$\GL_2(\Z/40\Z)$-generators: |
$\begin{bmatrix}3&24\\2&13\end{bmatrix}$, $\begin{bmatrix}9&38\\11&25\end{bmatrix}$, $\begin{bmatrix}21&6\\15&13\end{bmatrix}$, $\begin{bmatrix}27&30\\7&23\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
40.48.1-40.x.1.1, 40.48.1-40.x.1.2, 40.48.1-40.x.1.3, 40.48.1-40.x.1.4, 80.48.1-40.x.1.1, 80.48.1-40.x.1.2, 80.48.1-40.x.1.3, 80.48.1-40.x.1.4, 80.48.1-40.x.1.5, 80.48.1-40.x.1.6, 80.48.1-40.x.1.7, 80.48.1-40.x.1.8, 80.48.1-40.x.1.9, 80.48.1-40.x.1.10, 80.48.1-40.x.1.11, 80.48.1-40.x.1.12, 80.48.1-40.x.1.13, 80.48.1-40.x.1.14, 80.48.1-40.x.1.15, 80.48.1-40.x.1.16, 120.48.1-40.x.1.1, 120.48.1-40.x.1.2, 120.48.1-40.x.1.3, 120.48.1-40.x.1.4, 240.48.1-40.x.1.1, 240.48.1-40.x.1.2, 240.48.1-40.x.1.3, 240.48.1-40.x.1.4, 240.48.1-40.x.1.5, 240.48.1-40.x.1.6, 240.48.1-40.x.1.7, 240.48.1-40.x.1.8, 240.48.1-40.x.1.9, 240.48.1-40.x.1.10, 240.48.1-40.x.1.11, 240.48.1-40.x.1.12, 240.48.1-40.x.1.13, 240.48.1-40.x.1.14, 240.48.1-40.x.1.15, 240.48.1-40.x.1.16, 280.48.1-40.x.1.1, 280.48.1-40.x.1.2, 280.48.1-40.x.1.3, 280.48.1-40.x.1.4 |
Cyclic 40-isogeny field degree: |
$24$ |
Cyclic 40-torsion field degree: |
$384$ |
Full 40-torsion field degree: |
$30720$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 10 y^{2} - 4 z^{2} - w^{2} $ |
| $=$ | $40 x^{2} - z w$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} - 10 y^{2} z^{2} + 25 z^{4} $ |
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:
$\displaystyle X$ |
$=$ |
$\displaystyle x$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{4}y$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{20}w$ |
Maps to other modular curves
$j$-invariant map
of degree 24 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle 2^8\,\frac{(z^{2}+w^{2})^{3}}{w^{2}z^{4}}$ |
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.