$\GL_2(\Z/36\Z)$-generators: |
$\begin{bmatrix}1&10\\12&35\end{bmatrix}$, $\begin{bmatrix}11&4\\24&35\end{bmatrix}$, $\begin{bmatrix}25&23\\12&1\end{bmatrix}$, $\begin{bmatrix}31&1\\24&1\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
36.288.8-36.f.4.1, 36.288.8-36.f.4.2, 36.288.8-36.f.4.3, 36.288.8-36.f.4.4, 36.288.8-36.f.4.5, 36.288.8-36.f.4.6, 36.288.8-36.f.4.7, 36.288.8-36.f.4.8, 72.288.8-36.f.4.1, 72.288.8-36.f.4.2, 72.288.8-36.f.4.3, 72.288.8-36.f.4.4, 72.288.8-36.f.4.5, 72.288.8-36.f.4.6, 72.288.8-36.f.4.7, 72.288.8-36.f.4.8, 72.288.8-36.f.4.9, 72.288.8-36.f.4.10, 72.288.8-36.f.4.11, 72.288.8-36.f.4.12, 72.288.8-36.f.4.13, 72.288.8-36.f.4.14, 72.288.8-36.f.4.15, 72.288.8-36.f.4.16, 72.288.8-36.f.4.17, 72.288.8-36.f.4.18, 72.288.8-36.f.4.19, 72.288.8-36.f.4.20, 72.288.8-36.f.4.21, 72.288.8-36.f.4.22, 72.288.8-36.f.4.23, 72.288.8-36.f.4.24, 180.288.8-36.f.4.1, 180.288.8-36.f.4.2, 180.288.8-36.f.4.3, 180.288.8-36.f.4.4, 180.288.8-36.f.4.5, 180.288.8-36.f.4.6, 180.288.8-36.f.4.7, 180.288.8-36.f.4.8, 252.288.8-36.f.4.1, 252.288.8-36.f.4.2, 252.288.8-36.f.4.3, 252.288.8-36.f.4.4, 252.288.8-36.f.4.5, 252.288.8-36.f.4.6, 252.288.8-36.f.4.7, 252.288.8-36.f.4.8 |
Cyclic 36-isogeny field degree: |
$3$ |
Cyclic 36-torsion field degree: |
$36$ |
Full 36-torsion field degree: |
$2592$ |
Canonical model in $\mathbb{P}^{ 7 }$ defined by 20 equations
$ 0 $ | $=$ | $ x w - y z $ |
| $=$ | $z u - w t - w u$ |
| $=$ | $x v - x r + z t$ |
| $=$ | $y v - y r + w t$ |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 9 x^{8} + x^{4} y^{3} z - x^{4} z^{4} - y^{3} z^{5} $ |
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 |
$(-1:-1/3:0:0:2:1:0:0)$, $(1:1/3:0:0:2:1:0:0)$, $(0:0:0:0:0:1:0:0)$, $(0:0:0:0:0:0:1:1)$ |
Maps between models of this curve
Birational map from canonical model to plane model:
$\displaystyle X$ |
$=$ |
$\displaystyle w$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle u$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle v$ |
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
36.72.4.f.1
:
$\displaystyle X$ |
$=$ |
$\displaystyle x$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle z$ |
$\displaystyle W$ |
$=$ |
$\displaystyle -w$ |
Equation of the image curve:
$0$ |
$=$ |
$ YZ+XW $ |
|
$=$ |
$ X^{3}-9XY^{2}-Z^{2}W+W^{3} $ |
Hi
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Cover information
Click on a modular curve in the diagram to see information about it.
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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.