Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ x^{2} t - w t^{2} $ |
| $=$ | $x^{2} w - w^{2} t$ |
| $=$ | $x^{2} z - z w t$ |
| $=$ | $x^{2} y - y w t$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 2 x^{7} - 3 x^{4} y z^{2} + x y^{2} z^{4} + 4 y z^{6} $ |
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Weierstrass model Weierstrass model
| $ y^{2} + x^{4} y $ | $=$ | $ -6x^{4} + 4 $ |
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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 |
|---|
| $(0:0:0:0:1)$, $(0:-1:1:0:0)$, $(0:0:1:0:0)$, $(0:1:1:0:0)$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -2^6\,\frac{4xt^{6}+128y^{2}zt^{4}-12yz^{4}wt+48yz^{2}w^{2}t^{2}-140yz^{2}t^{4}+48ywt^{5}-z^{7}+12z^{3}t^{4}-32zwt^{5}}{twz^{4}y}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
32.48.3.c.2
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle 8z$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{2}t$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 2X^{7}-3X^{4}YZ^{2}+XY^{2}Z^{4}+4YZ^{6} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
32.48.3.c.2
:
| $\displaystyle X$ |
$=$ |
$\displaystyle t$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle 6x^{4}-8xzt^{2}-t^{4}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle x$ |
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.
