Canonical model in $\mathbb{P}^{ 7 }$ defined by 15 equations
| $ 0 $ | $=$ | $ y w - y t + z^{2} - z v - u^{2} + u v $ |
| $=$ | $2 x t - y w - z^{2} + z u + z r - u r$ |
| $=$ | $x^{2} - x t + y w + 2 t^{2}$ |
| $=$ | $x w + x t - y w + z v + w t - t^{2} - u v$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 324 x^{14} + 216 x^{13} z - 72 x^{12} y^{2} - 5508 x^{12} z^{2} + 408 x^{11} y^{2} z - 5664 x^{11} z^{3} + \cdots + 179776 z^{14} $ |
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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
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle -y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle t$ |
| $\displaystyle W$ |
$=$ |
$\displaystyle -w$ |
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.da.2
:
| $\displaystyle X$ |
$=$ |
$\displaystyle w$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle 2r$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle t$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 324X^{14}-72X^{12}Y^{2}-4X^{10}Y^{4}+216X^{13}Z+408X^{11}Y^{2}Z-84X^{9}Y^{4}Z-4X^{7}Y^{6}Z-5508X^{12}Z^{2}+580X^{10}Y^{2}Z^{2}+236X^{8}Y^{4}Z^{2}-4X^{6}Y^{6}Z^{2}-5664X^{11}Z^{3}-3488X^{9}Y^{2}Z^{3}+412X^{7}Y^{4}Z^{3}+4X^{5}Y^{6}Z^{3}+38140X^{10}Z^{4}-1583X^{8}Y^{2}Z^{4}-634X^{6}Y^{4}Z^{4}+3X^{4}Y^{6}Z^{4}+52928X^{9}Z^{5}+9532X^{7}Y^{2}Z^{5}-1350X^{5}Y^{4}Z^{5}-35X^{3}Y^{6}Z^{5}-XY^{8}Z^{5}-131356X^{8}Z^{6}-2486X^{6}Y^{2}Z^{6}+455X^{4}Y^{4}Z^{6}+90X^{2}Y^{6}Z^{6}+Y^{8}Z^{6}-240880X^{7}Z^{7}-4512X^{5}Y^{2}Z^{7}+2688X^{3}Y^{4}Z^{7}-2XY^{6}Z^{7}+210352X^{6}Z^{8}+25845X^{4}Y^{2}Z^{8}-535X^{2}Y^{4}Z^{8}-52Y^{6}Z^{8}+573528X^{5}Z^{9}-15280X^{3}Y^{2}Z^{9}-2020XY^{4}Z^{9}-58028X^{4}Z^{10}-53800X^{2}Y^{2}Z^{10}+852Y^{4}Z^{10}-679048X^{3}Z^{11}+12296XY^{2}Z^{11}-217564X^{2}Z^{12}+31264Y^{2}Z^{12}+313760XZ^{13}+179776Z^{14} $ |
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.
