Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x^{2} w + y^{2} z + z^{2} w $ |
| $=$ | $x y w - 4 w^{3}$ |
| $=$ | $x y z - 4 z w^{2}$ |
| $=$ | $x y^{2} - 4 y w^{2}$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 2 x^{4} y + x^{2} y^{2} z + z^{5} $ |
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Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{6} - 1 $ |
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This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the known rational points on this modular curve (one row per $j$-invariant).
Maps to other modular curves
$j$-invariant map
of degree 36 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -\frac{256x^{2}w^{6}-256y^{4}w^{4}-z^{8}+48z^{5}w^{3}-512z^{2}w^{6}}{w^{6}z^{2}}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
12.36.2.a.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle 2w$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 2X^{4}Y+X^{2}Y^{2}Z+Z^{5} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
12.36.2.a.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle \frac{1}{2}y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{8}y^{3}+\frac{1}{4}yzw$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
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.
