Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ 4 x y + z t $ |
| $=$ | $5 x z + y^{2}$ |
| $=$ | $5 z^{2} + 5 w^{2} - 4 t^{2}$ |
| $=$ | $20 x^{2} - y t$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 125 x^{6} - y^{2} z^{4} - z^{6} $ |
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Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ -x^{6} + 125 $ |
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This modular curve has 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
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle 2^8\,\frac{(5w^{2}-3t^{2})^{3}}{t^{4}(5w^{2}-4t^{2})}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
60.36.2.p.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{5}{2}w$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{5}{2}z$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 125X^{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
60.36.2.p.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle -\frac{1}{2}z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{8}z^{2}w$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle -\frac{1}{5}y$ |
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.
