$\GL_2(\Z/48\Z)$-generators: |
$\begin{bmatrix}7&2\\24&23\end{bmatrix}$, $\begin{bmatrix}7&20\\32&35\end{bmatrix}$, $\begin{bmatrix}9&22\\20&11\end{bmatrix}$, $\begin{bmatrix}9&34\\20&39\end{bmatrix}$, $\begin{bmatrix}43&22\\40&11\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
48.192.1-48.a.1.1, 48.192.1-48.a.1.2, 48.192.1-48.a.1.3, 48.192.1-48.a.1.4, 48.192.1-48.a.1.5, 48.192.1-48.a.1.6, 48.192.1-48.a.1.7, 48.192.1-48.a.1.8, 48.192.1-48.a.1.9, 48.192.1-48.a.1.10, 48.192.1-48.a.1.11, 48.192.1-48.a.1.12, 240.192.1-48.a.1.1, 240.192.1-48.a.1.2, 240.192.1-48.a.1.3, 240.192.1-48.a.1.4, 240.192.1-48.a.1.5, 240.192.1-48.a.1.6, 240.192.1-48.a.1.7, 240.192.1-48.a.1.8, 240.192.1-48.a.1.9, 240.192.1-48.a.1.10, 240.192.1-48.a.1.11, 240.192.1-48.a.1.12 |
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 $ | $=$ | $ 3 x^{2} - 2 z^{2} - w^{2} $ |
| $=$ | $3 x y + 3 y^{2} + z^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} - 3 x^{2} y^{2} + 9 z^{4} $ |
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{3}w$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{3}z$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle -2^8\,\frac{(z^{8}-4z^{4}w^{4}+w^{8})^{3}}{w^{4}z^{16}(2z^{2}-w^{2})(2z^{2}+w^{2})}$ |
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.