$\GL_2(\Z/20\Z)$-generators: |
$\begin{bmatrix}1&5\\1&18\end{bmatrix}$, $\begin{bmatrix}1&11\\13&14\end{bmatrix}$, $\begin{bmatrix}15&17\\9&2\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
20.48.1-20.a.1.1, 20.48.1-20.a.1.2, 40.48.1-20.a.1.1, 40.48.1-20.a.1.2, 60.48.1-20.a.1.1, 60.48.1-20.a.1.2, 120.48.1-20.a.1.1, 120.48.1-20.a.1.2, 140.48.1-20.a.1.1, 140.48.1-20.a.1.2, 220.48.1-20.a.1.1, 220.48.1-20.a.1.2, 260.48.1-20.a.1.1, 260.48.1-20.a.1.2, 280.48.1-20.a.1.1, 280.48.1-20.a.1.2 |
Cyclic 20-isogeny field degree: |
$6$ |
Cyclic 20-torsion field degree: |
$48$ |
Full 20-torsion field degree: |
$1920$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 5 x^{2} - y z $ |
| $=$ | $2 x^{2} - 5 y^{2} + 4 y z - z^{2} - w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 125 x^{4} - 22 x^{2} z^{2} + y^{2} z^{2} + z^{4} $ |
This modular curve has no real points, and therefore no rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
$\displaystyle X$ |
$=$ |
$\displaystyle x$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle w$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle z$ |
Maps to other modular curves
$j$-invariant map
of degree 24 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{118162368yz^{5}-1565676000yz^{3}w^{2}+10687500yzw^{4}-44478720z^{6}+1073051280z^{4}w^{2}-103095000z^{2}w^{4}+78125w^{6}}{z(68381yz^{4}-150250yz^{2}w^{2}+15625yw^{4}-25740z^{5}+1760z^{3}w^{2}+27500zw^{4})}$ |
Hi
|
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.