| $\GL_2(\Z/56\Z)$-generators: |
$\begin{bmatrix}9&24\\4&23\end{bmatrix}$, $\begin{bmatrix}28&31\\23&4\end{bmatrix}$, $\begin{bmatrix}41&32\\36&27\end{bmatrix}$, $\begin{bmatrix}43&40\\4&1\end{bmatrix}$ |
| Contains $-I$: |
yes |
| Quadratic refinements: |
56.96.1-56.gh.1.1, 56.96.1-56.gh.1.2, 112.96.1-56.gh.1.1, 112.96.1-56.gh.1.2, 112.96.1-56.gh.1.3, 112.96.1-56.gh.1.4, 168.96.1-56.gh.1.1, 168.96.1-56.gh.1.2, 280.96.1-56.gh.1.1, 280.96.1-56.gh.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 $ | $=$ | $ 7 x^{2} - 2 y^{2} + y z - z^{2} $ |
| $=$ | $7 x^{2} + 7 y^{2} - 7 y z + 7 z^{2} - 2 w^{2}$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 2 x^{4} + 20 x^{2} y^{2} - 21 x^{2} z^{2} + 162 y^{4} - 252 y^{2} z^{2} + 98 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:
| $\displaystyle X$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{2}{7}w$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle 2^4\cdot3^3\,\frac{204577493120yz^{11}-2355801424704yz^{9}w^{2}+4013602415424yz^{7}w^{4}-977551745184yz^{5}w^{6}-1078461855576yz^{3}w^{8}+394205219100yzw^{10}+319862218816z^{12}+288300555200z^{10}w^{2}-3475232331984z^{8}w^{4}+4030747860384z^{6}w^{6}-1284474721572z^{4}w^{8}-21033647460z^{2}w^{10}+36263467125w^{12}}{51144373280yz^{11}-83240633952yz^{9}w^{2}+20667884832yz^{7}w^{4}+1466275608yz^{5}w^{6}-2508257178yz^{3}w^{8}-49601160yzw^{10}+79965554704z^{12}-69006046144z^{10}w^{2}+47451808152z^{8}w^{4}-18661952232z^{6}w^{6}+3387815361z^{4}w^{8}+255997098z^{2}w^{10}+944784w^{12}}$ |
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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.
