$\GL_2(\Z/20\Z)$-generators: |
$\begin{bmatrix}3&6\\0&19\end{bmatrix}$, $\begin{bmatrix}9&14\\0&3\end{bmatrix}$, $\begin{bmatrix}17&13\\0&3\end{bmatrix}$, $\begin{bmatrix}19&18\\0&11\end{bmatrix}$ |
$\GL_2(\Z/20\Z)$-subgroup: |
$C_2\times C_{20}:C_4^2$ |
Contains $-I$: |
yes |
Quadratic refinements: |
20.144.1-20.g.1.1, 20.144.1-20.g.1.2, 20.144.1-20.g.1.3, 20.144.1-20.g.1.4, 40.144.1-20.g.1.1, 40.144.1-20.g.1.2, 40.144.1-20.g.1.3, 40.144.1-20.g.1.4, 40.144.1-20.g.1.5, 40.144.1-20.g.1.6, 40.144.1-20.g.1.7, 40.144.1-20.g.1.8, 40.144.1-20.g.1.9, 40.144.1-20.g.1.10, 40.144.1-20.g.1.11, 40.144.1-20.g.1.12, 60.144.1-20.g.1.1, 60.144.1-20.g.1.2, 60.144.1-20.g.1.3, 60.144.1-20.g.1.4, 120.144.1-20.g.1.1, 120.144.1-20.g.1.2, 120.144.1-20.g.1.3, 120.144.1-20.g.1.4, 120.144.1-20.g.1.5, 120.144.1-20.g.1.6, 120.144.1-20.g.1.7, 120.144.1-20.g.1.8, 120.144.1-20.g.1.9, 120.144.1-20.g.1.10, 120.144.1-20.g.1.11, 120.144.1-20.g.1.12, 140.144.1-20.g.1.1, 140.144.1-20.g.1.2, 140.144.1-20.g.1.3, 140.144.1-20.g.1.4, 220.144.1-20.g.1.1, 220.144.1-20.g.1.2, 220.144.1-20.g.1.3, 220.144.1-20.g.1.4, 260.144.1-20.g.1.1, 260.144.1-20.g.1.2, 260.144.1-20.g.1.3, 260.144.1-20.g.1.4, 280.144.1-20.g.1.1, 280.144.1-20.g.1.2, 280.144.1-20.g.1.3, 280.144.1-20.g.1.4, 280.144.1-20.g.1.5, 280.144.1-20.g.1.6, 280.144.1-20.g.1.7, 280.144.1-20.g.1.8, 280.144.1-20.g.1.9, 280.144.1-20.g.1.10, 280.144.1-20.g.1.11, 280.144.1-20.g.1.12 |
Cyclic 20-isogeny field degree: |
$1$ |
Cyclic 20-torsion field degree: |
$8$ |
Full 20-torsion field degree: |
$640$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 5 x z + y^{2} - y w $ |
| $=$ | $5 x^{2} + 2 y^{2} - 6 y w + 5 z^{2} + 5 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 25 x^{4} - 10 x^{3} y + x^{2} y^{2} + 10 x^{2} z^{2} - 6 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:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle 5w$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle 5z$ |
Maps to other modular curves
$j$-invariant map
of degree 72 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle -\frac{1}{5^3}\cdot\frac{614400000000xyz^{15}w-9356200000000xyz^{13}w^{3}+18012608000000xyz^{11}w^{5}-11257958400000xyz^{9}w^{7}+2775544080000xyz^{7}w^{9}-257026664000xyz^{5}w^{11}+6553616000xyz^{3}w^{13}-25600070xyzw^{15}+72000000000xz^{17}-10635600000000xz^{15}w^{2}+58635900000000xz^{13}w^{4}-73424372000000xz^{11}w^{6}+34178572000000xz^{9}w^{8}-6528143320000xz^{7}w^{10}+465920750000xz^{5}w^{12}-9062403500xz^{3}w^{14}+28160005xzw^{16}-374400000000yz^{16}w-708600000000yz^{14}w^{3}+13668820000000yz^{12}w^{5}-16819719200000yz^{10}w^{7}+6876181680000yz^{8}w^{9}-1066981520000yz^{6}w^{11}+53862252000yz^{4}w^{13}-568319250yz^{2}w^{15}+511999yw^{17}-64000000000z^{18}+6013200000000z^{16}w^{2}-17679700000000z^{14}w^{4}-491960000000z^{12}w^{6}+14715427200000z^{10}w^{8}-7313022200000z^{8}w^{10}+1177582028000z^{6}w^{12}-58316700500z^{4}w^{14}+591359625z^{2}w^{16}-512000w^{18}}{w^{2}z^{4}(4100000xyz^{9}w-76736000xyz^{7}w^{3}+76718800xyz^{5}w^{5}-9339910xyz^{3}w^{7}+122881xyzw^{9}+200000xz^{11}-85750000xz^{9}w^{2}+416354000xz^{7}w^{4}-208441500xz^{5}w^{6}+15769705xz^{3}w^{8}-143360xzw^{10}-2900000yz^{10}w-7570000yz^{8}w^{3}+108598400yz^{6}w^{5}-38510810yz^{4}w^{7}+1572842yz^{2}w^{9}-4096yw^{11}-200000z^{12}+50450000z^{10}w^{2}-100980000z^{8}w^{4}-106886900z^{6}w^{6}+43063745z^{4}w^{8}-1675259z^{2}w^{10}+4096w^{12})}$ |
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.