$\GL_2(\Z/56\Z)$-generators: |
$\begin{bmatrix}1&15\\2&43\end{bmatrix}$, $\begin{bmatrix}19&38\\50&41\end{bmatrix}$, $\begin{bmatrix}27&45\\18&41\end{bmatrix}$, $\begin{bmatrix}43&23\\4&5\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
56.96.1-56.gs.1.1, 56.96.1-56.gs.1.2, 112.96.1-56.gs.1.1, 112.96.1-56.gs.1.2, 112.96.1-56.gs.1.3, 112.96.1-56.gs.1.4, 168.96.1-56.gs.1.1, 168.96.1-56.gs.1.2, 280.96.1-56.gs.1.1, 280.96.1-56.gs.1.2 |
Cyclic 56-isogeny field degree: |
$32$ |
Cyclic 56-torsion field degree: |
$768$ |
Full 56-torsion field degree: |
$64512$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 14 x^{2} - 2 y^{2} + y z - z^{2} $ |
| $=$ | $9 y^{2} - 8 y z + 8 z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 4 x^{4} + 20 x^{2} y^{2} + 21 x^{2} z^{2} + 81 y^{4} + 126 y^{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:
$\displaystyle X$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle 2x$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{2}{7}w$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle 2^4\cdot3^3\,\frac{204577493120yz^{11}-507798361728yz^{9}w^{2}-26773032384yz^{7}w^{4}+21059958528yz^{5}w^{6}+3130016904yz^{3}w^{8}+248005800yzw^{10}+319862218816z^{12}+326120338880z^{10}w^{2}-117033998256z^{8}w^{4}-35840514528z^{6}w^{6}-1416409092z^{4}w^{8}-35823060z^{2}w^{10}-7381125w^{12}}{51144373280yz^{11}+125905270680yz^{9}w^{2}+56675653020yz^{7}w^{4}+8208542916yz^{5}w^{6}+660981384yz^{3}w^{8}+39680928yzw^{10}+79965554704z^{12}+10989492248z^{10}w^{2}-25725752199z^{8}w^{4}-8604561798z^{6}w^{6}-988721559z^{4}w^{8}-47396664z^{2}w^{10}+944784w^{12}}$ |
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.