Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 15 x^{2} y - x z^{2} + y w^{2} $ |
| $=$ | $15 x^{2} w + 60 x y z + w^{3}$ |
| $=$ | $60 y^{2} w + z w^{2}$ |
| $=$ | $60 y^{2} z + z^{2} w$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 2 x^{3} y - y^{2} z^{2} - 3375 z^{4} $ |
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Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{6} - 3375 $ |
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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
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle 2^8\,\frac{3375x^{7}w+450x^{5}w^{3}+45x^{3}w^{5}+xw^{7}-64yz^{7}-32yz^{4}w^{3}-8yzw^{6}}{w^{5}(15x^{3}+xw^{2}-4yzw)}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
60.36.2.d.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{15}{2}x$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{30}w$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 2X^{3}Y-Y^{2}Z^{2}-3375Z^{4} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
60.36.2.d.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle -y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{120}xw^{2}-y^{3}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{30}w$ |
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.
