$\GL_2(\Z/48\Z)$-generators: |
$\begin{bmatrix}17&19\\36&37\end{bmatrix}$, $\begin{bmatrix}17&35\\20&25\end{bmatrix}$, $\begin{bmatrix}23&31\\32&45\end{bmatrix}$, $\begin{bmatrix}25&9\\16&23\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
48.192.1-48.en.1.1, 48.192.1-48.en.1.2, 48.192.1-48.en.1.3, 48.192.1-48.en.1.4, 48.192.1-48.en.1.5, 48.192.1-48.en.1.6, 48.192.1-48.en.1.7, 48.192.1-48.en.1.8, 240.192.1-48.en.1.1, 240.192.1-48.en.1.2, 240.192.1-48.en.1.3, 240.192.1-48.en.1.4, 240.192.1-48.en.1.5, 240.192.1-48.en.1.6, 240.192.1-48.en.1.7, 240.192.1-48.en.1.8 |
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} - 2 y^{2} + z^{2} $ |
| $=$ | $9 x^{2} + 6 y^{2} + 3 z^{2} - w^{2}$ |
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known 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
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle -\frac{2}{3}\cdot\frac{(36z^{4}-144z^{3}w+108z^{2}w^{2}-24zw^{3}+w^{4})^{3}(36z^{4}+144z^{3}w+108z^{2}w^{2}+24zw^{3}+w^{4})^{3}}{w^{2}z^{2}(6z^{2}-w^{2})^{2}(6z^{2}+w^{2})^{8}}$ |
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.