Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 2 x^{2} y - x z^{2} - y w^{2} $ |
| $=$ | $2 x^{2} w + 8 x y z - w^{3}$ |
| $=$ | $8 y^{2} w + z w^{2}$ |
| $=$ | $8 y^{2} z + z^{2} w$ |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{3} y - 2 y^{2} z^{2} + z^{4} $ |
Weierstrass model Weierstrass model
$ y^{2} + x^{3} y $ | $=$ | $ 2 $ |
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 36 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle 2^9\,\frac{4x^{6}z^{2}-2x^{4}z^{2}w^{2}+2x^{2}z^{2}w^{4}+xyw^{6}-8z^{8}+4z^{5}w^{3}-z^{2}w^{6}}{w^{4}z^{2}(2x^{2}-w^{2})}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
24.36.2.a.1
:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{4}x$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{4}w$ |
Equation of the image curve:
$0$ |
$=$ |
$ X^{3}Y-2Y^{2}Z^{2}+Z^{4} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
24.36.2.a.1
:
$\displaystyle X$ |
$=$ |
$\displaystyle -y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{32}xw^{2}$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{4}w$ |
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.