$\GL_2(\Z/48\Z)$-generators: |
$\begin{bmatrix}19&24\\8&11\end{bmatrix}$, $\begin{bmatrix}23&34\\36&29\end{bmatrix}$, $\begin{bmatrix}25&12\\36&23\end{bmatrix}$, $\begin{bmatrix}33&2\\4&15\end{bmatrix}$, $\begin{bmatrix}37&18\\12&11\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
48.192.1-48.p.2.1, 48.192.1-48.p.2.2, 48.192.1-48.p.2.3, 48.192.1-48.p.2.4, 48.192.1-48.p.2.5, 48.192.1-48.p.2.6, 48.192.1-48.p.2.7, 48.192.1-48.p.2.8, 48.192.1-48.p.2.9, 48.192.1-48.p.2.10, 48.192.1-48.p.2.11, 48.192.1-48.p.2.12, 240.192.1-48.p.2.1, 240.192.1-48.p.2.2, 240.192.1-48.p.2.3, 240.192.1-48.p.2.4, 240.192.1-48.p.2.5, 240.192.1-48.p.2.6, 240.192.1-48.p.2.7, 240.192.1-48.p.2.8, 240.192.1-48.p.2.9, 240.192.1-48.p.2.10, 240.192.1-48.p.2.11, 240.192.1-48.p.2.12 |
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 z + 2 y^{2} + z^{2} $ |
| $=$ | $3 x^{2} - 4 x z - 4 y^{2} + 4 z^{2} - w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 4 x^{4} - 3 y^{2} z^{2} + 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:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{3}w$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle z$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{2^8}{3^4}\cdot\frac{(81z^{8}-54z^{6}w^{2}+45z^{4}w^{4}-12z^{2}w^{6}+w^{8})^{3}}{w^{4}z^{8}(3z^{2}-w^{2})^{4}(6z^{2}-w^{2})^{2}}$ |
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.