Canonical model in $\mathbb{P}^{ 7 }$ defined by 15 equations
| $ 0 $ | $=$ | $ x r + y v + 2 t u $ |
| $=$ | $x w - 2 y u - z v$ |
| $=$ | $x w - x u + y u + z v - t v - t r$ |
| $=$ | $2 x v - y v - y r - w t$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 1024 x^{10} - 1152 x^{8} y^{2} + 576 x^{8} z^{2} + 260 x^{6} y^{4} - 676 x^{6} y^{2} z^{2} + \cdots + 2 y^{4} z^{6} $ |
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This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
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
40.60.4.w.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle -x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle v$ |
| $\displaystyle W$ |
$=$ |
$\displaystyle -r$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 14X^{2}+2Y^{2}+Z^{2}-W^{2} $ |
|
$=$ |
$ 2X^{3}-2XY^{2}-XZ^{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.bk.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x+y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 1024X^{10}-1152X^{8}Y^{2}+576X^{8}Z^{2}+260X^{6}Y^{4}-676X^{6}Y^{2}Z^{2}+209X^{6}Z^{4}-140X^{4}Y^{6}+98X^{4}Y^{4}Z^{2}-86X^{4}Y^{2}Z^{4}+36X^{4}Z^{6}+12X^{2}Y^{8}-8X^{2}Y^{6}Z^{2}+13X^{2}Y^{4}Z^{4}-12X^{2}Y^{2}Z^{6}+4X^{2}Z^{8}-4Y^{10}+10Y^{8}Z^{2}-8Y^{6}Z^{4}+2Y^{4}Z^{6} $ |
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.
