Canonical model in $\mathbb{P}^{ 12 }$ defined by 55 equations
| $ 0 $ | $=$ | $ x b - y b - z a - z b - w a - r b - r c - s d $ |
| $=$ | $x a - y b + z a + z b + z c - w a + r a - r b + r c$ |
| $=$ | $x c + z a - z b - w a - u d - v d - r b - s d$ |
| $=$ | $y b + z b + w a + t s + u d + v d - r a$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 21952 x^{16} y^{8} + 32928 x^{16} y^{6} z^{2} - 54880 x^{16} y^{4} z^{4} - 57624 x^{16} y^{2} z^{6} + \cdots - 470596 z^{24} $ |
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This modular curve has no real points and no $\Q_p$ points for $p=17,29,37$, 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
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle x-y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle W$ |
$=$ |
$\displaystyle w$ |
| $\displaystyle T$ |
$=$ |
$\displaystyle -r$ |
| $\displaystyle U$ |
$=$ |
$\displaystyle t$ |
| $\displaystyle V$ |
$=$ |
$\displaystyle d$ |
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.bk.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle s$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{2}c$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle d$ |
Equation of the image curve:
| $0$ |
$=$ |
$ -21952X^{16}Y^{8}-3951360X^{14}Y^{10}-181367424X^{12}Y^{12}-320060160X^{10}Y^{14}-144027072X^{8}Y^{16}+32928X^{16}Y^{6}Z^{2}+5166560X^{14}Y^{8}Z^{2}+277752384X^{12}Y^{10}Z^{2}-496093248X^{10}Y^{12}Z^{2}-8375288544X^{8}Y^{14}Z^{2}-10174483872X^{6}Y^{16}Z^{2}-54880X^{16}Y^{4}Z^{4}-8007776X^{14}Y^{6}Z^{4}-339491712X^{12}Y^{8}Z^{4}+6864951744X^{10}Y^{10}Z^{4}+28430105760X^{8}Y^{12}Z^{4}-29769024096X^{6}Y^{14}Z^{4}-91169136576X^{4}Y^{16}Z^{4}-57624X^{16}Y^{2}Z^{6}-13264104X^{14}Y^{4}Z^{6}-597841776X^{12}Y^{6}Z^{6}-11352241040X^{10}Y^{8}Z^{6}+11726990856X^{8}Y^{10}Z^{6}+186988766328X^{6}Y^{12}Z^{6}+269170594560X^{4}Y^{14}Z^{6}-110900845440X^{2}Y^{16}Z^{6}-23716X^{16}Z^{8}-6282976X^{14}Y^{2}Z^{8}-243030872X^{12}Y^{4}Z^{8}-2177683536X^{10}Y^{6}Z^{8}-66885420420X^{8}Y^{8}Z^{8}-162707683824X^{6}Y^{10}Z^{8}-197442445536X^{4}Y^{12}Z^{8}+407980685952X^{2}Y^{14}Z^{8}-36294822144Y^{16}Z^{8}-751520X^{14}Z^{10}-4964400X^{12}Y^{2}Z^{10}-674466016X^{10}Y^{4}Z^{10}+14387920960X^{8}Y^{6}Z^{10}-42144020352X^{6}Y^{8}Z^{10}+26932617936X^{4}Y^{10}Z^{10}-525210721056X^{2}Y^{12}Z^{10}+145179288576Y^{14}Z^{10}-7238576X^{12}Z^{12}-14555632X^{10}Y^{2}Z^{12}-3838687048X^{8}Y^{4}Z^{12}+28625440928X^{6}Y^{6}Z^{12}-62936348328X^{4}Y^{8}Z^{12}+311238057312X^{2}Y^{10}Z^{12}-213736174848Y^{12}Z^{12}-21219604X^{10}Z^{14}+100945880X^{8}Y^{2}Z^{14}-7256737124X^{6}Y^{4}Z^{14}+35124692736X^{4}Y^{6}Z^{14}-120062715696X^{2}Y^{8}Z^{14}+147643751808Y^{10}Z^{14}-31246712X^{8}Z^{16}+311006136X^{6}Y^{2}Z^{16}-7143805060X^{4}Y^{4}Z^{16}+30877705152X^{2}Y^{6}Z^{16}-53695426176Y^{8}Z^{16}-26652472X^{6}Z^{18}+348935958X^{4}Y^{2}Z^{18}-4041579290X^{2}Y^{4}Z^{18}+10467745344Y^{6}Z^{18}-13524833X^{4}Z^{20}+179700444X^{2}Y^{2}Z^{20}-1010609712Y^{4}Z^{20}-3831996X^{2}Z^{22}+37916592Y^{2}Z^{22}-470596Z^{24} $ |
This modular curve minimally covers the modular curves listed below.
