$\GL_2(\Z/20\Z)$-generators: |
$\begin{bmatrix}4&1\\1&2\end{bmatrix}$, $\begin{bmatrix}7&16\\5&13\end{bmatrix}$, $\begin{bmatrix}19&7\\0&1\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
20.80.1-20.f.1.1, 20.80.1-20.f.1.2, 40.80.1-20.f.1.1, 40.80.1-20.f.1.2, 60.80.1-20.f.1.1, 60.80.1-20.f.1.2, 120.80.1-20.f.1.1, 120.80.1-20.f.1.2, 140.80.1-20.f.1.1, 140.80.1-20.f.1.2, 220.80.1-20.f.1.1, 220.80.1-20.f.1.2, 260.80.1-20.f.1.1, 260.80.1-20.f.1.2, 280.80.1-20.f.1.1, 280.80.1-20.f.1.2 |
Cyclic 20-isogeny field degree: |
$36$ |
Cyclic 20-torsion field degree: |
$288$ |
Full 20-torsion field degree: |
$1152$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 2 x^{2} + y^{2} + z^{2} - w^{2} $ |
| $=$ | $3 x^{2} - 2 y^{2} - y z + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 5 x^{4} - 10 x^{2} y^{2} - 5 x^{2} z^{2} + 9 y^{4} + 6 y^{2} z^{2} + z^{4} $ |
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle x$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Maps to other modular curves
$j$-invariant map
of degree 40 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle 5^2\,\frac{14413248yz^{9}+3618272yz^{7}w^{2}-192665844yz^{5}w^{4}-108353700yz^{3}w^{6}-5704776yzw^{8}+929664z^{10}+55987536z^{8}w^{2}-60864272z^{6}w^{4}-169518195z^{4}w^{6}-29801898z^{2}w^{8}-324135w^{10}}{208525yz^{9}-353150yz^{7}w^{2}+248675yz^{5}w^{4}-85750yz^{3}w^{6}+12005yzw^{8}+13450z^{10}-84075z^{8}w^{2}+78400z^{6}w^{4}-15925z^{4}w^{6}-6860z^{2}w^{8}+2401w^{10}}$ |
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.