| $\GL_2(\Z/40\Z)$-generators: |
$\begin{bmatrix}1&31\\15&22\end{bmatrix}$, $\begin{bmatrix}17&37\\34&35\end{bmatrix}$, $\begin{bmatrix}23&20\\10&13\end{bmatrix}$ |
| Contains $-I$: |
yes |
| Quadratic refinements: |
80.48.1-40.cn.2.1, 80.48.1-40.cn.2.2, 80.48.1-40.cn.2.3, 80.48.1-40.cn.2.4, 80.48.1-40.cn.2.5, 80.48.1-40.cn.2.6, 80.48.1-40.cn.2.7, 80.48.1-40.cn.2.8, 240.48.1-40.cn.2.1, 240.48.1-40.cn.2.2, 240.48.1-40.cn.2.3, 240.48.1-40.cn.2.4, 240.48.1-40.cn.2.5, 240.48.1-40.cn.2.6, 240.48.1-40.cn.2.7, 240.48.1-40.cn.2.8 |
| 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 $ | $=$ | $ x^{2} + y z $ |
| $=$ | $11 x^{2} - 25 y^{2} - 11 y z - 5 z^{2} + 10 w^{2}$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 25 x^{4} - 22 x^{2} z^{2} - 10 y^{2} z^{2} + 5 z^{4} $ |
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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:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle w$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle z$ |
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.
