Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ x z - x w - y t $ |
| $=$ | $3 x y + 2 z^{2} + 2 z w + t^{2}$ |
| $=$ | $4 x^{2} - y^{2} - z t + w t$ |
![Copy content]()
![Toggle raw display]()
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 4 x^{7} + 4 x^{5} y^{2} + 8 x^{4} y z^{2} + x^{3} y^{4} + 4 x y^{2} z^{4} + y^{5} z^{2} $ |
![Copy content]()
![Toggle raw display]()
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
Maps to other modular curves
$j$-invariant map
of degree 72 from the canonical model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle 2^3\cdot3^3\,\frac{167392xw^{8}t+2155528xw^{6}t^{3}+3533868xw^{4}t^{5}+1226460xw^{2}t^{7}+118773xt^{9}+360yz^{2}w^{7}+212760yz^{2}w^{5}t^{2}+831276yz^{2}w^{3}t^{4}+473508yz^{2}wt^{6}+24yzw^{8}+294848yzw^{6}t^{2}+1941048yzw^{4}t^{4}+2214924yzw^{2}t^{6}+319041yzt^{8}-384yw^{9}+164680yw^{7}t^{2}+2012616yw^{5}t^{4}+2965476yw^{3}t^{6}+585720ywt^{8}}{t(128xw^{8}+9728xw^{6}t^{2}+53784xw^{4}t^{4}+37356xw^{2}t^{6}+6129xt^{8}+432yz^{2}w^{5}t+7452yz^{2}w^{3}t^{3}+13707yz^{2}wt^{5}+496yzw^{6}t+12348yzw^{4}t^{3}+41601yzw^{2}t^{5}+16461yzt^{7}+128yw^{7}t+9360yw^{5}t^{3}+49020yw^{3}t^{5}+26235ywt^{7})}$ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
48.72.5.l.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle z$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 4X^{7}+4X^{5}Y^{2}+8X^{4}YZ^{2}+X^{3}Y^{4}+4XY^{2}Z^{4}+Y^{5}Z^{2} $ |
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.
