Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ - y u + z t $ |
| $=$ | $ - x u + y z$ |
| $=$ | $ - x t + y^{2}$ |
| $=$ | $x u + 4 y w + t u - t v$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{8} z^{2} + 8 x^{6} y^{2} z^{2} - 2 x^{6} z^{4} + 2 x^{4} y^{4} z^{2} + 4 x^{4} y^{2} z^{4} + \cdots + 5 y^{8} z^{2} $ |
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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 |
|---|
| $(0:0:1:-1/4:0:1:1)$, $(0:0:0:-1/4:0:0:1)$, $(0:0:0:1/4:0:0:1)$, $(0:0:-1:1/4:0:1:1)$ |
Maps to other modular curves
$j$-invariant map
of degree 144 from the canonical model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -\frac{(16w^{4}-16w^{2}v^{2}+v^{4})^{3}}{v^{2}w^{8}(4w-v)(4w+v)}$ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
40.144.7.fy.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle 2w$ |
Equation of the image curve:
| $0$ |
$=$ |
$ X^{8}Z^{2}+8X^{6}Y^{2}Z^{2}-2X^{6}Z^{4}+2X^{4}Y^{4}Z^{2}+4X^{4}Y^{2}Z^{4}-16X^{2}Y^{6}Z^{2}-2X^{2}Y^{4}Z^{4}+40Y^{10}+5Y^{8}Z^{2} $ |
This modular curve minimally covers the modular curves listed below.
