$\GL_2(\Z/20\Z)$-generators: |
$\begin{bmatrix}1&16\\10&3\end{bmatrix}$, $\begin{bmatrix}5&16\\16&19\end{bmatrix}$, $\begin{bmatrix}9&2\\6&15\end{bmatrix}$, $\begin{bmatrix}9&16\\6&13\end{bmatrix}$, $\begin{bmatrix}11&16\\14&1\end{bmatrix}$ |
$\GL_2(\Z/20\Z)$-subgroup: |
$D_{10}.C_2^5$ |
Contains $-I$: |
yes |
Quadratic refinements: |
20.144.1-20.a.2.1, 20.144.1-20.a.2.2, 20.144.1-20.a.2.3, 20.144.1-20.a.2.4, 40.144.1-20.a.2.1, 40.144.1-20.a.2.2, 40.144.1-20.a.2.3, 40.144.1-20.a.2.4, 60.144.1-20.a.2.1, 60.144.1-20.a.2.2, 60.144.1-20.a.2.3, 60.144.1-20.a.2.4, 120.144.1-20.a.2.1, 120.144.1-20.a.2.2, 120.144.1-20.a.2.3, 120.144.1-20.a.2.4, 140.144.1-20.a.2.1, 140.144.1-20.a.2.2, 140.144.1-20.a.2.3, 140.144.1-20.a.2.4, 220.144.1-20.a.2.1, 220.144.1-20.a.2.2, 220.144.1-20.a.2.3, 220.144.1-20.a.2.4, 260.144.1-20.a.2.1, 260.144.1-20.a.2.2, 260.144.1-20.a.2.3, 260.144.1-20.a.2.4, 280.144.1-20.a.2.1, 280.144.1-20.a.2.2, 280.144.1-20.a.2.3, 280.144.1-20.a.2.4 |
Cyclic 20-isogeny field degree: |
$2$ |
Cyclic 20-torsion field degree: |
$16$ |
Full 20-torsion field degree: |
$640$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 5 x^{2} - 3 z^{2} - 4 z w $ |
| $=$ | $5 x y + 5 y^{2} + 2 z^{2} + 3 z w + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 5 x^{4} + 6 x^{2} z^{2} + 4 x y 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:
$\displaystyle X$ |
$=$ |
$\displaystyle x$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle 4y$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle z$ |
Maps to other modular curves
$j$-invariant map
of degree 72 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle -\frac{(269z^{6}+172z^{5}w-1760z^{4}w^{2}-3840z^{3}w^{3}-3040z^{2}w^{4}-848zw^{5}+16w^{6})^{3}}{z(z+w)^{2}(2z+w)^{10}(3z+4w)^{5}}$ |
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.