x^{4} y^{12} + 1568 x^{4} y^{10} z^{2} - 56 x^{4} y^{8} z^{4} + \cdots + 36 y^{6} z^{10} $ |
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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 |
|---|
| $(0:0:0:0:0:0:1:1:-1:1:0)$, $(0:0:0:0:0:0:-1:-1:-1:1:0)$, $(-1/3:0:0:0:0:1/3:0:-1:1/3:1:0)$, $(1/3:0:0:0:0:-1/3:0:1:1/3:1:0)$ |
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
56.96.4.g.3
:
| $\displaystyle X$ |
$=$ |
$\displaystyle -y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle w$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle s$ |
| $\displaystyle W$ |
$=$ |
$\displaystyle -a$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 2X^{2}-4XY+2Y^{2}+3Z^{2}-2ZW-W^{2} $ |
|
$=$ |
$ 4X^{3}+2X^{2}Y+2XY^{2}-2XZ^{2}-YZ^{2}+2YZW+XW^{2} $ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
56.192.11.fj.4
:
| $\displaystyle X$ |
$=$ |
$\displaystyle b$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle 2x$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle 4y$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 64X^{6}Y^{6}Z^{4}+10976X^{4}Y^{12}+1568X^{4}Y^{10}Z^{2}-56X^{4}Y^{8}Z^{4}+72X^{4}Y^{6}Z^{6}-58X^{4}Y^{4}Z^{8}+4X^{4}Y^{2}Z^{10}-2X^{4}Z^{12}-38416X^{2}Y^{14}+5488X^{2}Y^{12}Z^{2}-9212X^{2}Y^{10}Z^{4}-420X^{2}Y^{8}Z^{6}-453X^{2}Y^{6}Z^{8}-22X^{2}Y^{4}Z^{10}-9X^{2}Y^{2}Z^{12}+38416Y^{14}Z^{2}-5488Y^{12}Z^{4}+2548Y^{10}Z^{6}-168Y^{8}Z^{8}+36Y^{6}Z^{10} $ |
This modular curve minimally covers the modular curves listed below.
