Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 5 x^{2} y + x z^{2} + y w^{2} $ |
| $=$ | $ - 5 x^{2} w + 20 x y z - w^{3}$ |
| $=$ | $20 y^{2} w + z w^{2}$ |
| $=$ | $20 y^{2} z + z^{2} w$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 2 x^{3} y + y^{2} z^{2} + 125 z^{4} $ |
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Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{6} - 125 $ |
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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
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle 2^8\,\frac{125x^{6}z^{2}+25x^{4}z^{2}w^{2}+10x^{2}z^{2}w^{4}+5xyw^{6}+16z^{8}+8z^{5}w^{3}+2z^{2}w^{6}}{w^{4}z^{2}(5x^{2}+w^{2})}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
60.36.2.a.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{5}{2}x$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{10}w$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 2X^{3}Y+Y^{2}Z^{2}+125Z^{4} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
60.36.2.a.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle -y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{40}xw^{2}+y^{3}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{10}w$ |
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.
