Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ x w + x t + y z - y t $ |
| $=$ | $14 y^{2} - z^{2} - z w + z t$ |
| $=$ | $14 x^{2} - 14 x y + z^{2} + z w - z t + w t$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 14 x^{6} y - x^{6} z - 196 x^{4} y^{3} + 5 x^{4} y z^{2} + 2744 x^{2} y^{5} - 98 x^{2} y^{3} z^{2} + \cdots + 4 y^{3} z^{4} $ |
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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 |
|---|
| $(0:0:1:0:1)$, $(0:0:-1:1:0)$, $(0:0:0:0:1)$, $(0:0:0:1:0)$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the canonical model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle 2^2\,\frac{1647072xyw^{10}-1647072xyw^{9}t-28010304xyw^{8}t^{2}-56796768xyw^{7}t^{3}-26286624xyw^{6}t^{4}+26286624xyw^{4}t^{6}+56796768xyw^{3}t^{7}+28010304xyw^{2}t^{8}+1647072xywt^{9}-1647072xyt^{10}+z^{2}w^{10}-235301z^{2}w^{9}t+703z^{2}w^{8}t^{2}-133534z^{2}w^{7}t^{3}+3978473z^{2}w^{6}t^{4}+6502705z^{2}w^{5}t^{5}+2100857z^{2}w^{4}t^{6}-4190446z^{2}w^{3}t^{7}-2000033z^{2}w^{2}t^{8}-352949z^{2}wt^{9}+117649z^{2}t^{10}-117647zw^{11}-6zw^{10}t+2119092zw^{9}t^{2}+1631923zw^{8}t^{3}+2107815zw^{7}t^{4}-1598344zw^{6}t^{5}-279272zw^{5}t^{6}-4287111zw^{4}t^{7}+424253zw^{3}t^{8}-236004zw^{2}t^{9}+235302zwt^{10}-zt^{11}+w^{12}-117653w^{11}t-234594w^{10}t^{2}+1176501w^{9}t^{3}+2507104w^{8}t^{4}+531643w^{7}t^{5}+6165496w^{6}t^{6}+531643w^{5}t^{7}+2507104w^{4}t^{8}+1176501w^{3}t^{9}-234594w^{2}t^{10}-117653wt^{11}+t^{12}}{5040xyw^{10}-1680xyw^{9}t-15176xyw^{8}t^{2}-21336xyw^{7}t^{3}-12600xyw^{6}t^{4}+12600xyw^{4}t^{6}+21336xyw^{3}t^{7}+15176xyw^{2}t^{8}+1680xywt^{9}-5040xyt^{10}-37z^{2}w^{10}-83z^{2}w^{9}t+115z^{2}w^{8}t^{2}-532z^{2}w^{7}t^{3}-638z^{2}w^{6}t^{4}-1242z^{2}w^{5}t^{5}-1538z^{2}w^{4}t^{6}-2056z^{2}w^{3}t^{7}-969z^{2}w^{2}t^{8}-203z^{2}wt^{9}+323z^{2}t^{10}-37zw^{11}-406zw^{10}t-38zw^{9}t^{2}+201zw^{8}t^{3}+1414zw^{7}t^{4}+192zw^{6}t^{5}-1092zw^{5}t^{6}-2038zw^{4}t^{7}+239zw^{3}t^{8}+1002zw^{2}t^{9}+886zwt^{10}-323zt^{11}-37w^{11}t+207w^{10}t^{2}+111w^{9}t^{3}-1612w^{8}t^{4}-4190w^{7}t^{5}-5422w^{6}t^{6}-4190w^{5}t^{7}-1612w^{4}t^{8}+111w^{3}t^{9}+207w^{2}t^{10}-37wt^{11}}$ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
56.96.5.d.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{14}z$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 14X^{6}Y-X^{6}Z-196X^{4}Y^{3}+5X^{4}YZ^{2}+2744X^{2}Y^{5}-98X^{2}Y^{3}Z^{2}-8X^{2}Y^{2}Z^{3}+2744Y^{6}Z+980Y^{5}Z^{2}+112Y^{4}Z^{3}+4Y^{3}Z^{4} $ |
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.
