Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ x z + x t - y w - y t $ |
| $=$ | $2 x^{2} + z^{2} - z w + z t$ |
| $=$ | $2 x y - 2 y^{2} + z^{2} - z w + z t + w t$ |
![Copy content]()
![Toggle raw display]()
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{6} y - 2 x^{6} z + 5 x^{4} y^{2} z - 4 x^{4} z^{3} + 8 x^{2} y^{3} z^{2} + 14 x^{2} y^{2} z^{3} + \cdots + 8 y z^{6} $ |
![Copy content]()
![Toggle raw display]()
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
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle 2^2\,\frac{235296y^{2}w^{10}-235296y^{2}w^{9}t-4001472y^{2}w^{8}t^{2}-8113824y^{2}w^{7}t^{3}-3755232y^{2}w^{6}t^{4}+3755232y^{2}w^{4}t^{6}+8113824y^{2}w^{3}t^{7}+4001472y^{2}w^{2}t^{8}+235296y^{2}wt^{9}-235296y^{2}t^{10}-117649z^{2}w^{10}+352949z^{2}w^{9}t+2000033z^{2}w^{8}t^{2}+4190446z^{2}w^{7}t^{3}-2100857z^{2}w^{6}t^{4}-6502705z^{2}w^{5}t^{5}-3978473z^{2}w^{4}t^{6}+133534z^{2}w^{3}t^{7}-703z^{2}w^{2}t^{8}+235301z^{2}wt^{9}-z^{2}t^{10}+zw^{11}-235302zw^{10}t+236004zw^{9}t^{2}-424253zw^{8}t^{3}+4287111zw^{7}t^{4}+279272zw^{6}t^{5}+1598344zw^{5}t^{6}-2107815zw^{4}t^{7}-1631923zw^{3}t^{8}-2119092zw^{2}t^{9}+6zwt^{10}+117647zt^{11}-w^{12}+5w^{11}t+352242w^{10}t^{2}+824235w^{9}t^{3}+1549808w^{8}t^{4}+1345973w^{7}t^{5}-6165496w^{6}t^{6}-2409259w^{5}t^{7}-6564016w^{4}t^{8}-3177237w^{3}t^{9}+116946w^{2}t^{10}+235301wt^{11}-t^{12}}{720y^{2}w^{10}-240y^{2}w^{9}t-2168y^{2}w^{8}t^{2}-3048y^{2}w^{7}t^{3}-1800y^{2}w^{6}t^{4}+1800y^{2}w^{4}t^{6}+3048y^{2}w^{3}t^{7}+2168y^{2}w^{2}t^{8}+240y^{2}wt^{9}-720y^{2}t^{10}-323z^{2}w^{10}+203z^{2}w^{9}t+969z^{2}w^{8}t^{2}+2056z^{2}w^{7}t^{3}+1538z^{2}w^{6}t^{4}+1242z^{2}w^{5}t^{5}+638z^{2}w^{4}t^{6}+532z^{2}w^{3}t^{7}-115z^{2}w^{2}t^{8}+83z^{2}wt^{9}+37z^{2}t^{10}+323zw^{11}-886zw^{10}t-1002zw^{9}t^{2}-239zw^{8}t^{3}+2038zw^{7}t^{4}+1092zw^{6}t^{5}-192zw^{5}t^{6}-1414zw^{4}t^{7}-201zw^{3}t^{8}+38zw^{2}t^{9}+406zwt^{10}+37zt^{11}-323w^{11}t-87w^{10}t^{2}+973w^{9}t^{3}+3136w^{8}t^{4}+5090w^{7}t^{5}+5422w^{6}t^{6}+3290w^{5}t^{7}+88w^{4}t^{8}-1195w^{3}t^{9}-327w^{2}t^{10}+397wt^{11}}$ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
56.96.5.b.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{2}z$ |
Equation of the image curve:
| $0$ |
$=$ |
$ X^{6}Y-2X^{6}Z+5X^{4}Y^{2}Z+8X^{2}Y^{3}Z^{2}-4X^{4}Z^{3}+14X^{2}Y^{2}Z^{3}+4Y^{4}Z^{3}+16Y^{3}Z^{4}-8X^{2}Z^{5}+20Y^{2}Z^{5}+8YZ^{6} $ |
The following modular covers realize this modular curve as a fiber product over $X(1)$.
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.
