| $\GL_2(\Z/56\Z)$-generators: |
$\begin{bmatrix}2&37\\41&12\end{bmatrix}$, $\begin{bmatrix}5&12\\6&43\end{bmatrix}$, $\begin{bmatrix}26&9\\3&50\end{bmatrix}$, $\begin{bmatrix}54&35\\31&44\end{bmatrix}$ |
| Contains $-I$: |
yes |
| Quadratic refinements: |
56.48.1-56.t.1.1, 56.48.1-56.t.1.2, 56.48.1-56.t.1.3, 56.48.1-56.t.1.4, 112.48.1-56.t.1.1, 112.48.1-56.t.1.2, 112.48.1-56.t.1.3, 112.48.1-56.t.1.4, 168.48.1-56.t.1.1, 168.48.1-56.t.1.2, 168.48.1-56.t.1.3, 168.48.1-56.t.1.4, 280.48.1-56.t.1.1, 280.48.1-56.t.1.2, 280.48.1-56.t.1.3, 280.48.1-56.t.1.4 |
| Cyclic 56-isogeny field degree: |
$32$ |
| Cyclic 56-torsion field degree: |
$768$ |
| Full 56-torsion field degree: |
$129024$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 14 y^{2} - 4 z^{2} - w^{2} $ |
| $=$ | $56 x^{2} + z w$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} - 14 y^{2} z^{2} + 49 z^{4} $ |
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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:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{4}y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{28}w$ |
Maps to other modular curves
$j$-invariant map
of degree 24 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle 2^8\,\frac{(z^{2}+w^{2})^{3}}{w^{2}z^{4}}$ |
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.
