Canonical model in $\mathbb{P}^{ 5 }$ defined by 9 equations
| $ 0 $ | $=$ | $ x w + y t + w u $ |
| $=$ | $x w - z t - w t$ |
| $=$ | $x y + x z - y t + z u$ |
| $=$ | $x y - x z - y u - z u + w t$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 2 x^{3} y^{3} z + x^{3} y^{2} z^{2} + 4 y^{7} - 2 y^{6} z + 3 y^{5} z^{2} + 5 y^{4} z^{3} - y^{3} z^{4} + \cdots + z^{7} $ |
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This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:-1:0:1:0:0)$, $(1/2:0:0:0:0:1)$, $(0:0:1:0:0:0)$, $(-1/2:0:0:0:1:0)$ |
Maps to other modular curves
$j$-invariant map
of degree 108 from the canonical model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -\frac{824526xt^{2}u^{9}-467638xtu^{10}+986094xu^{11}+81yw^{11}-391986yw^{8}u^{3}-88823yw^{5}u^{6}-721738yw^{2}u^{9}-13824z^{9}u^{3}-145152z^{6}u^{6}-539136z^{3}u^{9}-81zw^{11}+88722zw^{8}u^{3}+65021zw^{5}u^{6}+76922zw^{2}u^{9}+81w^{12}+14067w^{9}t^{3}+75789w^{9}t^{2}u+228231w^{9}tu^{2}+62505w^{9}u^{3}+790911w^{6}t^{3}u^{3}+314622w^{6}t^{2}u^{4}+662079w^{6}tu^{5}+71622w^{6}u^{6}+221998w^{3}t^{3}u^{6}+26532w^{3}t^{2}u^{7}+746999w^{3}tu^{8}+69742w^{3}u^{9}-27t^{12}-108t^{11}u-108t^{10}u^{2}-729t^{9}u^{3}-5085t^{8}u^{4}-110202t^{7}u^{5}-398428t^{6}u^{6}-858445t^{5}u^{7}-1053406t^{4}u^{8}-501418t^{3}u^{9}-536950t^{2}u^{10}-236872tu^{11}-493074u^{12}}{u^{3}(48xt^{2}u^{6}+48xtu^{7}+105yw^{5}u^{3}-24yw^{2}u^{6}+9zw^{5}u^{3}-24zw^{2}u^{6}-57w^{6}tu^{2}-126w^{3}t^{3}u^{3}+24w^{3}t^{2}u^{4}-48w^{3}tu^{5}+48w^{3}u^{6}+t^{9}+3t^{8}u-23t^{6}u^{3}-24t^{5}u^{4}-45t^{4}u^{5}-47t^{3}u^{6}-48t^{2}u^{7}-24tu^{8})}$ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
18.108.6.a.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{3}y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{3}w$ |
Equation of the image curve:
| $0$ |
$=$ |
$ -2X^{3}Y^{3}Z+X^{3}Y^{2}Z^{2}+4Y^{7}-2Y^{6}Z+3Y^{5}Z^{2}+5Y^{4}Z^{3}-Y^{3}Z^{4}+3Y^{2}Z^{5}+YZ^{6}+Z^{7} $ |
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.
