$\GL_2(\Z/48\Z)$-generators: |
$\begin{bmatrix}1&17\\16&31\end{bmatrix}$, $\begin{bmatrix}9&29\\28&33\end{bmatrix}$, $\begin{bmatrix}25&27\\0&31\end{bmatrix}$, $\begin{bmatrix}39&35\\4&31\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
48.192.1-48.y.2.1, 48.192.1-48.y.2.2, 48.192.1-48.y.2.3, 48.192.1-48.y.2.4, 48.192.1-48.y.2.5, 48.192.1-48.y.2.6, 48.192.1-48.y.2.7, 48.192.1-48.y.2.8, 240.192.1-48.y.2.1, 240.192.1-48.y.2.2, 240.192.1-48.y.2.3, 240.192.1-48.y.2.4, 240.192.1-48.y.2.5, 240.192.1-48.y.2.6, 240.192.1-48.y.2.7, 240.192.1-48.y.2.8 |
Cyclic 48-isogeny field degree: |
$8$ |
Cyclic 48-torsion field degree: |
$64$ |
Full 48-torsion field degree: |
$12288$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ x^{2} - 2 y^{2} - z^{2} + w^{2} $ |
| $=$ | $x^{2} + 4 y^{2} - 4 z^{2} + 2 w^{2}$ |
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
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle -\frac{2^4}{3}\cdot\frac{(81z^{8}-108z^{6}w^{2}-90z^{4}w^{4}+84z^{2}w^{6}+w^{8})^{3}}{w^{4}z^{2}(3z^{2}-2w^{2})(3z^{2}-w^{2})^{8}}$ |
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.