$\GL_2(\Z/40\Z)$-generators: |
$\begin{bmatrix}5&14\\18&23\end{bmatrix}$, $\begin{bmatrix}9&38\\19&7\end{bmatrix}$, $\begin{bmatrix}17&2\\20&3\end{bmatrix}$, $\begin{bmatrix}19&10\\6&17\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
40.96.1-40.fh.1.1, 40.96.1-40.fh.1.2, 40.96.1-40.fh.1.3, 40.96.1-40.fh.1.4, 120.96.1-40.fh.1.1, 120.96.1-40.fh.1.2, 120.96.1-40.fh.1.3, 120.96.1-40.fh.1.4, 280.96.1-40.fh.1.1, 280.96.1-40.fh.1.2, 280.96.1-40.fh.1.3, 280.96.1-40.fh.1.4 |
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 $ | $=$ | $ 5 x^{2} + 3 y^{2} + 2 y z + 2 z^{2} $ |
| $=$ | $5 x^{2} - 4 y^{2} - 6 y z - 6 z^{2} - 2 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} - 12 x^{2} y^{2} + 20 x^{2} z^{2} + 196 y^{4} - 420 y^{2} z^{2} + 225 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:
$\displaystyle X$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{2}x$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{5}w$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle 2^4\,\frac{233766000000yz^{11}-1167075000000yz^{9}w^{2}-4842127080000yz^{7}w^{4}-3116065820000yz^{5}w^{6}+591905564600yz^{3}w^{8}+515529514500yzw^{10}+535221000000z^{12}+1406295000000z^{10}w^{2}-1881774090000z^{8}w^{4}-4862237660000z^{6}w^{6}-2202519174100z^{4}w^{8}-87861473700z^{2}w^{10}+49147147049w^{12}}{2164500000yz^{11}+917000000yz^{9}w^{2}+959420000yz^{7}w^{4}+482601000yz^{5}w^{6}+152103350yz^{3}w^{8}-5546310yzw^{10}+4955750000z^{12}+5605100000z^{10}w^{2}+3307535000z^{8}w^{4}+1037575000z^{6}w^{6}+138717775z^{4}w^{8}-25834760z^{2}w^{10}+151263w^{12}}$ |
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.