Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ x w + x t + y z + y t $ |
| $=$ | $7 y^{2} + z^{2} - z w + z t$ |
| $=$ | $7 x^{2} + 7 x y - z^{2} + z w - z t - w t$ |
![Copy content]()
![Toggle raw display]()
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 343 x^{6} y - 343 x^{6} z + 245 x^{4} y^{2} z - 49 x^{4} z^{3} + 56 x^{2} y^{3} z^{2} + 49 x^{2} y^{2} z^{3} + \cdots + 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
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle 2^2\,\frac{823536xyw^{10}-823536xyw^{9}t-14005152xyw^{8}t^{2}-28398384xyw^{7}t^{3}-13143312xyw^{6}t^{4}+13143312xyw^{4}t^{6}+28398384xyw^{3}t^{7}+14005152xyw^{2}t^{8}+823536xywt^{9}-823536xyt^{10}+z^{2}w^{10}-235301z^{2}w^{9}t+703z^{2}w^{8}t^{2}-133534z^{2}w^{7}t^{3}+3978473z^{2}w^{6}t^{4}+6502705z^{2}w^{5}t^{5}+2100857z^{2}w^{4}t^{6}-4190446z^{2}w^{3}t^{7}-2000033z^{2}w^{2}t^{8}-352949z^{2}wt^{9}+117649z^{2}t^{10}+117647zw^{11}+6zw^{10}t-2119092zw^{9}t^{2}-1631923zw^{8}t^{3}-2107815zw^{7}t^{4}+1598344zw^{6}t^{5}+279272zw^{5}t^{6}+4287111zw^{4}t^{7}-424253zw^{3}t^{8}+236004zw^{2}t^{9}-235302zwt^{10}+zt^{11}+w^{12}-117653w^{11}t-234594w^{10}t^{2}+1176501w^{9}t^{3}+2507104w^{8}t^{4}+531643w^{7}t^{5}+6165496w^{6}t^{6}+531643w^{5}t^{7}+2507104w^{4}t^{8}+1176501w^{3}t^{9}-234594w^{2}t^{10}-117653wt^{11}+t^{12}}{2520xyw^{10}-840xyw^{9}t-7588xyw^{8}t^{2}-10668xyw^{7}t^{3}-6300xyw^{6}t^{4}+6300xyw^{4}t^{6}+10668xyw^{3}t^{7}+7588xyw^{2}t^{8}+840xywt^{9}-2520xyt^{10}-37z^{2}w^{10}-83z^{2}w^{9}t+115z^{2}w^{8}t^{2}-532z^{2}w^{7}t^{3}-638z^{2}w^{6}t^{4}-1242z^{2}w^{5}t^{5}-1538z^{2}w^{4}t^{6}-2056z^{2}w^{3}t^{7}-969z^{2}w^{2}t^{8}-203z^{2}wt^{9}+323z^{2}t^{10}+37zw^{11}+406zw^{10}t+38zw^{9}t^{2}-201zw^{8}t^{3}-1414zw^{7}t^{4}-192zw^{6}t^{5}+1092zw^{5}t^{6}+2038zw^{4}t^{7}-239zw^{3}t^{8}-1002zw^{2}t^{9}-886zwt^{10}+323zt^{11}-37w^{11}t+207w^{10}t^{2}+111w^{9}t^{3}-1612w^{8}t^{4}-4190w^{7}t^{5}-5422w^{6}t^{6}-4190w^{5}t^{7}-1612w^{4}t^{8}+111w^{3}t^{9}+207w^{2}t^{10}-37wt^{11}}$ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
28.96.5.d.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle z$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 343X^{6}Y-343X^{6}Z+245X^{4}Y^{2}Z+56X^{2}Y^{3}Z^{2}-49X^{4}Z^{3}+49X^{2}Y^{2}Z^{3}+4Y^{4}Z^{3}+8Y^{3}Z^{4}-7X^{2}Z^{5}+5Y^{2}Z^{5}+YZ^{6} $ |
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.
