Canonical model in $\mathbb{P}^{ 12 }$ defined by 55 equations
| $ 0 $ | $=$ | $ r c + r d - s d + a b - a d $ |
| $=$ | $r b - r d + s c - s d + a b + a c$ |
| $=$ | $x d - z b + z c - w c - t b - v a$ |
| $=$ | $z b - z d - w c - w d - u a - v r - v s + v a$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 64 x^{16} y^{8} - 672 x^{16} y^{6} z^{2} + 7840 x^{16} y^{4} z^{4} + 57624 x^{16} y^{2} z^{6} + \cdots + 7909306972 y^{4} z^{20} $ |
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This modular curve has no real points and no $\Q_p$ points for $p=5,29,37$, and therefore no rational points.
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
28.126.7.b.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle x-y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle W$ |
$=$ |
$\displaystyle w$ |
| $\displaystyle T$ |
$=$ |
$\displaystyle -t$ |
| $\displaystyle U$ |
$=$ |
$\displaystyle u$ |
| $\displaystyle V$ |
$=$ |
$\displaystyle v$ |
Equation of the image curve:
| $0$ |
$=$ |
$ XU-2ZU-YV+WV $ |
|
$=$ |
$ YU+2ZU-WU+3TU+XV-YV+2ZV+WV-TV $ |
|
$=$ |
$ 2XU+2ZU-WU+TU-XV+YV+4WV+2TV $ |
|
$=$ |
$ X^{2}-2XY-XW+4ZW-XT-YT+2ZT-WT-T^{2} $ |
|
$=$ |
$ X^{2}-XW+YW+2ZW+W^{2}+XT-YT-WT-3T^{2}+U^{2}-2UV-V^{2} $ |
|
$=$ |
$ X^{2}-XY+2XZ-YW-2ZW+W^{2}-YT-3WT-2T^{2} $ |
|
$=$ |
$ X^{2}-2XY+XZ-YZ+2Z^{2}+YW+ZW-XT-YT-3ZT+3WT-T^{2}+U^{2}-2UV-V^{2} $ |
|
$=$ |
$ X^{2}-2XY+Y^{2}-XZ+3YZ-2Z^{2}+XW-ZW-2XT+3YT-ZT+3WT+T^{2}+U^{2}-2UV-V^{2} $ |
|
$=$ |
$ X^{2}-3XY-XZ+YZ-2Z^{2}+2XW-3YW-ZW-W^{2}-2XT-YT-3ZT+WT-U^{2}+UV-V^{2} $ |
|
$=$ |
$ X^{2}-XY+Y^{2}+XZ-YZ-2Z^{2}-2XW+3YW-3ZW+2W^{2}-XT+3YT-ZT-2WT+U^{2}+2UV $ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
56.252.13.bb.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle a$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{2}d$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle v$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 64X^{16}Y^{8}+1280X^{14}Y^{10}+6528X^{12}Y^{12}+1280X^{10}Y^{14}+64X^{8}Y^{16}-672X^{16}Y^{6}Z^{2}-8032X^{14}Y^{8}Z^{2}-15424X^{12}Y^{10}Z^{2}+162624X^{10}Y^{12}Z^{2}-134432X^{8}Y^{14}Z^{2}+32X^{6}Y^{16}Z^{2}+7840X^{16}Y^{4}Z^{4}+187040X^{14}Y^{6}Z^{4}+2229056X^{12}Y^{8}Z^{4}+10573248X^{10}Y^{10}Z^{4}+18242336X^{8}Y^{12}Z^{4}-67424X^{6}Y^{14}Z^{4}+57624X^{16}Y^{2}Z^{6}+2267720X^{14}Y^{4}Z^{6}+29054704X^{12}Y^{6}Z^{6}+176148560X^{10}Y^{8}Z^{6}+511819896X^{8}Y^{10}Z^{6}+458387944X^{6}Y^{12}Z^{6}+166012X^{16}Z^{8}+9428384X^{14}Y^{2}Z^{8}+142348920X^{12}Y^{4}Z^{8}+990962672X^{10}Y^{6}Z^{8}+3040518012X^{8}Y^{8}Z^{8}+2932715856X^{6}Y^{10}Z^{8}+224772016X^{4}Y^{12}Z^{8}+15454208X^{14}Z^{10}+347637360X^{12}Y^{2}Z^{10}+2666180160X^{10}Y^{4}Z^{10}+7573637568X^{8}Y^{6}Z^{10}+7755959904X^{6}Y^{8}Z^{10}+2110690288X^{4}Y^{10}Z^{10}+312333056X^{12}Z^{12}+2233429408X^{10}Y^{2}Z^{12}+6012200040X^{8}Y^{4}Z^{12}+18648778288X^{6}Y^{6}Z^{12}+32421240824X^{4}Y^{8}Z^{12}-2166691212X^{10}Z^{14}-11852430856X^{8}Y^{2}Z^{14}-17053928444X^{6}Y^{4}Z^{14}-16246856304X^{4}Y^{6}Z^{14}+12267496528X^{2}Y^{8}Z^{14}+5059848192X^{8}Z^{16}+20390924680X^{6}Y^{2}Z^{16}+21422000516X^{4}Y^{4}Z^{16}+25180650768X^{2}Y^{6}Z^{16}-5165261696X^{6}Z^{18}-18804780862X^{4}Y^{2}Z^{18}-12993861454X^{2}Y^{4}Z^{18}+1977326743X^{4}Z^{20}+7909306972X^{2}Y^{2}Z^{20}+7909306972Y^{4}Z^{20} $ |
This modular curve minimally covers the modular curves listed below.
