$\GL_2(\Z/48\Z)$-generators: |
$\begin{bmatrix}1&25\\0&47\end{bmatrix}$, $\begin{bmatrix}25&20\\8&45\end{bmatrix}$, $\begin{bmatrix}43&39\\36&11\end{bmatrix}$, $\begin{bmatrix}45&22\\4&47\end{bmatrix}$, $\begin{bmatrix}47&36\\4&13\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
48.96.1-48.bn.2.1, 48.96.1-48.bn.2.2, 48.96.1-48.bn.2.3, 48.96.1-48.bn.2.4, 48.96.1-48.bn.2.5, 48.96.1-48.bn.2.6, 48.96.1-48.bn.2.7, 48.96.1-48.bn.2.8, 48.96.1-48.bn.2.9, 48.96.1-48.bn.2.10, 48.96.1-48.bn.2.11, 48.96.1-48.bn.2.12, 48.96.1-48.bn.2.13, 48.96.1-48.bn.2.14, 48.96.1-48.bn.2.15, 48.96.1-48.bn.2.16, 96.96.1-48.bn.2.1, 96.96.1-48.bn.2.2, 96.96.1-48.bn.2.3, 96.96.1-48.bn.2.4, 96.96.1-48.bn.2.5, 96.96.1-48.bn.2.6, 96.96.1-48.bn.2.7, 96.96.1-48.bn.2.8, 240.96.1-48.bn.2.1, 240.96.1-48.bn.2.2, 240.96.1-48.bn.2.3, 240.96.1-48.bn.2.4, 240.96.1-48.bn.2.5, 240.96.1-48.bn.2.6, 240.96.1-48.bn.2.7, 240.96.1-48.bn.2.8, 240.96.1-48.bn.2.9, 240.96.1-48.bn.2.10, 240.96.1-48.bn.2.11, 240.96.1-48.bn.2.12, 240.96.1-48.bn.2.13, 240.96.1-48.bn.2.14, 240.96.1-48.bn.2.15, 240.96.1-48.bn.2.16 |
Cyclic 48-isogeny field degree: |
$8$ |
Cyclic 48-torsion field degree: |
$128$ |
Full 48-torsion field degree: |
$24576$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 24 x y + 6 y^{2} + w^{2} $ |
| $=$ | $24 x^{2} - 6 x y - z^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} - 3 x^{2} y^{2} + 9 x^{2} z^{2} + 18 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:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{3}z$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{6}w$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{387072y^{2}z^{10}-13824y^{2}z^{8}w^{2}-5453568y^{2}z^{6}w^{4}+24107328y^{2}z^{4}w^{6}-18878616y^{2}z^{2}w^{8}+1572858y^{2}w^{10}-131072z^{12}+196608z^{10}w^{2}+303360z^{8}w^{4}-191744z^{6}w^{6}-173568z^{4}w^{8}-1049280z^{2}w^{10}+131071w^{12}}{w^{2}z^{2}(384y^{2}z^{6}-1056y^{2}z^{4}w^{2}+168y^{2}z^{2}w^{4}-6y^{2}w^{6}+512z^{6}w^{2}-272z^{4}w^{4}+32z^{2}w^{6}-w^{8})}$ |
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.