Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ x^{2} - x u + x v + z w $ |
| $=$ | $x^{2} + x z + x v - y u + z^{2}$ |
| $=$ | $x y - x z - x u + y v$ |
| $=$ | $x z - x w - y w$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - x^{6} y^{2} z - x^{6} z^{3} + x^{4} y^{5} - 3 x^{4} y^{4} z + 4 x^{4} y^{3} z^{2} - 2 x^{4} y^{2} z^{3} + \cdots + y^{5} z^{4} $ |
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This modular curve has 6 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(-1:0:0:0:0:0:1)$, $(1:1:0:0:1/2:1:0)$, $(0:0:0:0:1:0:0)$, $(0:0:0:1:1/2:-1:1)$, $(0:0:0:0:0:0:1)$, $(-1:1:0:1:1/2:0:1)$ |
Maps to other modular curves
Map
of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve
$X_0(34)$
:
| $\displaystyle X$ |
$=$ |
$\displaystyle -y+t$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle w-t+u$ |
Equation of the image curve:
| $0$ |
$=$ |
$ X^{3}Y+X^{2}Y^{2}+XY^{3}-2Y^{4}-X^{3}Z+2XY^{2}Z+Y^{3}Z+Y^{2}Z^{2}-XZ^{3}+YZ^{3} $ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
$X_0(68)$
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle t$ |
Equation of the image curve:
| $0$ |
$=$ |
$ -X^{6}Y^{2}Z-X^{6}Z^{3}+X^{4}Y^{5}-3X^{4}Y^{4}Z+4X^{4}Y^{3}Z^{2}-2X^{4}Y^{2}Z^{3}+X^{4}YZ^{4}+3X^{2}Y^{4}Z^{3}-2X^{2}Y^{3}Z^{4}+Y^{5}Z^{4} $ |
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.
