$\GL_2(\Z/40\Z)$-generators: |
$\begin{bmatrix}5&22\\24&23\end{bmatrix}$, $\begin{bmatrix}11&2\\30&17\end{bmatrix}$, $\begin{bmatrix}15&4\\9&13\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
80.96.1-40.gt.1.1, 80.96.1-40.gt.1.2, 80.96.1-40.gt.1.3, 80.96.1-40.gt.1.4, 80.96.1-40.gt.1.5, 80.96.1-40.gt.1.6, 80.96.1-40.gt.1.7, 80.96.1-40.gt.1.8, 240.96.1-40.gt.1.1, 240.96.1-40.gt.1.2, 240.96.1-40.gt.1.3, 240.96.1-40.gt.1.4, 240.96.1-40.gt.1.5, 240.96.1-40.gt.1.6, 240.96.1-40.gt.1.7, 240.96.1-40.gt.1.8 |
Cyclic 40-isogeny field degree: |
$24$ |
Cyclic 40-torsion field degree: |
$384$ |
Full 40-torsion field degree: |
$15360$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 3 y^{2} - 2 y z + 2 z^{2} + w^{2} $ |
| $=$ | $5 x^{2} + y^{2} + y z - z^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} - 4 x^{2} y^{2} + 15 x^{2} z^{2} + 9 y^{4} - 30 y^{2} z^{2} + 25 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:
$\displaystyle X$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle x$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{5}w$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle -\frac{2^4}{3^8}\cdot\frac{19712000000yz^{11}+29568000000yz^{9}w^{2}+16277760000yz^{7}w^{4}+4005504000yz^{5}w^{6}+356270400yz^{3}w^{8}-5715360yzw^{10}-1472000000z^{12}-678400000z^{10}w^{2}-254880000z^{8}w^{4}-636480000z^{6}w^{6}-313707600z^{4}w^{8}-42593040z^{2}w^{10}+83349w^{12}}{w^{8}(100yz^{3}+30yzw^{2}-25z^{4}-5z^{2}w^{2}-3w^{4})}$ |
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.