$\GL_2(\Z/40\Z)$-generators: |
$\begin{bmatrix}9&14\\16&27\end{bmatrix}$, $\begin{bmatrix}11&5\\14&37\end{bmatrix}$, $\begin{bmatrix}31&17\\24&19\end{bmatrix}$, $\begin{bmatrix}35&26\\26&15\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
80.144.1-40.bc.1.1, 80.144.1-40.bc.1.2, 80.144.1-40.bc.1.3, 80.144.1-40.bc.1.4, 80.144.1-40.bc.1.5, 80.144.1-40.bc.1.6, 80.144.1-40.bc.1.7, 80.144.1-40.bc.1.8, 80.144.1-40.bc.1.9, 80.144.1-40.bc.1.10, 80.144.1-40.bc.1.11, 80.144.1-40.bc.1.12, 80.144.1-40.bc.1.13, 80.144.1-40.bc.1.14, 80.144.1-40.bc.1.15, 80.144.1-40.bc.1.16, 240.144.1-40.bc.1.1, 240.144.1-40.bc.1.2, 240.144.1-40.bc.1.3, 240.144.1-40.bc.1.4, 240.144.1-40.bc.1.5, 240.144.1-40.bc.1.6, 240.144.1-40.bc.1.7, 240.144.1-40.bc.1.8, 240.144.1-40.bc.1.9, 240.144.1-40.bc.1.10, 240.144.1-40.bc.1.11, 240.144.1-40.bc.1.12, 240.144.1-40.bc.1.13, 240.144.1-40.bc.1.14, 240.144.1-40.bc.1.15, 240.144.1-40.bc.1.16 |
Cyclic 40-isogeny field degree: |
$4$ |
Cyclic 40-torsion field degree: |
$64$ |
Full 40-torsion field degree: |
$10240$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ x^{2} - 3 x z + y^{2} - 5 z^{2} + 2 w^{2} $ |
| $=$ | $5 x^{2} + 5 x z + y^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 5 x^{4} - 10 x^{2} y^{2} + 2 x^{2} z^{2} + z^{4} $ |
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:
$\displaystyle X$ |
$=$ |
$\displaystyle x$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle w$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle y$ |
Maps to other modular curves
$j$-invariant map
of degree 72 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle 2^6\,\frac{108xz^{17}-864xz^{15}w^{2}-2880xz^{13}w^{4}+56576xz^{11}w^{6}-271360xz^{9}w^{8}+663552xz^{7}w^{10}-905216xz^{5}w^{12}+655360xz^{3}w^{14}-196608xzw^{16}+405z^{18}-6912z^{16}w^{2}+48240z^{14}w^{4}-177568z^{12}w^{6}+364544z^{10}w^{8}-384000z^{8}w^{10}+105472z^{6}w^{12}+163840z^{4}w^{14}-147456z^{2}w^{16}+32768w^{18}}{z^{10}(z^{2}-2w^{2})^{2}(4xz^{3}-16xzw^{2}+15z^{4}-46z^{2}w^{2}+16w^{4})}$ |
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.