$\GL_2(\Z/40\Z)$-generators: |
$\begin{bmatrix}11&16\\10&7\end{bmatrix}$, $\begin{bmatrix}33&30\\6&7\end{bmatrix}$, $\begin{bmatrix}37&0\\24&33\end{bmatrix}$, $\begin{bmatrix}37&32\\18&17\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
40.96.1-40.bm.1.1, 40.96.1-40.bm.1.2, 40.96.1-40.bm.1.3, 40.96.1-40.bm.1.4, 120.96.1-40.bm.1.1, 120.96.1-40.bm.1.2, 120.96.1-40.bm.1.3, 120.96.1-40.bm.1.4, 280.96.1-40.bm.1.1, 280.96.1-40.bm.1.2, 280.96.1-40.bm.1.3, 280.96.1-40.bm.1.4 |
Cyclic 40-isogeny field degree: |
$12$ |
Cyclic 40-torsion field degree: |
$192$ |
Full 40-torsion field degree: |
$15360$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 2 x y + w^{2} $ |
| $=$ | $5 x^{2} + 5 y^{2} + z^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 4 x^{4} + 5 x^{2} y^{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:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle \frac{2}{5}z$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{2^6}{5^4}\cdot\frac{(2z^{2}-5w^{2})^{3}(2z^{2}+5w^{2})^{3}}{w^{8}z^{4}}$ |
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.