$\GL_2(\Z/48\Z)$-generators: |
$\begin{bmatrix}19&0\\4&17\end{bmatrix}$, $\begin{bmatrix}41&13\\8&27\end{bmatrix}$, $\begin{bmatrix}41&14\\4&11\end{bmatrix}$, $\begin{bmatrix}41&25\\8&15\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
48.192.1-48.cx.2.1, 48.192.1-48.cx.2.2, 48.192.1-48.cx.2.3, 48.192.1-48.cx.2.4, 48.192.1-48.cx.2.5, 48.192.1-48.cx.2.6, 48.192.1-48.cx.2.7, 48.192.1-48.cx.2.8, 240.192.1-48.cx.2.1, 240.192.1-48.cx.2.2, 240.192.1-48.cx.2.3, 240.192.1-48.cx.2.4, 240.192.1-48.cx.2.5, 240.192.1-48.cx.2.6, 240.192.1-48.cx.2.7, 240.192.1-48.cx.2.8 |
Cyclic 48-isogeny field degree: |
$8$ |
Cyclic 48-torsion field degree: |
$128$ |
Full 48-torsion field degree: |
$12288$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 2 x^{2} + 2 x y + 2 y^{2} + z^{2} $ |
| $=$ | $3 x^{2} - 2 z^{2} + 3 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 1764 x^{4} + 60 x^{2} y^{2} - 84 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:
$\displaystyle X$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle 6y$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle 6w$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle -\frac{2^8}{3}\cdot\frac{(z^{8}-48z^{6}w^{2}+180z^{4}w^{4}-216z^{2}w^{6}+81w^{8})^{3}}{w^{2}z^{16}(2z^{2}-3w^{2})^{2}(4z^{2}-3w^{2})}$ |
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.