Canonical model in $\mathbb{P}^{ 7 }$ defined by 15 equations
| $ 0 $ | $=$ | $ x v + x r - z^{2} + 2 z r - w t + w u - w v $ |
| $=$ | $x w - x t - x u + z t - z u - z r + w t - w u$ |
| $=$ | $x t - x u + x v + y z + y t - y u - y r + z t$ |
| $=$ | $x z + x u - x v - x r - y w + 2 y u$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 9025 x^{13} - 53789 x^{12} y - 150233 x^{11} y^{2} - 10108 x^{11} y z - 262916 x^{10} y^{3} + \cdots + 16 y^{5} z^{8} $ |
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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
$X_0(38)$
:
| $\displaystyle X$ |
$=$ |
$\displaystyle -z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
| $\displaystyle W$ |
$=$ |
$\displaystyle x+y+z$ |
Equation of the image curve:
| $0$ |
$=$ |
$ X^{2}-XY+2XZ-YZ+2XW-YW+ZW $ |
|
$=$ |
$ Y^{3}-X^{2}Z+XZ^{2}+X^{2}W-XYW+YZW+2XW^{2}+YW^{2}-ZW^{2} $ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
38.120.8.e.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle t$ |
Equation of the image curve:
| $0$ |
$=$ |
$ -9025X^{13}-53789X^{12}Y-150233X^{11}Y^{2}-10108X^{11}YZ-262916X^{10}Y^{3}-58995X^{10}Y^{2}Z-3249X^{10}YZ^{2}-339952X^{9}Y^{4}-160047X^{9}Y^{3}Z-18962X^{9}Y^{2}Z^{2}-343186X^{8}Y^{5}-263953X^{8}Y^{4}Z-53056X^{8}Y^{3}Z^{2}-722X^{8}Y^{2}Z^{3}-263481X^{7}Y^{6}-302647X^{7}Y^{5}Z-94505X^{7}Y^{4}Z^{2}-3479X^{7}Y^{3}Z^{3}-361X^{7}Y^{2}Z^{4}-148351X^{6}Y^{7}-249471X^{6}Y^{6}Z-128616X^{6}Y^{5}Z^{2}-9126X^{6}Y^{4}Z^{3}-246X^{6}Y^{3}Z^{4}-61335X^{5}Y^{8}-155889X^{5}Y^{7}Z-132817X^{5}Y^{6}Z^{2}-12701X^{5}Y^{5}Z^{3}+2333X^{5}Y^{4}Z^{4}-8230X^{4}Y^{9}-73604X^{4}Y^{8}Z-97959X^{4}Y^{7}Z^{2}-16692X^{4}Y^{6}Z^{3}+7592X^{4}Y^{5}Z^{4}+312X^{4}Y^{4}Z^{5}-1560X^{3}Y^{10}-25724X^{3}Y^{9}Z-49785X^{3}Y^{8}Z^{2}-13263X^{3}Y^{7}Z^{3}+9439X^{3}Y^{6}Z^{4}+652X^{3}Y^{5}Z^{5}-8X^{3}Y^{4}Z^{6}+102X^{2}Y^{11}-3292X^{2}Y^{10}Z-17508X^{2}Y^{9}Z^{2}-5910X^{2}Y^{8}Z^{3}+5816X^{2}Y^{7}Z^{4}+1452X^{2}Y^{6}Z^{5}-372X^{2}Y^{5}Z^{6}+204XY^{11}Z-2108XY^{10}Z^{2}-1802XY^{9}Z^{3}+1630XY^{8}Z^{4}+972XY^{7}Z^{5}-364XY^{6}Z^{6}+102Y^{11}Z^{2}-376Y^{10}Z^{3}+252Y^{9}Z^{4}+232Y^{8}Z^{5}-76Y^{7}Z^{6}-48Y^{6}Z^{7}+16Y^{5}Z^{8} $ |
This modular curve minimally covers the modular curves listed below.
