$\GL_2(\Z/48\Z)$-generators: |
$\begin{bmatrix}11&7\\32&45\end{bmatrix}$, $\begin{bmatrix}11&28\\16&27\end{bmatrix}$, $\begin{bmatrix}23&45\\28&23\end{bmatrix}$, $\begin{bmatrix}47&8\\4&45\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
48.192.1-48.bd.1.1, 48.192.1-48.bd.1.2, 48.192.1-48.bd.1.3, 48.192.1-48.bd.1.4, 48.192.1-48.bd.1.5, 48.192.1-48.bd.1.6, 48.192.1-48.bd.1.7, 48.192.1-48.bd.1.8, 96.192.1-48.bd.1.1, 96.192.1-48.bd.1.2, 96.192.1-48.bd.1.3, 96.192.1-48.bd.1.4, 240.192.1-48.bd.1.1, 240.192.1-48.bd.1.2, 240.192.1-48.bd.1.3, 240.192.1-48.bd.1.4, 240.192.1-48.bd.1.5, 240.192.1-48.bd.1.6, 240.192.1-48.bd.1.7, 240.192.1-48.bd.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 $ | $=$ | $ 2 x^{2} + 2 x y + 2 y^{2} - 3 z^{2} $ |
| $=$ | $2 x^{2} + 2 x y + 8 y^{2} + 9 z^{2} - w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 49 x^{4} - 20 x^{2} y^{2} - 42 x^{2} z^{2} + 4 y^{4} + 6 y^{2} z^{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:
$\displaystyle X$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle x$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{3}w$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle -\frac{1}{3^8}\cdot\frac{(1296z^{8}-3456z^{6}w^{2}+720z^{4}w^{4}-48z^{2}w^{6}+w^{8})^{3}}{w^{2}z^{16}(12z^{2}-w^{2})^{2}(24z^{2}-w^{2})}$ |
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.