$\GL_2(\Z/48\Z)$-generators: |
$\begin{bmatrix}7&12\\36&13\end{bmatrix}$, $\begin{bmatrix}27&14\\40&43\end{bmatrix}$, $\begin{bmatrix}27&29\\28&43\end{bmatrix}$, $\begin{bmatrix}27&47\\8&29\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
48.192.1-48.y.1.1, 48.192.1-48.y.1.2, 48.192.1-48.y.1.3, 48.192.1-48.y.1.4, 48.192.1-48.y.1.5, 48.192.1-48.y.1.6, 48.192.1-48.y.1.7, 48.192.1-48.y.1.8, 240.192.1-48.y.1.1, 240.192.1-48.y.1.2, 240.192.1-48.y.1.3, 240.192.1-48.y.1.4, 240.192.1-48.y.1.5, 240.192.1-48.y.1.6, 240.192.1-48.y.1.7, 240.192.1-48.y.1.8 |
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 $ | $=$ | $ 2 x^{2} - y^{2} + 2 z^{2} + 2 w^{2} $ |
| $=$ | $3 x^{2} - 2 z^{2} + 2 z w - 2 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} - 10 x^{2} y^{2} - 2 x^{2} z^{2} + 25 y^{4} + 4 y^{2} z^{2} + 4 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:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle x$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle -\frac{1}{3}\cdot\frac{(z^{8}-88z^{7}w+172z^{6}w^{2}-232z^{5}w^{3}+310z^{4}w^{4}-232z^{3}w^{5}+172z^{2}w^{6}-88zw^{7}+w^{8})^{3}}{(z-w)^{2}(z+w)^{4}(z^{2}-zw+w^{2})^{8}(5z^{2}-2zw+5w^{2})}$ |
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.