Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ x z + x w - y z - y t $ |
| $=$ | $14 x y - z^{2} - z w - z t + w t$ |
| $=$ | $14 x^{2} + 14 y^{2} - z^{2} - z w - z t + 2 w t$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{5} z + 28 x^{4} y^{2} - 2 x^{4} z^{2} - 70 x^{3} y^{2} z + 3 x^{3} z^{3} + 112 x^{2} y^{2} z^{2} + \cdots + 28 y^{2} 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 |
|---|
| $(0:0:-1:1:0)$, $(0:0:-1:0:1)$, $(0:0:0:1:0)$, $(0:0:0:0:1)$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the canonical model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle 2^2\,\frac{1647072y^{2}w^{10}+1647072y^{2}w^{9}t-28010304y^{2}w^{8}t^{2}+56796768y^{2}w^{7}t^{3}-26286624y^{2}w^{6}t^{4}+26286624y^{2}w^{4}t^{6}-56796768y^{2}w^{3}t^{7}+28010304y^{2}w^{2}t^{8}-1647072y^{2}wt^{9}-1647072y^{2}t^{10}+z^{2}w^{10}+235301z^{2}w^{9}t+703z^{2}w^{8}t^{2}+133534z^{2}w^{7}t^{3}+3978473z^{2}w^{6}t^{4}-6502705z^{2}w^{5}t^{5}+2100857z^{2}w^{4}t^{6}+4190446z^{2}w^{3}t^{7}-2000033z^{2}w^{2}t^{8}+352949z^{2}wt^{9}+117649z^{2}t^{10}-117647zw^{11}+6zw^{10}t+2119092zw^{9}t^{2}-1631923zw^{8}t^{3}+2107815zw^{7}t^{4}+1598344zw^{6}t^{5}-279272zw^{5}t^{6}+4287111zw^{4}t^{7}+424253zw^{3}t^{8}+236004zw^{2}t^{9}+235302zwt^{10}+zt^{11}+w^{12}+235301w^{11}t-116946w^{10}t^{2}-3177237w^{9}t^{3}+6564016w^{8}t^{4}-2409259w^{7}t^{5}+6165496w^{6}t^{6}+1345973w^{5}t^{7}-1549808w^{4}t^{8}+824235w^{3}t^{9}-352242w^{2}t^{10}+5wt^{11}+t^{12}}{5040y^{2}w^{10}+1680y^{2}w^{9}t-15176y^{2}w^{8}t^{2}+21336y^{2}w^{7}t^{3}-12600y^{2}w^{6}t^{4}+12600y^{2}w^{4}t^{6}-21336y^{2}w^{3}t^{7}+15176y^{2}w^{2}t^{8}-1680y^{2}wt^{9}-5040y^{2}t^{10}-37z^{2}w^{10}+83z^{2}w^{9}t+115z^{2}w^{8}t^{2}+532z^{2}w^{7}t^{3}-638z^{2}w^{6}t^{4}+1242z^{2}w^{5}t^{5}-1538z^{2}w^{4}t^{6}+2056z^{2}w^{3}t^{7}-969z^{2}w^{2}t^{8}+203z^{2}wt^{9}+323z^{2}t^{10}-37zw^{11}+406zw^{10}t-38zw^{9}t^{2}-201zw^{8}t^{3}+1414zw^{7}t^{4}-192zw^{6}t^{5}-1092zw^{5}t^{6}+2038zw^{4}t^{7}+239zw^{3}t^{8}-1002zw^{2}t^{9}+886zwt^{10}+323zt^{11}+397w^{11}t+327w^{10}t^{2}-1195w^{9}t^{3}-88w^{8}t^{4}+3290w^{7}t^{5}-5422w^{6}t^{6}+5090w^{5}t^{7}-3136w^{4}t^{8}+973w^{3}t^{9}+87w^{2}t^{10}-323wt^{11}}$ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
56.96.5.c.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{14}t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle y$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 28X^{4}Y^{2}+X^{5}Z-70X^{3}Y^{2}Z+196XY^{4}Z-2X^{4}Z^{2}+112X^{2}Y^{2}Z^{2}+3X^{3}Z^{3}-70XY^{2}Z^{3}-2X^{2}Z^{4}+28Y^{2}Z^{4}+XZ^{5} $ |
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.
