Canonical model in $\mathbb{P}^{ 10 }$ defined by 36 equations
| $ 0 $ | $=$ | $ u b + v^{2} $ |
| $=$ | $x b - z v$ |
| $=$ | $x v + z u$ |
| $=$ | $y b - w v + t v$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 112 x^{13} y^{2} z^{3} + 3136 x^{12} y^{4} z^{2} - 1960 x^{12} y^{2} z^{4} - 16 x^{12} z^{6} + \cdots - 246016 z^{18} $ |
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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 |
|---|
| $(0:0:0:0:0:-1:-1:-2:-2:1:1)$, $(0:0:0:0:0:-1:1:0:0:-3:1)$, $(0:0:0:0:0:-1:1:0:-4:1:1)$, $(0:0:0:0:0:-1:-1:0:0:1:1)$ |
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
$X_0(56)$
:
| $\displaystyle X$ |
$=$ |
$\displaystyle -x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle -y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle W$ |
$=$ |
$\displaystyle -w$ |
| $\displaystyle T$ |
$=$ |
$\displaystyle t$ |
Equation of the image curve:
| $0$ |
$=$ |
$ YZ+XW+XT $ |
|
$=$ |
$ X^{2}+Y^{2}-XZ-YW-2W^{2}-WT $ |
|
$=$ |
$ 2X^{2}-Y^{2}+XZ+Z^{2}-YW-YT $ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
56.192.11.fc.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle \frac{1}{2}s+b$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{4}t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{4}r-\frac{1}{4}s$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 4302592X^{8}Y^{10}-489419840X^{6}Y^{12}-82802307392X^{4}Y^{14}-5442710038528X^{2}Y^{16}+42313823813632Y^{18}+21952X^{11}Y^{6}Z-960400X^{9}Y^{8}Z-608010032X^{7}Y^{10}Z-61783607648X^{5}Y^{12}Z-1691043431168X^{3}Y^{14}Z+41747120816128XY^{16}Z+3136X^{12}Y^{4}Z^{2}-367696X^{10}Y^{6}Z^{2}-103684784X^{8}Y^{8}Z^{2}-5064150784X^{6}Y^{10}Z^{2}+369225856832X^{4}Y^{12}Z^{2}+27272792581888X^{2}Y^{14}Z^{2}-152632007327744Y^{16}Z^{2}+112X^{13}Y^{2}Z^{3}-66836X^{11}Y^{4}Z^{3}-3510948X^{9}Y^{6}Z^{3}+1036790216X^{7}Y^{8}Z^{3}+153239234288X^{5}Y^{10}Z^{3}+3308086582528X^{3}Y^{12}Z^{3}-115911002382336XY^{14}Z^{3}-1960X^{12}Y^{2}Z^{4}+197568X^{10}Y^{4}Z^{4}+95233264X^{8}Y^{6}Z^{4}+4056691184X^{6}Y^{8}Z^{4}-730690644432X^{4}Y^{10}Z^{4}-35152376710528X^{2}Y^{12}Z^{4}+202650293321728Y^{14}Z^{4}+17367X^{11}Y^{2}Z^{5}+2481654X^{9}Y^{4}Z^{5}-496862940X^{7}Y^{6}Z^{5}-88261912480X^{5}Y^{8}Z^{5}-694468466944X^{3}Y^{10}Z^{5}+107238482817536XY^{12}Z^{5}-16X^{12}Z^{6}-86821X^{10}Y^{2}Z^{6}-24851820X^{8}Y^{4}Z^{6}-750075144X^{6}Y^{6}Z^{6}+384016093664X^{4}Y^{8}Z^{6}+14227473472128X^{2}Y^{10}Z^{6}-122253401618432Y^{12}Z^{6}+232X^{11}Z^{7}+238000X^{9}Y^{2}Z^{7}+93331182X^{7}Y^{4}Z^{7}+16709955696X^{5}Y^{6}Z^{7}-389107058816X^{3}Y^{8}Z^{7}-38342660475392XY^{10}Z^{7}-2073X^{10}Z^{8}-350952X^{8}Y^{2}Z^{8}-242208372X^{6}Y^{4}Z^{8}-67149594316X^{4}Y^{6}Z^{8}-1686162755200X^{2}Y^{8}Z^{8}+34405543547648Y^{10}Z^{8}+10092X^{9}Z^{9}+482006X^{7}Y^{2}Z^{9}+735135534X^{5}Y^{4}Z^{9}+107984044784X^{3}Y^{6}Z^{9}+5340072013696XY^{8}Z^{9}-28990X^{8}Z^{10}-7254198X^{6}Y^{2}Z^{10}-2222165484X^{4}Y^{4}Z^{10}+16609843600X^{2}Y^{6}Z^{10}-4571521210368Y^{8}Z^{10}+18436X^{7}Z^{11}+40348357X^{5}Y^{2}Z^{11}+5272797488X^{3}Y^{4}Z^{11}-291190053056XY^{6}Z^{11}+121031X^{6}Z^{12}-99642011X^{4}Y^{2}Z^{12}-10419385480X^{2}Y^{4}Z^{12}+282549109248Y^{6}Z^{12}-219200X^{5}Z^{13}+121338000X^{3}Y^{2}Z^{13}+15432635456XY^{4}Z^{13}-179992X^{4}Z^{14}-25151224X^{2}Y^{2}Z^{14}-10902140928Y^{4}Z^{14}+485664X^{3}Z^{15}-142113104XY^{2}Z^{15}+198768X^{2}Z^{16}+152797904Y^{2}Z^{16}-432512XZ^{17}-246016Z^{18} $ |
This modular curve minimally covers the modular curves listed below.
