$\GL_2(\Z/40\Z)$-generators: |
$\begin{bmatrix}5&7\\27&10\end{bmatrix}$, $\begin{bmatrix}8&1\\33&22\end{bmatrix}$, $\begin{bmatrix}13&31\\8&27\end{bmatrix}$, $\begin{bmatrix}29&10\\31&21\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
40.80.1-40.v.1.1, 40.80.1-40.v.1.2, 40.80.1-40.v.1.3, 40.80.1-40.v.1.4, 120.80.1-40.v.1.1, 120.80.1-40.v.1.2, 120.80.1-40.v.1.3, 120.80.1-40.v.1.4, 280.80.1-40.v.1.1, 280.80.1-40.v.1.2, 280.80.1-40.v.1.3, 280.80.1-40.v.1.4 |
Cyclic 40-isogeny field degree: |
$72$ |
Cyclic 40-torsion field degree: |
$1152$ |
Full 40-torsion field degree: |
$18432$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 4 x^{2} - y^{2} - z^{2} - 2 w^{2} $ |
| $=$ | $2 x^{2} + y^{2} - y z + 3 z^{2} + 4 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 20 x^{4} - 60 x^{2} y^{2} + 10 x^{2} z^{2} + 49 y^{4} - 14 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:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle 2x$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle 2w$ |
Maps to other modular curves
$j$-invariant map
of degree 40 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle 2^3\cdot3^3\cdot5^2\,\frac{(2z^{2}+3w^{2})(236yz^{7}+2874yz^{5}w^{2}+8352yz^{3}w^{4}+6912yzw^{6}-712z^{8}-4018z^{6}w^{2}-4509z^{4}w^{4}+5616z^{2}w^{6}+8640w^{8})}{1475yz^{9}+3300yz^{7}w^{2}-2700yz^{5}w^{4}-10800yz^{3}w^{6}-6480yzw^{8}-4450z^{10}-23350z^{8}w^{2}-45600z^{6}w^{4}-37800z^{4}w^{6}-8640z^{2}w^{8}+2592w^{10}}$ |
Hi
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Cover information
Click on a modular curve in the diagram to see information about it.
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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.