$\GL_2(\Z/48\Z)$-generators: |
$\begin{bmatrix}3&7\\20&3\end{bmatrix}$, $\begin{bmatrix}19&31\\44&19\end{bmatrix}$, $\begin{bmatrix}41&27\\4&17\end{bmatrix}$, $\begin{bmatrix}47&35\\24&29\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
48.192.1-48.bx.2.1, 48.192.1-48.bx.2.2, 48.192.1-48.bx.2.3, 48.192.1-48.bx.2.4, 48.192.1-48.bx.2.5, 48.192.1-48.bx.2.6, 48.192.1-48.bx.2.7, 48.192.1-48.bx.2.8, 240.192.1-48.bx.2.1, 240.192.1-48.bx.2.2, 240.192.1-48.bx.2.3, 240.192.1-48.bx.2.4, 240.192.1-48.bx.2.5, 240.192.1-48.bx.2.6, 240.192.1-48.bx.2.7, 240.192.1-48.bx.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 $ | $=$ | $ x^{2} - y^{2} + 2 z^{2} + w^{2} $ |
| $=$ | $4 x^{2} + 2 y^{2} + w^{2}$ |
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
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{1}{3}\cdot\frac{(16z^{8}+384z^{6}w^{2}+720z^{4}w^{4}+432z^{2}w^{6}+81w^{8})^{3}}{w^{2}z^{16}(4z^{2}+3w^{2})^{2}(8z^{2}+3w^{2})}$ |
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.