Canonical model in $\mathbb{P}^{ 8 }$ defined by 21 equations
| $ 0 $ | $=$ | $ x t + x u + x v - y z + y t - 2 y u + z s - w s - r s $ |
| $=$ | $2 x t - x u - x r - y w - y t - y u + z s - v s - r s$ |
| $=$ | $x^{2} + x w + x s - z w - z u + z s - w u - t^{2} + t u - u^{2} + s^{2}$ |
| $=$ | $3 x w + x t + x u - 2 x v + 2 y w + z s - w s + r s$ |
| $=$ | $\cdots$ |
![Copy content]()
![Toggle raw display]()
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 315 x^{13} + 752 x^{12} y - 2412 x^{12} z + 528 x^{11} y^{2} - 5160 x^{11} y z + 8316 x^{11} z^{2} + \cdots + 315 z^{13} $ |
![Copy content]()
![Toggle raw display]()
This modular curve has 3 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:0:0:0:0:-1:0:0:1)$, $(0:0:0:0:-1:0:0:0:1)$, $(0:0:0:0:1:1:0:0:1)$ |
Maps to other modular curves
Map
of degree 3 from the canonical model of this modular curve to the canonical model of the modular curve
36.48.3.d.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle -y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle s$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 9X^{4}-2XY^{3}-4X^{2}YZ-Y^{2}Z^{2}-2XZ^{3} $ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
36.144.9.cr.2
:
| $\displaystyle X$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{3}{2}s$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 315X^{13}+752X^{12}Y+528X^{11}Y^{2}+96X^{10}Y^{3}+8X^{9}Y^{4}-2412X^{12}Z-5160X^{11}YZ-2904X^{10}Y^{2}Z-48X^{9}Y^{3}Z+36X^{8}Y^{4}Z+8316X^{11}Z^{2}+13416X^{10}YZ^{2}+5064X^{9}Y^{2}Z^{2}-1152X^{8}Y^{3}Z^{2}-108X^{7}Y^{4}Z^{2}-18477X^{10}Z^{3}-16648X^{9}YZ^{3}-1008X^{8}Y^{2}Z^{3}+3168X^{7}Y^{3}Z^{3}-300X^{6}Y^{4}Z^{3}+30015X^{9}Z^{4}+8928X^{8}YZ^{4}-5472X^{7}Y^{2}Z^{4}-3888X^{6}Y^{3}Z^{4}+828X^{5}Y^{4}Z^{4}-38475X^{8}Z^{5}+2880X^{7}YZ^{5}+3528X^{6}Y^{2}Z^{5}+1584X^{5}Y^{3}Z^{5}-216X^{4}Y^{4}Z^{5}+39150X^{7}Z^{6}-13416X^{6}YZ^{6}+3528X^{5}Y^{2}Z^{6}+2160X^{4}Y^{3}Z^{6}-624X^{3}Y^{4}Z^{6}-30834X^{6}Z^{7}+22320X^{5}YZ^{7}-5472X^{4}Y^{2}Z^{7}-3744X^{3}Y^{3}Z^{7}+468X^{2}Y^{4}Z^{7}+16200X^{5}Z^{8}-16344X^{4}YZ^{8}-1008X^{3}Y^{2}Z^{8}+2736X^{2}Y^{3}Z^{8}-108XY^{4}Z^{8}-2790X^{4}Z^{9}+848X^{3}YZ^{9}+5064X^{2}Y^{2}Z^{9}-912XY^{3}Z^{9}+8Y^{4}Z^{9}-3897X^{3}Z^{10}+6288X^{2}YZ^{10}-2904XY^{2}Z^{10}+96Y^{3}Z^{10}+3942X^{2}Z^{11}-3864XYZ^{11}+528Y^{2}Z^{11}-1683XZ^{12}+752YZ^{12}+315Z^{13} $ |
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.
