Canonical model in $\mathbb{P}^{ 7 }$ defined by 15 equations
| $ 0 $ | $=$ | $ x^{2} - t^{2} - 2 u v + v^{2} $ |
| $=$ | $2 x y - x r + z v - t v - t r$ |
| $=$ | $x y - y t + 2 z v + w v$ |
| $=$ | $x u - x r - z u + z v - 2 t u$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 16 x^{10} - 8 x^{8} z^{2} - 16 x^{6} y^{4} + 32 x^{6} y^{2} z^{2} - 9 x^{6} z^{4} - 13 x^{4} y^{4} z^{2} + \cdots + 2 y^{2} z^{8} $ |
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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 |
|---|
| $(0:0:0:0:-1:0:-1:1)$, $(0:0:0:0:0:1:0:0)$, $(0:0:0:0:0:1/2:1:1)$, $(0:0:0:0:1:0:-1:1)$ |
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
20.60.4.h.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle u-v$ |
| $\displaystyle W$ |
$=$ |
$\displaystyle v-r$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 7X^{2}+Y^{2}+2ZW+W^{2} $ |
|
$=$ |
$ X^{3}-XY^{2}+XZ^{2}+YZ^{2}+YZW $ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
40.120.8.bj.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{2}y+\frac{1}{2}w$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle z$ |
Equation of the image curve:
| $0$ |
$=$ |
$ -16X^{10}-8X^{8}Z^{2}-16X^{6}Y^{4}+32X^{6}Y^{2}Z^{2}-9X^{6}Z^{4}-13X^{4}Y^{4}Z^{2}+26X^{4}Y^{2}Z^{4}-2X^{4}Z^{6}-4X^{2}Y^{8}+16X^{2}Y^{6}Z^{2}-26X^{2}Y^{4}Z^{4}+20X^{2}Y^{2}Z^{6}-X^{2}Z^{8}-Y^{4}Z^{6}+2Y^{2}Z^{8} $ |
The following modular covers realize this modular curve as a fiber product over $X(1)$.
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.
