Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x^{2} w + y w^{2} $ |
| $=$ | $x^{2} z + y z w$ |
| $=$ | $x^{2} y + y^{2} w$ |
| $=$ | $x^{3} + x y w$ |
| $=$ | $\cdots$ |
![Copy content]()
![Toggle raw display]()
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{5} + x y^{2} z^{2} - 2 y z^{4} $ |
![Copy content]()
![Toggle raw display]()
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{6} - 1 $ |
![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 36 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle 2^6\,\frac{12xyz^{5}w-76xz^{3}w^{4}-16y^{2}w^{6}-48yz^{4}w^{3}-z^{8}+16z^{2}w^{6}}{wz^{5}yx}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
24.36.2.cw.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle 2z$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{2}w$ |
Equation of the image curve:
| $0$ |
$=$ |
$ X^{5}+XY^{2}Z^{2}-2YZ^{4} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
24.36.2.cw.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle -\frac{1}{2}w$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle -xzw+\frac{1}{8}w^{3}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle -x$ |
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.
