$\GL_2(\Z/48\Z)$-generators: |
$\begin{bmatrix}1&19\\40&23\end{bmatrix}$, $\begin{bmatrix}23&24\\32&23\end{bmatrix}$, $\begin{bmatrix}35&23\\36&31\end{bmatrix}$, $\begin{bmatrix}39&47\\32&5\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
48.192.1-48.ea.2.1, 48.192.1-48.ea.2.2, 48.192.1-48.ea.2.3, 48.192.1-48.ea.2.4, 48.192.1-48.ea.2.5, 48.192.1-48.ea.2.6, 48.192.1-48.ea.2.7, 48.192.1-48.ea.2.8, 240.192.1-48.ea.2.1, 240.192.1-48.ea.2.2, 240.192.1-48.ea.2.3, 240.192.1-48.ea.2.4, 240.192.1-48.ea.2.5, 240.192.1-48.ea.2.6, 240.192.1-48.ea.2.7, 240.192.1-48.ea.2.8 |
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} - 4 y^{2} - z^{2} $ |
| $=$ | $5 x^{2} + 2 y^{2} - z^{2} - w^{2}$ |
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
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{2}{3}\cdot\frac{(81z^{8}+3240z^{6}w^{2}+4824z^{4}w^{4}+1440z^{2}w^{6}+16w^{8})^{3}}{w^{2}z^{2}(3z^{2}-2w^{2})^{8}(3z^{2}+2w^{2})^{2}}$ |
Hi
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Cover information
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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.