Canonical model in $\mathbb{P}^{ 7 }$ defined by 15 equations
| $ 0 $ | $=$ | $ x t + x v - x r - z u - w t - w u - w v - w r $ |
| $=$ | $x^{2} - x w + y z + 2 w^{2}$ |
| $=$ | $x t - x v + x r - y t - z t - z v - w u$ |
| $=$ | $2 x^{2} - x z + 2 x w + y w - z w + t^{2} - u v - v^{2}$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 512 x^{14} - 52 x^{12} y^{2} - 880 x^{12} y z - 1808 x^{12} z^{2} + 4 x^{10} y^{4} + 60 x^{10} y^{3} z + \cdots - 8 y z^{13} $ |
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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 |
|---|
| $(0:0:0:0:-1/2:0:1/2:1)$, $(1/3:2/3:-1/3:1/3:-1/3:0:1:0)$, $(-1/3:-2/3:1/3:-1/3:-1/3:0:1:0)$, $(0:0:0:0:1:0:1:0)$ |
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
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.dd.2
:
| $\displaystyle X$ |
$=$ |
$\displaystyle w$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{2}r$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle t$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 512X^{14}-52X^{12}Y^{2}+4X^{10}Y^{4}-880X^{12}YZ+60X^{10}Y^{3}Z-1808X^{12}Z^{2}+564X^{10}Y^{2}Z^{2}+28X^{8}Y^{4}Z^{2}+3608X^{10}YZ^{3}+280X^{8}Y^{3}Z^{3}-12X^{6}Y^{5}Z^{3}+1632X^{10}Z^{4}-2369X^{8}Y^{2}Z^{4}-141X^{6}Y^{4}Z^{4}-18X^{4}Y^{6}Z^{4}-2124X^{8}YZ^{5}+1353X^{6}Y^{3}Z^{5}-192X^{4}Y^{5}Z^{5}+12X^{2}Y^{7}Z^{5}-1044X^{8}Z^{6}+2524X^{6}Y^{2}Z^{6}-1215X^{4}Y^{4}Z^{6}+131X^{2}Y^{6}Z^{6}-2Y^{8}Z^{6}+2538X^{6}YZ^{7}-2588X^{4}Y^{3}Z^{7}+581X^{2}Y^{5}Z^{7}-22Y^{7}Z^{7}+916X^{6}Z^{8}-2849X^{4}Y^{2}Z^{8}+1231X^{2}Y^{4}Z^{8}-94Y^{6}Z^{8}-1422X^{4}YZ^{9}+1395X^{2}Y^{3}Z^{9}-202Y^{5}Z^{9}-224X^{4}Z^{10}+818X^{2}Y^{2}Z^{10}-240Y^{4}Z^{10}+216X^{2}YZ^{11}-160Y^{3}Z^{11}+16X^{2}Z^{12}-56Y^{2}Z^{12}-8YZ^{13} $ |
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.
