$\GL_2(\Z/40\Z)$-generators: |
$\begin{bmatrix}5&8\\22&7\end{bmatrix}$, $\begin{bmatrix}21&0\\12&3\end{bmatrix}$, $\begin{bmatrix}25&32\\32&29\end{bmatrix}$, $\begin{bmatrix}27&0\\4&29\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
40.192.1-40.ca.2.1, 40.192.1-40.ca.2.2, 40.192.1-40.ca.2.3, 40.192.1-40.ca.2.4, 40.192.1-40.ca.2.5, 40.192.1-40.ca.2.6, 40.192.1-40.ca.2.7, 40.192.1-40.ca.2.8, 80.192.1-40.ca.2.1, 80.192.1-40.ca.2.2, 80.192.1-40.ca.2.3, 80.192.1-40.ca.2.4, 120.192.1-40.ca.2.1, 120.192.1-40.ca.2.2, 120.192.1-40.ca.2.3, 120.192.1-40.ca.2.4, 120.192.1-40.ca.2.5, 120.192.1-40.ca.2.6, 120.192.1-40.ca.2.7, 120.192.1-40.ca.2.8, 240.192.1-40.ca.2.1, 240.192.1-40.ca.2.2, 240.192.1-40.ca.2.3, 240.192.1-40.ca.2.4, 280.192.1-40.ca.2.1, 280.192.1-40.ca.2.2, 280.192.1-40.ca.2.3, 280.192.1-40.ca.2.4, 280.192.1-40.ca.2.5, 280.192.1-40.ca.2.6, 280.192.1-40.ca.2.7, 280.192.1-40.ca.2.8 |
Cyclic 40-isogeny field degree: |
$6$ |
Cyclic 40-torsion field degree: |
$96$ |
Full 40-torsion field degree: |
$7680$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 2 x^{2} + 2 y^{2} + z^{2} $ |
| $=$ | $3 x^{2} - 2 y^{2} + z w - w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 4 x^{4} - 12 x^{2} y^{2} + 6 x^{2} z^{2} + 9 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 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{2^4}{5^2}\cdot\frac{(61z^{8}-176z^{7}w+532z^{6}w^{2}-728z^{5}w^{3}+420z^{4}w^{4}-112z^{3}w^{5}+112z^{2}w^{6}-64zw^{7}+16w^{8})^{3}}{z^{4}(z-2w)^{4}(z^{2}+zw-w^{2})^{4}(3z^{2}-2zw+2w^{2})^{4}}$ |
Hi
|
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.