Embedded model Embedded model in $\mathbb{P}^{6}$
| $ 0 $ | $=$ | $ - y v^{2} + u^{2} v $ |
| $=$ | $ - x v^{2} + u^{3}$ |
| $=$ | $x u v + t v^{2}$ |
| $=$ | $x y v + t u v$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 8 x^{11} - 2 x y^{2} z^{8} + y z^{10} $ |
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Weierstrass model Weierstrass model
| $ y^{2} + x^{6} y $ | $=$ | $ 16 $ |
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This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Embedded model |
|---|
| $(0:0:0:0:0:0:1)$, $(0:0:0:1:0:0:0)$, $(0:0:-1:1:0:0:0)$, $(0:0:1:1:0:0:0)$ |
Maps to other modular curves
$j$-invariant map
of degree 72 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle 2^6\,\frac{12zw^{4}tv+13zwuv^{4}+76zt^{3}v^{3}-4zv^{6}-w^{7}-48w^{3}t^{2}v^{2}}{vtw^{4}z}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
48.72.5.b.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle u$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle 2w$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle v$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 8X^{11}-2XY^{2}Z^{8}+YZ^{10} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
48.72.5.b.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle v$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle 4wuv^{4}-v^{6}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle -u$ |
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.
