Embedded model Embedded model in $\mathbb{P}^{6}$
| $ 0 $ | $=$ | $ x w v + t u v $ |
| $=$ | $ - x w v + w^{2} t$ |
| $=$ | $x w v - y v^{2}$ |
| $=$ | $x^{2} v + t^{2} u$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{11} + x y^{2} z^{8} + 54 y z^{10} $ |
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Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{12} - 729 $ |
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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 72 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{2^4}{3^2}\cdot\frac{432yz^{4}uv-76yu^{3}v^{3}-4yv^{6}+216z^{7}-288z^{3}u^{2}v^{2}-13zwuv^{4}}{vuz^{4}y}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
48.72.5.h.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle t$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle 36z$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{6}v$ |
Equation of the image curve:
| $0$ |
$=$ |
$ X^{11}+XY^{2}Z^{8}+54YZ^{10} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
48.72.5.h.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle -\frac{1}{2}v$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle -\frac{3}{4}ztv^{4}-\frac{1}{64}v^{6}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle -t$ |
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.
