$\GL_2(\Z/20\Z)$-generators: |
$\begin{bmatrix}9&3\\0&13\end{bmatrix}$, $\begin{bmatrix}11&10\\0&3\end{bmatrix}$, $\begin{bmatrix}11&10\\0&19\end{bmatrix}$, $\begin{bmatrix}13&12\\0&11\end{bmatrix}$ |
$\GL_2(\Z/20\Z)$-subgroup: |
$C_2\times C_{20}:C_4^2$ |
Contains $-I$: |
yes |
Quadratic refinements: |
20.144.1-20.g.2.1, 20.144.1-20.g.2.2, 20.144.1-20.g.2.3, 20.144.1-20.g.2.4, 40.144.1-20.g.2.1, 40.144.1-20.g.2.2, 40.144.1-20.g.2.3, 40.144.1-20.g.2.4, 40.144.1-20.g.2.5, 40.144.1-20.g.2.6, 40.144.1-20.g.2.7, 40.144.1-20.g.2.8, 40.144.1-20.g.2.9, 40.144.1-20.g.2.10, 40.144.1-20.g.2.11, 40.144.1-20.g.2.12, 60.144.1-20.g.2.1, 60.144.1-20.g.2.2, 60.144.1-20.g.2.3, 60.144.1-20.g.2.4, 120.144.1-20.g.2.1, 120.144.1-20.g.2.2, 120.144.1-20.g.2.3, 120.144.1-20.g.2.4, 120.144.1-20.g.2.5, 120.144.1-20.g.2.6, 120.144.1-20.g.2.7, 120.144.1-20.g.2.8, 120.144.1-20.g.2.9, 120.144.1-20.g.2.10, 120.144.1-20.g.2.11, 120.144.1-20.g.2.12, 140.144.1-20.g.2.1, 140.144.1-20.g.2.2, 140.144.1-20.g.2.3, 140.144.1-20.g.2.4, 220.144.1-20.g.2.1, 220.144.1-20.g.2.2, 220.144.1-20.g.2.3, 220.144.1-20.g.2.4, 260.144.1-20.g.2.1, 260.144.1-20.g.2.2, 260.144.1-20.g.2.3, 260.144.1-20.g.2.4, 280.144.1-20.g.2.1, 280.144.1-20.g.2.2, 280.144.1-20.g.2.3, 280.144.1-20.g.2.4, 280.144.1-20.g.2.5, 280.144.1-20.g.2.6, 280.144.1-20.g.2.7, 280.144.1-20.g.2.8, 280.144.1-20.g.2.9, 280.144.1-20.g.2.10, 280.144.1-20.g.2.11, 280.144.1-20.g.2.12 |
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 $ | $=$ | $ x^{2} + x z + y w $ |
| $=$ | $x^{2} - 3 x z - y^{2} + 3 y w - 5 z^{2} - w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} + 2 x^{3} y + x^{2} y^{2} + 2 x^{2} z^{2} + 6 x y z^{2} + 5 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:
$\displaystyle X$ |
$=$ |
$\displaystyle x$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Maps to other modular curves
$j$-invariant map
of degree 72 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{5120014xyz^{15}w-262144640xyz^{13}w^{3}+2056213312xyz^{11}w^{5}-4440870528xyz^{9}w^{7}+3602546688xyz^{7}w^{9}-1152806912xyz^{5}w^{11}+119759360xyz^{3}w^{13}-1572864xyzw^{15}-511999xz^{17}+113663850xz^{15}w^{2}-2154490080xz^{13}w^{4}+8535852160xz^{11}w^{6}-11001890688xz^{9}w^{8}+5382310144xz^{7}w^{10}-874804480xz^{5}w^{12}+9070080xz^{3}w^{14}+958464xzw^{16}+5632001yz^{16}w-362496140yz^{14}w^{3}+3727366000yz^{12}w^{5}-10445029312yz^{10}w^{7}+10937143040yz^{8}w^{9}-4699159808yz^{6}w^{11}+750539520yz^{4}w^{13}-27227136yz^{2}w^{15}+36864yw^{17}-512000z^{18}+118271925z^{16}w^{2}-2332668020z^{14}w^{4}+9420656224z^{12}w^{6}-11700835520z^{10}w^{8}+4708936704z^{8}w^{10}-31485440z^{6}w^{12}-226300160z^{4}w^{14}+15393792z^{2}w^{16}-32768w^{18}}{w^{4}z^{2}(122881xyz^{9}w-1867982xyz^{7}w^{3}+3068752xyz^{5}w^{5}-613888xyz^{3}w^{7}+6560xyzw^{9}-20480xz^{11}+1572842xz^{9}w^{2}-7702162xz^{7}w^{4}+4343936xz^{5}w^{6}-60560xz^{3}w^{8}-4640xzw^{10}+143360yz^{10}w-3153941yz^{8}w^{3}+8337660yz^{6}w^{5}-3330832yz^{4}w^{7}+137200yz^{2}w^{9}-64yw^{11}-20480z^{12}+1675259z^{10}w^{2}-8612749z^{8}w^{4}+4275476z^{6}w^{6}+807840z^{4}w^{8}-80720z^{2}w^{10}+64w^{12})}$ |
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.