$\GL_2(\Z/56\Z)$-generators: |
$\begin{bmatrix}17&44\\40&45\end{bmatrix}$, $\begin{bmatrix}17&48\\2&23\end{bmatrix}$, $\begin{bmatrix}35&40\\48&11\end{bmatrix}$, $\begin{bmatrix}39&28\\22&45\end{bmatrix}$, $\begin{bmatrix}51&42\\4&53\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
56.96.1-56.s.2.1, 56.96.1-56.s.2.2, 56.96.1-56.s.2.3, 56.96.1-56.s.2.4, 56.96.1-56.s.2.5, 56.96.1-56.s.2.6, 56.96.1-56.s.2.7, 56.96.1-56.s.2.8, 56.96.1-56.s.2.9, 56.96.1-56.s.2.10, 56.96.1-56.s.2.11, 56.96.1-56.s.2.12, 112.96.1-56.s.2.1, 112.96.1-56.s.2.2, 112.96.1-56.s.2.3, 112.96.1-56.s.2.4, 112.96.1-56.s.2.5, 112.96.1-56.s.2.6, 112.96.1-56.s.2.7, 112.96.1-56.s.2.8, 168.96.1-56.s.2.1, 168.96.1-56.s.2.2, 168.96.1-56.s.2.3, 168.96.1-56.s.2.4, 168.96.1-56.s.2.5, 168.96.1-56.s.2.6, 168.96.1-56.s.2.7, 168.96.1-56.s.2.8, 168.96.1-56.s.2.9, 168.96.1-56.s.2.10, 168.96.1-56.s.2.11, 168.96.1-56.s.2.12, 280.96.1-56.s.2.1, 280.96.1-56.s.2.2, 280.96.1-56.s.2.3, 280.96.1-56.s.2.4, 280.96.1-56.s.2.5, 280.96.1-56.s.2.6, 280.96.1-56.s.2.7, 280.96.1-56.s.2.8, 280.96.1-56.s.2.9, 280.96.1-56.s.2.10, 280.96.1-56.s.2.11, 280.96.1-56.s.2.12 |
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 + w^{2} $ |
| $=$ | $7 x y + 7 y^{2} + z^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 2 x^{4} + 7 x^{2} y^{2} + 21 x^{2} z^{2} + 49 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:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{7}w$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{7}z$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle -2^4\,\frac{441y^{2}z^{10}+1323y^{2}z^{8}w^{2}+126y^{2}z^{6}w^{4}-126y^{2}z^{4}w^{6}-1323y^{2}z^{2}w^{8}-441y^{2}w^{10}+31z^{12}+60z^{10}w^{2}-48z^{8}w^{4}-64z^{6}w^{6}-255z^{4}w^{8}-192z^{2}w^{10}-32w^{12}}{w^{4}z^{4}(7y^{2}z^{2}-7y^{2}w^{2}+z^{4})}$ |
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.