$\GL_2(\Z/40\Z)$-generators: |
$\begin{bmatrix}17&1\\29&6\end{bmatrix}$, $\begin{bmatrix}17&7\\10&33\end{bmatrix}$, $\begin{bmatrix}17&16\\34&13\end{bmatrix}$, $\begin{bmatrix}29&20\\10&29\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
40.80.1-40.j.1.1, 40.80.1-40.j.1.2, 40.80.1-40.j.1.3, 40.80.1-40.j.1.4, 120.80.1-40.j.1.1, 120.80.1-40.j.1.2, 120.80.1-40.j.1.3, 120.80.1-40.j.1.4, 280.80.1-40.j.1.1, 280.80.1-40.j.1.2, 280.80.1-40.j.1.3, 280.80.1-40.j.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} - 2 y^{2} - z^{2} - w^{2} $ |
| $=$ | $2 x^{2} + 4 y^{2} + 3 z^{2} + z w + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} - 14 x^{2} y^{2} + 10 x^{2} z^{2} + 49 y^{4} - 60 y^{2} z^{2} + 20 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:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle x$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{2}w$ |
Maps to other modular curves
$j$-invariant map
of degree 40 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle -5^4\,\frac{(z-w)^{3}(4z-3w)(4z^{2}-zw+w^{2})^{3}}{(z^{2}+zw-w^{2})^{5}}$ |
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.