$\GL_2(\Z/40\Z)$-generators: |
$\begin{bmatrix}21&28\\9&11\end{bmatrix}$, $\begin{bmatrix}25&28\\26&13\end{bmatrix}$, $\begin{bmatrix}25&36\\1&3\end{bmatrix}$, $\begin{bmatrix}39&0\\4&19\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
40.96.1-40.dx.1.1, 40.96.1-40.dx.1.2, 40.96.1-40.dx.1.3, 40.96.1-40.dx.1.4, 120.96.1-40.dx.1.1, 120.96.1-40.dx.1.2, 120.96.1-40.dx.1.3, 120.96.1-40.dx.1.4, 280.96.1-40.dx.1.1, 280.96.1-40.dx.1.2, 280.96.1-40.dx.1.3, 280.96.1-40.dx.1.4 |
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 $ | $=$ | $ x^{2} + x z - 5 y^{2} - z^{2} $ |
| $=$ | $8 x^{2} + 3 x z + 5 y^{2} - 3 z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 81 x^{4} + 22 x^{2} y^{2} + 135 x^{2} z^{2} + y^{4} + 10 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:
$\displaystyle X$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle 9y$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{3}{5}w$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{2^4\cdot3^3}{5^2}\cdot\frac{2136706000000xz^{11}-2948328000000xz^{9}w^{2}+1478324520000xz^{7}w^{4}-319202856000xz^{5}w^{6}+27085120200xz^{3}w^{8}-694416240xzw^{10}-1027861000000z^{12}+1589913400000z^{10}w^{2}-939471210000z^{8}w^{4}+260990100000z^{6}w^{6}-33433908300z^{4}w^{8}+1672661340z^{2}w^{10}-20253807w^{12}}{21367060000xz^{11}+455535000xz^{9}w^{2}-819468900xz^{7}w^{4}-78177960xz^{5}w^{6}+3359232xz^{3}w^{8}+708588xzw^{10}-10278610000z^{12}+1453027000z^{10}w^{2}+519256575z^{8}w^{4}-11983950z^{6}w^{6}-8388603z^{4}w^{8}-669222z^{2}w^{10}-59049w^{12}}$ |
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.