Canonical model in $\mathbb{P}^{ 7 }$ defined by 15 equations
| $ 0 $ | $=$ | $ x t + x u + x r - y t - y v + y r + w t + w u + w r $ |
| $=$ | $x t - x v + x r + z u - z v - w t + w u + w r$ |
| $=$ | $x t + x v - x r - y t - z t + z v + w u - w v$ |
| $=$ | $x^{2} - x w + y z + 2 w^{2}$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 40000000 x^{14} - 812500 x^{12} y^{2} + 13750000 x^{12} y z - 28250000 x^{12} z^{2} + 12500 x^{10} y^{4} + \cdots + 8 y z^{13} $ |
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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.bl.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
| $\displaystyle W$ |
$=$ |
$\displaystyle z$ |
Equation of the image curve:
| $0$ |
$=$ |
$ X^{2}-XZ+2Z^{2}+YW $ |
|
$=$ |
$ 2X^{2}Z-Y^{2}Z-2Z^{3}+XYW-YZW+2XW^{2}+2ZW^{2} $ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
40.120.8.dc.2
:
| $\displaystyle X$ |
$=$ |
$\displaystyle w$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{2}r$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle t$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 40000000X^{14}-812500X^{12}Y^{2}+12500X^{10}Y^{4}+13750000X^{12}YZ-187500X^{10}Y^{3}Z-28250000X^{12}Z^{2}+1762500X^{10}Y^{2}Z^{2}+17500X^{8}Y^{4}Z^{2}-11275000X^{10}YZ^{3}-175000X^{8}Y^{3}Z^{3}+1500X^{6}Y^{5}Z^{3}+5100000X^{10}Z^{4}-1480625X^{8}Y^{2}Z^{4}-17625X^{6}Y^{4}Z^{4}-450X^{4}Y^{6}Z^{4}+1327500X^{8}YZ^{5}-169125X^{6}Y^{3}Z^{5}+4800X^{4}Y^{5}Z^{5}-60X^{2}Y^{7}Z^{5}-652500X^{8}Z^{6}+315500X^{6}Y^{2}Z^{6}-30375X^{4}Y^{4}Z^{6}+655X^{2}Y^{6}Z^{6}-2Y^{8}Z^{6}-317250X^{6}YZ^{7}+64700X^{4}Y^{3}Z^{7}-2905X^{2}Y^{5}Z^{7}+22Y^{7}Z^{7}+114500X^{6}Z^{8}-71225X^{4}Y^{2}Z^{8}+6155X^{2}Y^{4}Z^{8}-94Y^{6}Z^{8}+35550X^{4}YZ^{9}-6975X^{2}Y^{3}Z^{9}+202Y^{5}Z^{9}-5600X^{4}Z^{10}+4090X^{2}Y^{2}Z^{10}-240Y^{4}Z^{10}-1080X^{2}YZ^{11}+160Y^{3}Z^{11}+80X^{2}Z^{12}-56Y^{2}Z^{12}+8YZ^{13} $ |
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.
