$\GL_2(\Z/8\Z)$-generators: |
$\begin{bmatrix}1&0\\0&3\end{bmatrix}$, $\begin{bmatrix}3&4\\0&5\end{bmatrix}$, $\begin{bmatrix}7&2\\4&1\end{bmatrix}$, $\begin{bmatrix}7&4\\0&5\end{bmatrix}$ |
$\GL_2(\Z/8\Z)$-subgroup: |
$C_2^2\times D_4$ |
Contains $-I$: |
yes |
Quadratic refinements: |
8.96.1-8.e.1.1, 8.96.1-8.e.1.2, 8.96.1-8.e.1.3, 8.96.1-8.e.1.4, 8.96.1-8.e.1.5, 8.96.1-8.e.1.6, 16.96.1-8.e.1.1, 16.96.1-8.e.1.2, 16.96.1-8.e.1.3, 16.96.1-8.e.1.4, 24.96.1-8.e.1.1, 24.96.1-8.e.1.2, 24.96.1-8.e.1.3, 24.96.1-8.e.1.4, 24.96.1-8.e.1.5, 24.96.1-8.e.1.6, 40.96.1-8.e.1.1, 40.96.1-8.e.1.2, 40.96.1-8.e.1.3, 40.96.1-8.e.1.4, 40.96.1-8.e.1.5, 40.96.1-8.e.1.6, 48.96.1-8.e.1.1, 48.96.1-8.e.1.2, 48.96.1-8.e.1.3, 48.96.1-8.e.1.4, 56.96.1-8.e.1.1, 56.96.1-8.e.1.2, 56.96.1-8.e.1.3, 56.96.1-8.e.1.4, 56.96.1-8.e.1.5, 56.96.1-8.e.1.6, 80.96.1-8.e.1.1, 80.96.1-8.e.1.2, 80.96.1-8.e.1.3, 80.96.1-8.e.1.4, 88.96.1-8.e.1.1, 88.96.1-8.e.1.2, 88.96.1-8.e.1.3, 88.96.1-8.e.1.4, 88.96.1-8.e.1.5, 88.96.1-8.e.1.6, 104.96.1-8.e.1.1, 104.96.1-8.e.1.2, 104.96.1-8.e.1.3, 104.96.1-8.e.1.4, 104.96.1-8.e.1.5, 104.96.1-8.e.1.6, 112.96.1-8.e.1.1, 112.96.1-8.e.1.2, 112.96.1-8.e.1.3, 112.96.1-8.e.1.4, 120.96.1-8.e.1.1, 120.96.1-8.e.1.2, 120.96.1-8.e.1.3, 120.96.1-8.e.1.4, 120.96.1-8.e.1.5, 120.96.1-8.e.1.6, 136.96.1-8.e.1.1, 136.96.1-8.e.1.2, 136.96.1-8.e.1.3, 136.96.1-8.e.1.4, 136.96.1-8.e.1.5, 136.96.1-8.e.1.6, 152.96.1-8.e.1.1, 152.96.1-8.e.1.2, 152.96.1-8.e.1.3, 152.96.1-8.e.1.4, 152.96.1-8.e.1.5, 152.96.1-8.e.1.6, 168.96.1-8.e.1.1, 168.96.1-8.e.1.2, 168.96.1-8.e.1.3, 168.96.1-8.e.1.4, 168.96.1-8.e.1.5, 168.96.1-8.e.1.6, 176.96.1-8.e.1.1, 176.96.1-8.e.1.2, 176.96.1-8.e.1.3, 176.96.1-8.e.1.4, 184.96.1-8.e.1.1, 184.96.1-8.e.1.2, 184.96.1-8.e.1.3, 184.96.1-8.e.1.4, 184.96.1-8.e.1.5, 184.96.1-8.e.1.6, 208.96.1-8.e.1.1, 208.96.1-8.e.1.2, 208.96.1-8.e.1.3, 208.96.1-8.e.1.4, 232.96.1-8.e.1.1, 232.96.1-8.e.1.2, 232.96.1-8.e.1.3, 232.96.1-8.e.1.4, 232.96.1-8.e.1.5, 232.96.1-8.e.1.6, 240.96.1-8.e.1.1, 240.96.1-8.e.1.2, 240.96.1-8.e.1.3, 240.96.1-8.e.1.4, 248.96.1-8.e.1.1, 248.96.1-8.e.1.2, 248.96.1-8.e.1.3, 248.96.1-8.e.1.4, 248.96.1-8.e.1.5, 248.96.1-8.e.1.6, 264.96.1-8.e.1.1, 264.96.1-8.e.1.2, 264.96.1-8.e.1.3, 264.96.1-8.e.1.4, 264.96.1-8.e.1.5, 264.96.1-8.e.1.6, 272.96.1-8.e.1.1, 272.96.1-8.e.1.2, 272.96.1-8.e.1.3, 272.96.1-8.e.1.4, 280.96.1-8.e.1.1, 280.96.1-8.e.1.2, 280.96.1-8.e.1.3, 280.96.1-8.e.1.4, 280.96.1-8.e.1.5, 280.96.1-8.e.1.6, 296.96.1-8.e.1.1, 296.96.1-8.e.1.2, 296.96.1-8.e.1.3, 296.96.1-8.e.1.4, 296.96.1-8.e.1.5, 296.96.1-8.e.1.6, 304.96.1-8.e.1.1, 304.96.1-8.e.1.2, 304.96.1-8.e.1.3, 304.96.1-8.e.1.4, 312.96.1-8.e.1.1, 312.96.1-8.e.1.2, 312.96.1-8.e.1.3, 312.96.1-8.e.1.4, 312.96.1-8.e.1.5, 312.96.1-8.e.1.6, 328.96.1-8.e.1.1, 328.96.1-8.e.1.2, 328.96.1-8.e.1.3, 328.96.1-8.e.1.4, 328.96.1-8.e.1.5, 328.96.1-8.e.1.6 |
Cyclic 8-isogeny field degree: |
$2$ |
Cyclic 8-torsion field degree: |
$8$ |
Full 8-torsion field degree: |
$32$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 2 x y + z^{2} $ |
| $=$ | $2 x^{2} - 2 x y + 4 y^{2} + 2 z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} + x^{2} y^{2} + 3 x^{2} z^{2} + 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:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{2}w$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{2}z$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle -2^4\,\frac{126y^{2}z^{10}+378y^{2}z^{8}w^{2}+36y^{2}z^{6}w^{4}-36y^{2}z^{4}w^{6}-378y^{2}z^{2}w^{8}-126y^{2}w^{10}+31z^{12}+60z^{10}w^{2}-48z^{8}w^{4}-64z^{6}w^{6}-255z^{4}w^{8}-192z^{2}w^{10}-32w^{12}}{w^{4}z^{4}(2y^{2}z^{2}-2y^{2}w^{2}+z^{4})}$ |
Hi
|
Cover information
Click on a modular curve in the diagram to see information about it.
|
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.