Canonical model in $\mathbb{P}^{ 8 }$ defined by 21 equations
| $ 0 $ | $=$ | $ x z + x t + x r + x s - y v - y r - y s + u v + u r + u s $ |
| $=$ | $x^{2} + x z - x w - x t + x r - x s - z t + t^{2} - t r + t s$ |
| $=$ | $x z + x t + x u - 2 x v + x r - x s - y v - y r - y s - u v + v^{2} + v s$ |
| $=$ | $x^{2} - x y - 2 x w - x u - x r - x s + y v - z v + t v + u v - v^{2}$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 2304 x^{12} + 288 x^{11} y - 5184 x^{11} z + 408 x^{10} y^{2} + 5292 x^{10} y z + 20196 x^{10} z^{2} + \cdots + 63 z^{12} $ |
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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 |
|---|
| $(1:0:1:1:2:-3:-1:2:1)$, $(-1/2:0:-1/2:-1/2:-1:0:-1:1/2:1)$, $(1:0:-1/2:-1/2:1/2:0:-1:-1:1)$, $(1:0:-1/2:-1/2:1/2:0:2:-1:1)$ |
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
$X_0(42)$
:
| $\displaystyle X$ |
$=$ |
$\displaystyle -x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle -z$ |
| $\displaystyle W$ |
$=$ |
$\displaystyle -w$ |
| $\displaystyle T$ |
$=$ |
$\displaystyle -z+t-r+s$ |
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.3
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle 3s$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle v$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 2304X^{12}+288X^{11}Y+408X^{10}Y^{2}+48X^{9}Y^{3}+25X^{8}Y^{4}+2X^{7}Y^{5}+X^{6}Y^{6}-5184X^{11}Z+5292X^{10}YZ-303X^{9}Y^{2}Z+687X^{8}Y^{3}Z-19X^{7}Y^{4}Z+33X^{6}Y^{5}Z-2X^{5}Y^{6}Z+20196X^{10}Z^{2}-10899X^{9}YZ^{2}+7263X^{8}Y^{2}Z^{2}-1398X^{7}Y^{3}Z^{2}+520X^{6}Y^{4}Z^{2}-75X^{5}Y^{5}Z^{2}+3X^{4}Y^{6}Z^{2}-29718X^{9}Z^{3}+37152X^{8}YZ^{3}-13815X^{7}Y^{2}Z^{3}+5295X^{6}Y^{3}Z^{3}-1102X^{5}Y^{4}Z^{3}+103X^{4}Y^{5}Z^{3}-2X^{3}Y^{6}Z^{3}+53703X^{8}Z^{4}-50787X^{7}YZ^{4}+28494X^{6}Y^{2}Z^{4}-8589X^{5}Y^{3}Z^{4}+1321X^{4}Y^{4}Z^{4}-75X^{3}Y^{5}Z^{4}+X^{2}Y^{6}Z^{4}-49248X^{7}Z^{5}+66528X^{6}YZ^{5}-33669X^{5}Y^{2}Z^{5}+8247X^{4}Y^{3}Z^{5}-904X^{3}Y^{4}Z^{5}+33X^{2}Y^{5}Z^{5}+41607X^{6}Z^{6}-52434X^{5}YZ^{6}+24120X^{4}Y^{2}Z^{6}-4755X^{3}Y^{3}Z^{6}+385X^{2}Y^{4}Z^{6}-7XY^{5}Z^{6}-13365X^{5}Z^{7}+21060X^{4}YZ^{7}-9819X^{3}Y^{2}Z^{7}+1740X^{2}Y^{3}Z^{7}-91XY^{4}Z^{7}-9639X^{4}Z^{8}+1485X^{3}YZ^{8}+2079X^{2}Y^{2}Z^{8}-399XY^{3}Z^{8}+7Y^{4}Z^{8}+14643X^{3}Z^{9}-4734X^{2}YZ^{9}-294XY^{2}Z^{9}+42Y^{3}Z^{9}-6642X^{2}Z^{10}+1323XYZ^{10}+21Y^{2}Z^{10}+756XZ^{11}-126YZ^{11}+63Z^{12} $ |
This modular curve minimally covers the modular curves listed below.
