$\GL_2(\Z/32\Z)$-generators: |
$\begin{bmatrix}3&19\\0&5\end{bmatrix}$, $\begin{bmatrix}5&31\\0&3\end{bmatrix}$, $\begin{bmatrix}15&11\\0&23\end{bmatrix}$, $\begin{bmatrix}23&24\\16&11\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
32.192.1-32.c.1.1, 32.192.1-32.c.1.2, 32.192.1-32.c.1.3, 32.192.1-32.c.1.4, 32.192.1-32.c.1.5, 32.192.1-32.c.1.6, 32.192.1-32.c.1.7, 32.192.1-32.c.1.8, 96.192.1-32.c.1.1, 96.192.1-32.c.1.2, 96.192.1-32.c.1.3, 96.192.1-32.c.1.4, 96.192.1-32.c.1.5, 96.192.1-32.c.1.6, 96.192.1-32.c.1.7, 96.192.1-32.c.1.8, 160.192.1-32.c.1.1, 160.192.1-32.c.1.2, 160.192.1-32.c.1.3, 160.192.1-32.c.1.4, 160.192.1-32.c.1.5, 160.192.1-32.c.1.6, 160.192.1-32.c.1.7, 160.192.1-32.c.1.8, 224.192.1-32.c.1.1, 224.192.1-32.c.1.2, 224.192.1-32.c.1.3, 224.192.1-32.c.1.4, 224.192.1-32.c.1.5, 224.192.1-32.c.1.6, 224.192.1-32.c.1.7, 224.192.1-32.c.1.8 |
Cyclic 32-isogeny field degree: |
$2$ |
Cyclic 32-torsion field degree: |
$32$ |
Full 32-torsion field degree: |
$4096$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 2 x y - z^{2} $ |
| $=$ | $4 x^{2} + y^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} + x^{2} y^{2} + 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:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle w$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle z$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle 2^4\,\frac{90y^{2}z^{20}w^{2}-33165y^{2}z^{16}w^{6}+155880y^{2}z^{12}w^{10}-135180y^{2}z^{8}w^{14}+40950y^{2}z^{4}w^{18}-4095y^{2}w^{22}+z^{24}-2757z^{20}w^{4}+82086z^{16}w^{8}-102148z^{12}w^{12}+36861z^{8}w^{16}-4107z^{4}w^{20}+w^{24}}{w^{2}z^{8}(2y^{2}z^{12}-9y^{2}z^{8}w^{4}+6y^{2}z^{4}w^{8}-y^{2}w^{12}+7z^{12}w^{2}-14z^{8}w^{6}+7z^{4}w^{10}-w^{14})}$ |
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.