$\GL_2(\Z/40\Z)$-generators: |
$\begin{bmatrix}7&37\\8&25\end{bmatrix}$, $\begin{bmatrix}11&13\\8&37\end{bmatrix}$, $\begin{bmatrix}29&28\\32&25\end{bmatrix}$, $\begin{bmatrix}29&35\\8&39\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
40.192.1-40.cj.1.1, 40.192.1-40.cj.1.2, 40.192.1-40.cj.1.3, 40.192.1-40.cj.1.4, 80.192.1-40.cj.1.1, 80.192.1-40.cj.1.2, 80.192.1-40.cj.1.3, 80.192.1-40.cj.1.4, 120.192.1-40.cj.1.1, 120.192.1-40.cj.1.2, 120.192.1-40.cj.1.3, 120.192.1-40.cj.1.4, 240.192.1-40.cj.1.1, 240.192.1-40.cj.1.2, 240.192.1-40.cj.1.3, 240.192.1-40.cj.1.4, 280.192.1-40.cj.1.1, 280.192.1-40.cj.1.2, 280.192.1-40.cj.1.3, 280.192.1-40.cj.1.4 |
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 $ | $=$ | $ 5 x y + z w $ |
| $=$ | $5 x^{2} - 5 y^{2} - z^{2} - w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 5 x^{4} + 25 x^{2} y^{2} + x^{2} z^{2} - 5 y^{2} z^{2} $ |
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{5}z$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle -2^2\,\frac{(z^{2}-2zw-w^{2})^{3}(z^{2}+2zw-w^{2})^{3}(z^{4}+10z^{2}w^{2}+w^{4})^{3}}{w^{4}z^{4}(z^{2}+w^{2})^{8}}$ |
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.