$\GL_2(\Z/48\Z)$-generators: |
$\begin{bmatrix}17&17\\32&7\end{bmatrix}$, $\begin{bmatrix}17&42\\0&37\end{bmatrix}$, $\begin{bmatrix}19&33\\32&25\end{bmatrix}$, $\begin{bmatrix}25&42\\8&41\end{bmatrix}$, $\begin{bmatrix}45&26\\40&29\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
48.192.1-48.bk.2.1, 48.192.1-48.bk.2.2, 48.192.1-48.bk.2.3, 48.192.1-48.bk.2.4, 48.192.1-48.bk.2.5, 48.192.1-48.bk.2.6, 48.192.1-48.bk.2.7, 48.192.1-48.bk.2.8, 48.192.1-48.bk.2.9, 48.192.1-48.bk.2.10, 48.192.1-48.bk.2.11, 48.192.1-48.bk.2.12, 96.192.1-48.bk.2.1, 96.192.1-48.bk.2.2, 96.192.1-48.bk.2.3, 96.192.1-48.bk.2.4, 96.192.1-48.bk.2.5, 96.192.1-48.bk.2.6, 96.192.1-48.bk.2.7, 96.192.1-48.bk.2.8, 240.192.1-48.bk.2.1, 240.192.1-48.bk.2.2, 240.192.1-48.bk.2.3, 240.192.1-48.bk.2.4, 240.192.1-48.bk.2.5, 240.192.1-48.bk.2.6, 240.192.1-48.bk.2.7, 240.192.1-48.bk.2.8, 240.192.1-48.bk.2.9, 240.192.1-48.bk.2.10, 240.192.1-48.bk.2.11, 240.192.1-48.bk.2.12 |
Cyclic 48-isogeny field degree: |
$4$ |
Cyclic 48-torsion field degree: |
$64$ |
Full 48-torsion field degree: |
$12288$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ y^{2} + z^{2} - z w + w^{2} $ |
| $=$ | $3 x^{2} + 3 y^{2} - 2 z^{2} - z w - 2 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 225 x^{4} + 30 x^{2} y^{2} - 6 x^{2} z^{2} + y^{4} - y^{2} z^{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:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle 3x$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle 3w$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle -\frac{1}{3}\cdot\frac{(z^{8}-88z^{7}w+172z^{6}w^{2}-232z^{5}w^{3}+310z^{4}w^{4}-232z^{3}w^{5}+172z^{2}w^{6}-88zw^{7}+w^{8})^{3}}{(z-w)^{2}(z+w)^{4}(z^{2}-zw+w^{2})^{8}(5z^{2}-2zw+5w^{2})}$ |
Hi
|
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.