| $\GL_2(\Z/56\Z)$-generators: |
$\begin{bmatrix}8&21\\39&4\end{bmatrix}$, $\begin{bmatrix}16&25\\29&48\end{bmatrix}$, $\begin{bmatrix}27&20\\20&15\end{bmatrix}$, $\begin{bmatrix}32&3\\31&8\end{bmatrix}$ |
| Contains $-I$: |
yes |
| Quadratic refinements: |
56.96.1-56.gx.1.1, 56.96.1-56.gx.1.2, 112.96.1-56.gx.1.1, 112.96.1-56.gx.1.2, 112.96.1-56.gx.1.3, 112.96.1-56.gx.1.4, 168.96.1-56.gx.1.1, 168.96.1-56.gx.1.2, 280.96.1-56.gx.1.1, 280.96.1-56.gx.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} + y^{2} + 3 y z - 3 z^{2} + w^{2} $ |
| $=$ | $7 x^{2} - 4 y^{2} - 5 y z + 5 z^{2} - w^{2}$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} - 18 x^{2} y^{2} - 56 x^{2} z^{2} + 25 y^{4} + 210 y^{2} z^{2} + 441 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:
| $\displaystyle X$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{7}w$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle 2^4\,\frac{272442601581696yz^{11}-313472358763200yz^{9}w^{2}+140385513441600yz^{7}w^{4}-30378902204000yz^{5}w^{6}+3147418225000yz^{3}w^{8}-123460837500yzw^{10}-189801162726336z^{12}+254795306439744z^{10}w^{2}-138078565989840z^{8}w^{4}+38219269743200z^{6}w^{6}-5582604422500z^{4}w^{8}+390953062500z^{2}w^{10}-9138146875w^{12}}{2522616681312yz^{11}-1262810752000yz^{9}w^{2}+215915207200yz^{7}w^{4}-14495523000yz^{5}w^{6}+334731250yz^{3}w^{8}-1706250yzw^{10}-1757418173392z^{12}+1216857455968z^{10}w^{2}-304119975880z^{8}w^{4}+32760547400z^{6}w^{6}-1405473125z^{4}w^{8}+17937500z^{2}w^{10}-28125w^{12}}$ |
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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.
