$\GL_2(\Z/20\Z)$-generators: |
$\begin{bmatrix}3&1\\1&6\end{bmatrix}$, $\begin{bmatrix}15&14\\14&7\end{bmatrix}$, $\begin{bmatrix}17&8\\13&11\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
20.80.1-20.b.1.1, 20.80.1-20.b.1.2, 40.80.1-20.b.1.1, 40.80.1-20.b.1.2, 60.80.1-20.b.1.1, 60.80.1-20.b.1.2, 120.80.1-20.b.1.1, 120.80.1-20.b.1.2, 140.80.1-20.b.1.1, 140.80.1-20.b.1.2, 220.80.1-20.b.1.1, 220.80.1-20.b.1.2, 260.80.1-20.b.1.1, 260.80.1-20.b.1.2, 280.80.1-20.b.1.1, 280.80.1-20.b.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 $ | $=$ | $ x^{2} - y^{2} + y z - z^{2} - z w - w^{2} $ |
| $=$ | $2 x^{2} + y^{2} - y z + y w + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} + 2 x^{3} z + x^{2} y^{2} + 4 x^{2} z^{2} + 6 x y^{2} z + 3 x z^{3} + 4 y^{4} + 4 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:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle x$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Maps to other modular curves
$j$-invariant map
of degree 40 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle 5^2\,\frac{130977yz^{9}+811701yz^{8}w+1555038yz^{7}w^{2}-1193994yz^{6}w^{3}-11058444yz^{5}w^{4}-22727736yz^{4}w^{5}-23968584yz^{3}w^{6}-13649280yz^{2}w^{7}-3770880yzw^{8}-352000yw^{9}-180711z^{10}-1596996z^{9}w-6595398z^{8}w^{2}-16605540z^{7}w^{3}-27787347z^{6}w^{4}-31657446z^{5}w^{5}-23824509z^{4}w^{6}-10170504z^{3}w^{7}-844160z^{2}w^{8}+1121920zw^{9}+326400w^{10}}{405yz^{8}w+810yz^{7}w^{2}+540yz^{6}w^{3}-405yz^{5}w^{4}-1350yz^{4}w^{5}-1080yz^{3}w^{6}-435yz^{2}w^{7}-15yzw^{8}+55yw^{9}-81z^{10}-405z^{9}w-270z^{8}w^{2}+810z^{7}w^{3}+2295z^{6}w^{4}+2457z^{5}w^{5}+1005z^{4}w^{6}-330z^{3}w^{7}-685z^{2}w^{8}-295zw^{9}-51w^{10}}$ |
Hi
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Cover information
Click on a modular curve in the diagram to see information about it.
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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.