Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ - x w + y^{2} $ |
| $=$ | $4 z^{2} + w^{2} + t^{2}$ |
| $=$ | $4 x y + z w$ |
| $=$ | $4 x^{2} + y z$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{6} + y^{2} z^{4} + z^{6} $ |
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Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ -x^{6} - 1 $ |
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This modular curve has no real points and no $\Q_p$ points for $p=7$, and therefore no rational points.
Maps to other modular curves
$j$-invariant map
of degree 36 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle 2^6\,\frac{(3w^{2}-t^{2})^{3}}{w^{2}(w^{2}+t^{2})^{2}}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
12.36.2.i.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{2}t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{2}w$ |
Equation of the image curve:
| $0$ |
$=$ |
$ X^{6}+Y^{2}Z^{4}+Z^{6} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
12.36.2.i.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle -\frac{1}{2}w$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle -\frac{1}{8}w^{2}t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle -y$ |
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.
