Canonical model in $\mathbb{P}^{ 12 }$ defined by 55 equations
| $ 0 $ | $=$ | $ 2 r s - r a + r b - s a - 2 s b + a b + c d $ |
| $=$ | $2 x r + x s + x a + x c + x d + y b - y c + z r + v a + v c$ |
| $=$ | $x r - x s + 2 y s - y b - y d + z r - z s + v r - v s$ |
| $=$ | $x r - 2 x s + x c + z a - z c + t c + u s$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 7381125 x^{8} y^{12} + 14762250 x^{8} y^{10} z^{2} + 10333575 x^{8} y^{8} z^{4} + 2843100 x^{8} y^{6} z^{6} + \cdots + z^{20} $ |
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This modular curve has no real points and no $\Q_p$ points for $p=13$, 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
30.120.7.h.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle -z$ |
| $\displaystyle W$ |
$=$ |
$\displaystyle w$ |
| $\displaystyle T$ |
$=$ |
$\displaystyle t$ |
| $\displaystyle U$ |
$=$ |
$\displaystyle u$ |
| $\displaystyle V$ |
$=$ |
$\displaystyle v$ |
Equation of the image curve:
| $0$ |
$=$ |
$ XY-2Y^{2}+YZ+Z^{2}-XW+YW-YT+YV-WV $ |
|
$=$ |
$ 2X^{2}+XY-2XZ-2Z^{2}+2XW-YW+ZW+W^{2}+XT-YT+WT-XU-YU+ZU+WU-TU-U^{2}-ZV $ |
|
$=$ |
$ X^{2}-3XZ+YZ+Z^{2}+2XW-2ZW-XT-WT-T^{2}-XU+YU-3ZU-2TU+XV+YV-WV-UV $ |
|
$=$ |
$ X^{2}-XY-3YZ+YW-2ZW-YT-ZT-WT-T^{2}-XU-3ZU-2TU+XV-YV+TV-UV $ |
|
$=$ |
$ 2XZ+YZ+2XW-2YW+ZW+2XT-2YT+ZT+WT+T^{2}+YU+2ZU+TU+YV+ZV-WV $ |
|
$=$ |
$ 2XY+2XZ+Z^{2}+2XW-ZW+XT+2ZT-WT+YU-2ZU-TU-WV $ |
|
$=$ |
$ X^{2}-XZ+YZ+XW-2YW+ZW+XT-2ZT-WT-XU+2YU+TU+XV-YV+2ZV+WV-TV-UV $ |
|
$=$ |
$ 2XY-3XZ-2YZ-2Z^{2}-XW-ZW-2XT-2YT-2ZT-T^{2}+YU-ZU-TU-ZV-WV $ |
|
$=$ |
$ 2X^{2}-2XZ+YZ-Z^{2}-XW-XT+2ZT+T^{2}-2XU+2YU-2XV+YV-3ZV-WV-2TV+2UV $ |
|
$=$ |
$ X^{2}-XZ+YZ-2YW-W^{2}+3XT-2YT-ZT+T^{2}+3XU+YU+ZU-WU+2TU+U^{2}-4XV+YV-WV-TV-UV-V^{2} $ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
60.240.13.nn.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle d$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{3}{2}x$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{3}{2}r$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 7381125X^{8}Y^{12}+147622500X^{6}Y^{14}+615093750X^{4}Y^{16}+3143812500X^{2}Y^{18}+512578125Y^{20}-262440000X^{5}Y^{14}Z-2624400000X^{3}Y^{16}Z+2187000000XY^{18}Z+14762250X^{8}Y^{10}Z^{2}+285403500X^{6}Y^{12}Z^{2}+852930000X^{4}Y^{14}Z^{2}+5476612500X^{2}Y^{16}Z^{2}+3348843750Y^{18}Z^{2}-594864000X^{5}Y^{12}Z^{3}-5773680000X^{3}Y^{14}Z^{3}+2916000000XY^{16}Z^{3}+10333575X^{8}Y^{8}Z^{4}+186988500X^{6}Y^{10}Z^{4}+199017000X^{4}Y^{12}Z^{4}+5955930000X^{2}Y^{14}Z^{4}+7255828125Y^{16}Z^{4}-484056000X^{5}Y^{10}Z^{5}-4443984000X^{3}Y^{12}Z^{5}+1788480000XY^{14}Z^{5}+2843100X^{8}Y^{6}Z^{6}+43083900X^{6}Y^{8}Z^{6}-147258000X^{4}Y^{10}Z^{6}+4701402000X^{2}Y^{12}Z^{6}+8688465000Y^{14}Z^{6}-169516800X^{5}Y^{8}Z^{7}-1372464000X^{3}Y^{10}Z^{7}+303264000XY^{12}Z^{7}+229635X^{8}Y^{4}Z^{8}+801900X^{6}Y^{6}Z^{8}-53192700X^{4}Y^{8}Z^{8}+2072223000X^{2}Y^{10}Z^{8}+5900384250Y^{12}Z^{8}-22680000X^{5}Y^{6}Z^{9}-113788800X^{3}Y^{8}Z^{9}-25488000XY^{10}Z^{9}+7290X^{8}Y^{2}Z^{10}-160380X^{6}Y^{4}Z^{10}+1911600X^{4}Y^{6}Z^{10}+388690200X^{2}Y^{8}Z^{10}+2103826500Y^{10}Z^{10}-362880X^{5}Y^{4}Z^{11}+11491200X^{3}Y^{6}Z^{11}+64224000XY^{8}Z^{11}+81X^{8}Z^{12}-8100X^{6}Y^{2}Z^{12}+348840X^{4}Y^{4}Z^{12}+15848400X^{2}Y^{6}Z^{12}+335074050Y^{8}Z^{12}+8640X^{5}Y^{2}Z^{13}+328320X^{3}Y^{4}Z^{13}+26035200XY^{6}Z^{13}-108X^{6}Z^{14}+9360X^{4}Y^{2}Z^{14}+347600X^{2}Y^{4}Z^{14}+13900200Y^{6}Z^{14}-5760X^{3}Y^{2}Z^{15}+480000XY^{4}Z^{15}+54X^{4}Z^{16}-4620X^{2}Y^{2}Z^{16}+176665Y^{4}Z^{16}+960XY^{2}Z^{17}-12X^{2}Z^{18}+710Y^{2}Z^{18}+Z^{20} $ |
This modular curve minimally covers the modular curves listed below.
This modular curve is minimally covered by the modular curves in the database listed below.
