Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ x w + y z $ |
| $=$ | $9 x y + z t - w^{2} - t^{2}$ |
| $=$ | $3 x^{2} - 12 y^{2} + z w$ |
![Copy content]()
![Toggle raw display]()
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 3 x^{5} z^{2} + x^{4} y^{3} + 4 x^{2} y^{5} + 36 x^{2} y z^{4} - 24 x y^{4} z^{2} + 4 y^{7} $ |
![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
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -2^2\cdot3^3\,\frac{3z^{10}-6z^{9}t-72z^{8}t^{2}+342z^{7}t^{3}-792z^{6}t^{4}+854z^{5}t^{5}-1130z^{4}t^{6}+792z^{3}t^{7}-52z^{2}t^{8}-416zw^{8}t+1856zw^{6}t^{3}-3216zw^{4}t^{5}-928zw^{2}t^{7}+80zt^{9}-800w^{8}t^{2}+1504w^{6}t^{4}+1680w^{4}t^{6}-832w^{2}t^{8}-16t^{10}}{z^{5}t^{5}+2z^{4}t^{6}-9z^{3}t^{7}+10z^{2}t^{8}-13zw^{8}t-8zw^{6}t^{3}+6zw^{4}t^{5}+4zw^{2}t^{7}-5zt^{9}+6w^{10}+5w^{8}t^{2}-4w^{6}t^{4}-6w^{4}t^{6}-2w^{2}t^{8}+t^{10}}$ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
24.72.5.m.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{6}t$ |
Equation of the image curve:
| $0$ |
$=$ |
$ -3X^{5}Z^{2}+X^{4}Y^{3}+4X^{2}Y^{5}+36X^{2}YZ^{4}-24XY^{4}Z^{2}+4Y^{7} $ |
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.
