$\GL_2(\Z/40\Z)$-generators: |
$\begin{bmatrix}9&13\\28&19\end{bmatrix}$, $\begin{bmatrix}19&33\\10&17\end{bmatrix}$, $\begin{bmatrix}21&23\\12&17\end{bmatrix}$, $\begin{bmatrix}31&0\\38&3\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
80.144.1-40.bn.1.1, 80.144.1-40.bn.1.2, 80.144.1-40.bn.1.3, 80.144.1-40.bn.1.4, 80.144.1-40.bn.1.5, 80.144.1-40.bn.1.6, 80.144.1-40.bn.1.7, 80.144.1-40.bn.1.8, 80.144.1-40.bn.1.9, 80.144.1-40.bn.1.10, 80.144.1-40.bn.1.11, 80.144.1-40.bn.1.12, 80.144.1-40.bn.1.13, 80.144.1-40.bn.1.14, 80.144.1-40.bn.1.15, 80.144.1-40.bn.1.16, 240.144.1-40.bn.1.1, 240.144.1-40.bn.1.2, 240.144.1-40.bn.1.3, 240.144.1-40.bn.1.4, 240.144.1-40.bn.1.5, 240.144.1-40.bn.1.6, 240.144.1-40.bn.1.7, 240.144.1-40.bn.1.8, 240.144.1-40.bn.1.9, 240.144.1-40.bn.1.10, 240.144.1-40.bn.1.11, 240.144.1-40.bn.1.12, 240.144.1-40.bn.1.13, 240.144.1-40.bn.1.14, 240.144.1-40.bn.1.15, 240.144.1-40.bn.1.16 |
Cyclic 40-isogeny field degree: |
$4$ |
Cyclic 40-torsion field degree: |
$64$ |
Full 40-torsion field degree: |
$10240$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 2 x^{2} + z^{2} + z w $ |
| $=$ | $4 x^{2} + 10 y^{2} - 2 z^{2} - 6 z w - 5 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 5 x^{4} + 4 x^{2} z^{2} - 10 y^{2} z^{2} + 4 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:
$\displaystyle X$ |
$=$ |
$\displaystyle x$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{2}z$ |
Maps to other modular curves
$j$-invariant map
of degree 72 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{(16z^{6}+80z^{5}w+160z^{4}w^{2}+160z^{3}w^{3}+80z^{2}w^{4}+20zw^{5}+5w^{6})^{3}}{w^{5}z^{2}(z+w)^{10}(4z+5w)}$ |
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.