Embedded model Embedded model in $\mathbb{P}^{6}$
| $ 0 $ | $=$ | $ x^{2} t + u^{3} $ |
| $=$ | $x y v - t u v$ |
| $=$ | $x y t - t^{2} u$ |
| $=$ | $x y^{2} - w u v$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 2 x^{7} y - x^{2} y^{2} z^{4} - z^{8} $ |
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Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ -x^{12} + 1 $ |
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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
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -2^7\,\frac{6xz^{4}uv-8xu^{3}v^{3}+xv^{6}+2z^{7}-6z^{3}u^{2}v^{2}-4ztuv^{4}}{v^{2}u(xu^{2}v+z^{3}u+ztv^{2})}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
24.72.5.a.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle w$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{2}z$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{2}u$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 2X^{7}Y-X^{2}Y^{2}Z^{4}-Z^{8} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
24.72.5.a.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle -\frac{1}{2}u$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{32}zwu^{4}-w^{6}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle -w$ |
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.
