$\GL_2(\Z/40\Z)$-generators: |
$\begin{bmatrix}13&20\\24&9\end{bmatrix}$, $\begin{bmatrix}21&6\\30&27\end{bmatrix}$, $\begin{bmatrix}23&12\\15&29\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
80.96.1-40.fo.1.1, 80.96.1-40.fo.1.2, 80.96.1-40.fo.1.3, 80.96.1-40.fo.1.4, 80.96.1-40.fo.1.5, 80.96.1-40.fo.1.6, 80.96.1-40.fo.1.7, 80.96.1-40.fo.1.8, 240.96.1-40.fo.1.1, 240.96.1-40.fo.1.2, 240.96.1-40.fo.1.3, 240.96.1-40.fo.1.4, 240.96.1-40.fo.1.5, 240.96.1-40.fo.1.6, 240.96.1-40.fo.1.7, 240.96.1-40.fo.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 $ | $=$ | $ 5 x^{2} + y^{2} - y z - z^{2} $ |
| $=$ | $5 x^{2} - 2 y^{2} + 7 y z + 7 z^{2} + 2 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} + 14 x^{2} y^{2} + 30 x^{2} z^{2} + 9 y^{4} + 60 y^{2} z^{2} + 100 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
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle 2^4\cdot3^3\,\frac{442442000000yz^{11}+360165000000yz^{9}w^{2}+113705640000yz^{7}w^{4}+17223300000yz^{5}w^{6}+1230568200yz^{3}w^{8}+32148900yzw^{10}+342827000000z^{12}+349031000000z^{10}w^{2}+142429770000z^{8}w^{4}+29386980000z^{6}w^{6}+3145929300z^{4}w^{8}+155641500z^{2}w^{10}+2250423w^{12}}{110610500000yz^{11}+45214500000yz^{9}w^{2}+6483780000yz^{7}w^{4}+382941000yz^{5}w^{6}+8355150yz^{3}w^{8}+43740yzw^{10}+85706750000z^{12}+52523600000z^{10}w^{2}+11517165000z^{8}w^{4}+1102995000z^{6}w^{6}+44075475z^{4}w^{8}+566190z^{2}w^{10}+972w^{12}}$ |
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.