$\GL_2(\Z/56\Z)$-generators: |
$\begin{bmatrix}7&50\\6&23\end{bmatrix}$, $\begin{bmatrix}21&16\\50&27\end{bmatrix}$, $\begin{bmatrix}43&32\\4&31\end{bmatrix}$, $\begin{bmatrix}45&52\\30&19\end{bmatrix}$, $\begin{bmatrix}53&18\\12&55\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
56.96.1-56.s.1.1, 56.96.1-56.s.1.2, 56.96.1-56.s.1.3, 56.96.1-56.s.1.4, 56.96.1-56.s.1.5, 56.96.1-56.s.1.6, 56.96.1-56.s.1.7, 56.96.1-56.s.1.8, 56.96.1-56.s.1.9, 56.96.1-56.s.1.10, 56.96.1-56.s.1.11, 56.96.1-56.s.1.12, 112.96.1-56.s.1.1, 112.96.1-56.s.1.2, 112.96.1-56.s.1.3, 112.96.1-56.s.1.4, 112.96.1-56.s.1.5, 112.96.1-56.s.1.6, 112.96.1-56.s.1.7, 112.96.1-56.s.1.8, 168.96.1-56.s.1.1, 168.96.1-56.s.1.2, 168.96.1-56.s.1.3, 168.96.1-56.s.1.4, 168.96.1-56.s.1.5, 168.96.1-56.s.1.6, 168.96.1-56.s.1.7, 168.96.1-56.s.1.8, 168.96.1-56.s.1.9, 168.96.1-56.s.1.10, 168.96.1-56.s.1.11, 168.96.1-56.s.1.12, 280.96.1-56.s.1.1, 280.96.1-56.s.1.2, 280.96.1-56.s.1.3, 280.96.1-56.s.1.4, 280.96.1-56.s.1.5, 280.96.1-56.s.1.6, 280.96.1-56.s.1.7, 280.96.1-56.s.1.8, 280.96.1-56.s.1.9, 280.96.1-56.s.1.10, 280.96.1-56.s.1.11, 280.96.1-56.s.1.12 |
Cyclic 56-isogeny field degree: |
$16$ |
Cyclic 56-torsion field degree: |
$384$ |
Full 56-torsion field degree: |
$64512$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 7 x^{2} - 7 x y + 2 z^{2} $ |
| $=$ | $7 x^{2} + 7 x y + 14 y^{2} + 2 z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 49 x^{4} + 7 x^{2} y^{2} + 21 x^{2} z^{2} + 2 z^{4} $ |
This modular curve has no real points, and therefore no rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{2}w$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle -\frac{903168y^{2}z^{10}+677376y^{2}z^{8}w^{2}+16128y^{2}z^{6}w^{4}-4032y^{2}z^{4}w^{6}-10584y^{2}z^{2}w^{8}-882y^{2}w^{10}+131072z^{12}+196608z^{10}w^{2}+65280z^{8}w^{4}+4096z^{6}w^{6}+768z^{4}w^{8}-240z^{2}w^{10}-31w^{12}}{w^{4}z^{4}(56y^{2}z^{2}-14y^{2}w^{2}-w^{4})}$ |
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.