$\GL_2(\Z/40\Z)$-generators: |
$\begin{bmatrix}1&4\\4&15\end{bmatrix}$, $\begin{bmatrix}11&0\\20&3\end{bmatrix}$, $\begin{bmatrix}13&16\\30&1\end{bmatrix}$, $\begin{bmatrix}27&24\\34&7\end{bmatrix}$, $\begin{bmatrix}33&0\\26&7\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
40.96.1-40.y.1.1, 40.96.1-40.y.1.2, 40.96.1-40.y.1.3, 40.96.1-40.y.1.4, 40.96.1-40.y.1.5, 40.96.1-40.y.1.6, 40.96.1-40.y.1.7, 40.96.1-40.y.1.8, 40.96.1-40.y.1.9, 40.96.1-40.y.1.10, 40.96.1-40.y.1.11, 40.96.1-40.y.1.12, 40.96.1-40.y.1.13, 40.96.1-40.y.1.14, 40.96.1-40.y.1.15, 40.96.1-40.y.1.16, 120.96.1-40.y.1.1, 120.96.1-40.y.1.2, 120.96.1-40.y.1.3, 120.96.1-40.y.1.4, 120.96.1-40.y.1.5, 120.96.1-40.y.1.6, 120.96.1-40.y.1.7, 120.96.1-40.y.1.8, 120.96.1-40.y.1.9, 120.96.1-40.y.1.10, 120.96.1-40.y.1.11, 120.96.1-40.y.1.12, 120.96.1-40.y.1.13, 120.96.1-40.y.1.14, 120.96.1-40.y.1.15, 120.96.1-40.y.1.16, 280.96.1-40.y.1.1, 280.96.1-40.y.1.2, 280.96.1-40.y.1.3, 280.96.1-40.y.1.4, 280.96.1-40.y.1.5, 280.96.1-40.y.1.6, 280.96.1-40.y.1.7, 280.96.1-40.y.1.8, 280.96.1-40.y.1.9, 280.96.1-40.y.1.10, 280.96.1-40.y.1.11, 280.96.1-40.y.1.12, 280.96.1-40.y.1.13, 280.96.1-40.y.1.14, 280.96.1-40.y.1.15, 280.96.1-40.y.1.16 |
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 $ | $=$ | $ 5 x^{2} + 5 x y - 2 w^{2} $ |
| $=$ | $5 x y - 5 y^{2} + z^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 2 x^{4} - 10 x^{2} y^{2} - 15 x^{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:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{5}w$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{5}z$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle -2^2\,\frac{315y^{2}z^{10}+1890y^{2}z^{8}w^{2}+360y^{2}z^{6}w^{4}-720y^{2}z^{4}w^{6}-15120y^{2}z^{2}w^{8}-10080y^{2}w^{10}-31z^{12}-120z^{10}w^{2}+192z^{8}w^{4}+512z^{6}w^{6}+4080z^{4}w^{8}+6144z^{2}w^{10}+2048w^{12}}{w^{4}z^{4}(5y^{2}z^{2}-10y^{2}w^{2}-z^{4})}$ |
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.