Canonical model in $\mathbb{P}^{ 8 }$ defined by 21 equations
| $ 0 $ | $=$ | $ x^{2} - x y + x z - x t - y^{2} - y z - y u + y v - w r + u r $ |
| $=$ | $x^{2} - x y - x z + y u + y s - w u - w s - u v + u r - v s + r s$ |
| $=$ | $x y - x z + x r - x s + y^{2} + y z - y u + y v - w r + u v - u r + u s - v^{2} - 2 r s$ |
| $=$ | $x^{2} + 2 x y + 2 x z + x u - x v + x r + x s + 2 y^{2} + 2 y z - s^{2}$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 1250 x^{13} + 400 x^{12} y + 375 x^{12} z + 520 x^{11} y^{2} + 760 x^{11} y z - 825 x^{11} z^{2} + \cdots - y^{5} z^{8} $ |
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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 |
|---|
| $(0:-1/2:-1/2:1/2:-1:0:-1:0:1)$, $(0:0:0:0:0:1:0:0:0)$, $(0:1/2:1/2:-1/2:0:-1:0:0:1)$, $(0:0:0:0:0:0:0:1:0)$ |
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
$X_0(42)$
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle -y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle -z$ |
| $\displaystyle W$ |
$=$ |
$\displaystyle -w$ |
| $\displaystyle T$ |
$=$ |
$\displaystyle -x-y-z-u+v-r$ |
Equation of the image curve:
| $0$ |
$=$ |
$ X^{2}-XW+XT+YT $ |
|
$=$ |
$ X^{2}+XZ-Z^{2}+2XW-2ZW-W^{2}+YT $ |
|
$=$ |
$ X^{2}+Y^{2}+2YZ+Z^{2}+2YW+2ZW-YT-ZT $ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
42.192.9.c.4
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle r$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle s$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 1250X^{13}+400X^{12}Y+520X^{11}Y^{2}+680X^{10}Y^{3}+346X^{9}Y^{4}+80X^{8}Y^{5}+8X^{7}Y^{6}+375X^{12}Z+760X^{11}YZ-526X^{10}Y^{2}Z-1124X^{9}Y^{3}Z-625X^{8}Y^{4}Z-148X^{7}Y^{5}Z-12X^{6}Y^{6}Z-825X^{11}Z^{2}-934X^{10}YZ^{2}-145X^{9}Y^{2}Z^{2}+848X^{8}Y^{3}Z^{2}+639X^{7}Y^{4}Z^{2}+172X^{6}Y^{5}Z^{2}+18X^{5}Y^{6}Z^{2}+15X^{10}Z^{3}+431X^{9}YZ^{3}+568X^{8}Y^{2}Z^{3}-200X^{7}Y^{3}Z^{3}-332X^{6}Y^{4}Z^{3}-125X^{5}Y^{5}Z^{3}-13X^{4}Y^{6}Z^{3}+282X^{9}Z^{4}+71X^{8}YZ^{4}-397X^{7}Y^{2}Z^{4}-239X^{6}Y^{3}Z^{4}+57X^{5}Y^{4}Z^{4}+51X^{4}Y^{5}Z^{4}+9X^{3}Y^{6}Z^{4}-66X^{8}Z^{5}-299X^{7}YZ^{5}+102X^{6}Y^{2}Z^{5}+215X^{5}Y^{3}Z^{5}+65X^{4}Y^{4}Z^{5}-10X^{3}Y^{5}Z^{5}-3X^{2}Y^{6}Z^{5}-48X^{7}Z^{6}+88X^{6}YZ^{6}+99X^{5}Y^{2}Z^{6}-88X^{4}Y^{3}Z^{6}-45X^{3}Y^{4}Z^{6}-6X^{2}Y^{5}Z^{6}+XY^{6}Z^{6}+15X^{6}Z^{7}+13X^{5}YZ^{7}-51X^{4}Y^{2}Z^{7}+4X^{3}Y^{3}Z^{7}+16X^{2}Y^{4}Z^{7}+3XY^{5}Z^{7}+3X^{5}Z^{8}-13X^{4}YZ^{8}+13X^{3}Y^{2}Z^{8}-X^{2}Y^{3}Z^{8}-XY^{4}Z^{8}-Y^{5}Z^{8}-X^{4}Z^{9}+3X^{3}YZ^{9}-3X^{2}Y^{2}Z^{9}+XY^{3}Z^{9} $ |
This modular curve minimally covers the modular curves listed below.
