$\GL_2(\Z/40\Z)$-generators: |
$\begin{bmatrix}3&16\\4&25\end{bmatrix}$, $\begin{bmatrix}28&5\\27&21\end{bmatrix}$, $\begin{bmatrix}33&24\\4&13\end{bmatrix}$, $\begin{bmatrix}35&33\\17&16\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
40.48.1-40.bw.1.1, 40.48.1-40.bw.1.2, 40.48.1-40.bw.1.3, 40.48.1-40.bw.1.4, 120.48.1-40.bw.1.1, 120.48.1-40.bw.1.2, 120.48.1-40.bw.1.3, 120.48.1-40.bw.1.4, 280.48.1-40.bw.1.1, 280.48.1-40.bw.1.2, 280.48.1-40.bw.1.3, 280.48.1-40.bw.1.4 |
Cyclic 40-isogeny field degree: |
$12$ |
Cyclic 40-torsion field degree: |
$192$ |
Full 40-torsion field degree: |
$30720$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 4 x^{2} - 5 y^{2} - 4 y z - z^{2} - 2 w^{2} $ |
| $=$ | $6 x^{2} + 5 y^{2} + 5 y z + z^{2} + 2 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 125 x^{4} - 44 x^{2} z^{2} + 2 y^{2} z^{2} + 4 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:
$\displaystyle X$ |
$=$ |
$\displaystyle x$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle w$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{2}z$ |
Maps to other modular curves
$j$-invariant map
of degree 24 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle 2^3\,\frac{14770296yz^{5}-391419000yz^{3}w^{2}+5343750yzw^{4}+5559840z^{6}-268262820z^{4}w^{2}+51547500z^{2}w^{4}-78125w^{6}}{z(68381yz^{4}-300500yz^{2}w^{2}+62500yw^{4}+25740z^{5}-3520z^{3}w^{2}-110000zw^{4})}$ |
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.