Canonical model in $\mathbb{P}^{ 7 }$ defined by 15 equations
| $ 0 $ | $=$ | $ x y - x r - y^{2} - u r + r^{2} $ |
| $=$ | $x^{2} - x y - x r + y u + t u + u v + u r$ |
| $=$ | $x y + y t + y v - t r - u r - v r$ |
| $=$ | $x y - x r + y^{2} + y z + y w + y r + t r$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 2 x^{5} y^{6} z - 8 x^{5} y^{5} z^{2} + 10 x^{5} y^{4} z^{3} + 4 x^{5} y^{3} z^{4} - 20 x^{5} y^{2} z^{5} + \cdots - 4 z^{12} $ |
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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 |
|---|
| $(0:0:0:0:0:0:1:0)$, $(1/2:1/2:0:0:-1/2:1/2:0:1)$, $(0:0:0:0:0:1:-1:1)$, $(0:0:0:0:0:1:0:0)$, $(1:0:0:0:-1:1:0:0)$, $(0:-1:0:0:0:0:0:1)$ |
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
30.60.4.b.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x-y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle -x-w-t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle y+t+v$ |
| $\displaystyle W$ |
$=$ |
$\displaystyle z-w$ |
Equation of the image curve:
| $0$ |
$=$ |
$ X^{2}-4XY+XZ+3YZ-XW+2W^{2} $ |
|
$=$ |
$ X^{3}-X^{2}Y+X^{2}Z-2XYZ-Y^{2}Z+YZ^{2}-2X^{2}W-XYW-XZW+XW^{2}+ZW^{2} $ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
30.120.8.a.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle v$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle r$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 4Y^{12}+2X^{5}Y^{6}Z+10X^{4}Y^{7}Z+10X^{3}Y^{8}Z-20X^{2}Y^{9}Z-20XY^{10}Z-6Y^{11}Z-8X^{5}Y^{5}Z^{2}-15X^{4}Y^{6}Z^{2}+45X^{3}Y^{7}Z^{2}+120X^{2}Y^{8}Z^{2}+10XY^{9}Z^{2}-26Y^{10}Z^{2}+10X^{5}Y^{4}Z^{3}-45X^{4}Y^{5}Z^{3}-175X^{3}Y^{6}Z^{3}-25X^{2}Y^{7}Z^{3}+190XY^{8}Z^{3}+26Y^{9}Z^{3}+4X^{5}Y^{3}Z^{4}+115X^{4}Y^{4}Z^{4}+25X^{3}Y^{5}Z^{4}-405X^{2}Y^{6}Z^{4}-125XY^{7}Z^{4}+96Y^{8}Z^{4}-20X^{5}Y^{2}Z^{5}-45X^{4}Y^{3}Z^{5}+325X^{3}Y^{4}Z^{5}+265X^{2}Y^{5}Z^{5}-375XY^{6}Z^{5}-63Y^{7}Z^{5}+16X^{5}YZ^{6}-80X^{4}Y^{2}Z^{6}-230X^{3}Y^{3}Z^{6}+405X^{2}Y^{4}Z^{6}+305XY^{5}Z^{6}-126Y^{6}Z^{6}-4X^{5}Z^{7}+80X^{4}YZ^{7}-120X^{3}Y^{2}Z^{7}-380X^{2}Y^{3}Z^{7}+245XY^{4}Z^{7}+102Y^{5}Z^{7}-20X^{4}Z^{8}+160X^{3}YZ^{8}-80X^{2}Y^{2}Z^{8}-270XY^{3}Z^{8}+60Y^{4}Z^{8}-40X^{3}Z^{9}+160X^{2}YZ^{9}-20XY^{2}Z^{9}-71Y^{3}Z^{9}-40X^{2}Z^{10}+80XYZ^{10}-20XZ^{11}+16YZ^{11}-4Z^{12} $ |
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.
