$\GL_2(\Z/40\Z)$-generators: |
$\begin{bmatrix}5&19\\2&39\end{bmatrix}$, $\begin{bmatrix}23&36\\2&5\end{bmatrix}$, $\begin{bmatrix}31&13\\24&25\end{bmatrix}$, $\begin{bmatrix}31&25\\10&1\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
40.96.1-40.cm.1.1, 40.96.1-40.cm.1.2, 40.96.1-40.cm.1.3, 40.96.1-40.cm.1.4, 120.96.1-40.cm.1.1, 120.96.1-40.cm.1.2, 120.96.1-40.cm.1.3, 120.96.1-40.cm.1.4, 280.96.1-40.cm.1.1, 280.96.1-40.cm.1.2, 280.96.1-40.cm.1.3, 280.96.1-40.cm.1.4 |
Cyclic 40-isogeny field degree: |
$24$ |
Cyclic 40-torsion field degree: |
$384$ |
Full 40-torsion field degree: |
$15360$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 3 x^{2} + 2 x y + 2 y^{2} + z^{2} $ |
| $=$ | $2 x^{2} - 2 x y - 2 y^{2} - 3 z^{2} + 2 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 484 x^{4} + 180 x^{2} y^{2} - 132 x^{2} z^{2} + 25 y^{4} - 20 y^{2} z^{2} + 9 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:
$\displaystyle X$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle 2y$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle 2w$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle 2^6\,\frac{(z^{2}+w^{2})^{3}(3z^{2}-w^{2})^{3}}{z^{4}(z-w)^{4}(z+w)^{4}}$ |
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.