$\GL_2(\Z/8\Z)$-generators: |
$\begin{bmatrix}1&4\\0&1\end{bmatrix}$, $\begin{bmatrix}3&6\\0&1\end{bmatrix}$, $\begin{bmatrix}7&4\\0&7\end{bmatrix}$, $\begin{bmatrix}7&6\\4&1\end{bmatrix}$ |
$\GL_2(\Z/8\Z)$-subgroup: |
$C_2^4$ |
Contains $-I$: |
yes |
Quadratic refinements: |
8.192.1-8.a.1.1, 8.192.1-8.a.1.2, 8.192.1-8.a.1.3, 8.192.1-8.a.1.4, 8.192.1-8.a.1.5, 8.192.1-8.a.1.6, 24.192.1-8.a.1.1, 24.192.1-8.a.1.2, 24.192.1-8.a.1.3, 24.192.1-8.a.1.4, 24.192.1-8.a.1.5, 24.192.1-8.a.1.6, 40.192.1-8.a.1.1, 40.192.1-8.a.1.2, 40.192.1-8.a.1.3, 40.192.1-8.a.1.4, 40.192.1-8.a.1.5, 40.192.1-8.a.1.6, 56.192.1-8.a.1.1, 56.192.1-8.a.1.2, 56.192.1-8.a.1.3, 56.192.1-8.a.1.4, 56.192.1-8.a.1.5, 56.192.1-8.a.1.6, 88.192.1-8.a.1.1, 88.192.1-8.a.1.2, 88.192.1-8.a.1.3, 88.192.1-8.a.1.4, 88.192.1-8.a.1.5, 88.192.1-8.a.1.6, 104.192.1-8.a.1.1, 104.192.1-8.a.1.2, 104.192.1-8.a.1.3, 104.192.1-8.a.1.4, 104.192.1-8.a.1.5, 104.192.1-8.a.1.6, 120.192.1-8.a.1.1, 120.192.1-8.a.1.2, 120.192.1-8.a.1.3, 120.192.1-8.a.1.4, 120.192.1-8.a.1.5, 120.192.1-8.a.1.6, 136.192.1-8.a.1.1, 136.192.1-8.a.1.2, 136.192.1-8.a.1.3, 136.192.1-8.a.1.4, 136.192.1-8.a.1.5, 136.192.1-8.a.1.6, 152.192.1-8.a.1.1, 152.192.1-8.a.1.2, 152.192.1-8.a.1.3, 152.192.1-8.a.1.4, 152.192.1-8.a.1.5, 152.192.1-8.a.1.6, 168.192.1-8.a.1.1, 168.192.1-8.a.1.2, 168.192.1-8.a.1.3, 168.192.1-8.a.1.4, 168.192.1-8.a.1.5, 168.192.1-8.a.1.6, 184.192.1-8.a.1.1, 184.192.1-8.a.1.2, 184.192.1-8.a.1.3, 184.192.1-8.a.1.4, 184.192.1-8.a.1.5, 184.192.1-8.a.1.6, 232.192.1-8.a.1.1, 232.192.1-8.a.1.2, 232.192.1-8.a.1.3, 232.192.1-8.a.1.4, 232.192.1-8.a.1.5, 232.192.1-8.a.1.6, 248.192.1-8.a.1.1, 248.192.1-8.a.1.2, 248.192.1-8.a.1.3, 248.192.1-8.a.1.4, 248.192.1-8.a.1.5, 248.192.1-8.a.1.6, 264.192.1-8.a.1.1, 264.192.1-8.a.1.2, 264.192.1-8.a.1.3, 264.192.1-8.a.1.4, 264.192.1-8.a.1.5, 264.192.1-8.a.1.6, 280.192.1-8.a.1.1, 280.192.1-8.a.1.2, 280.192.1-8.a.1.3, 280.192.1-8.a.1.4, 280.192.1-8.a.1.5, 280.192.1-8.a.1.6, 296.192.1-8.a.1.1, 296.192.1-8.a.1.2, 296.192.1-8.a.1.3, 296.192.1-8.a.1.4, 296.192.1-8.a.1.5, 296.192.1-8.a.1.6, 312.192.1-8.a.1.1, 312.192.1-8.a.1.2, 312.192.1-8.a.1.3, 312.192.1-8.a.1.4, 312.192.1-8.a.1.5, 312.192.1-8.a.1.6, 328.192.1-8.a.1.1, 328.192.1-8.a.1.2, 328.192.1-8.a.1.3, 328.192.1-8.a.1.4, 328.192.1-8.a.1.5, 328.192.1-8.a.1.6 |
Cyclic 8-isogeny field degree: |
$2$ |
Cyclic 8-torsion field degree: |
$8$ |
Full 8-torsion field degree: |
$16$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ x^{2} - z^{2} + w^{2} $ |
| $=$ | $x^{2} + 2 y^{2} + z^{2}$ |
This modular curve has no real points, and therefore no rational points.
Maps to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle 2^4\,\frac{(16z^{8}-32z^{6}w^{2}+20z^{4}w^{4}-4z^{2}w^{6}+w^{8})^{3}}{w^{8}z^{4}(z-w)^{2}(z+w)^{2}(2z^{2}-w^{2})^{4}}$ |
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.