| $\GL_2(\Z/56\Z)$-generators: |
$\begin{bmatrix}2&15\\55&6\end{bmatrix}$, $\begin{bmatrix}6&55\\11&50\end{bmatrix}$, $\begin{bmatrix}7&16\\48&35\end{bmatrix}$, $\begin{bmatrix}16&11\\23&20\end{bmatrix}$ |
| Contains $-I$: |
yes |
| Quadratic refinements: |
56.96.1-56.di.1.1, 56.96.1-56.di.1.2, 56.96.1-56.di.1.3, 56.96.1-56.di.1.4, 112.96.1-56.di.1.1, 112.96.1-56.di.1.2, 112.96.1-56.di.1.3, 112.96.1-56.di.1.4, 168.96.1-56.di.1.1, 168.96.1-56.di.1.2, 168.96.1-56.di.1.3, 168.96.1-56.di.1.4, 280.96.1-56.di.1.1, 280.96.1-56.di.1.2, 280.96.1-56.di.1.3, 280.96.1-56.di.1.4 |
| 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 $ | $=$ | $ 14 x^{2} - 2 y^{2} + y z - z^{2} $ |
| $=$ | $15 y^{2} - 4 y z + 4 z^{2} - w^{2}$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 4 x^{4} - 52 x^{2} y^{2} + 21 x^{2} z^{2} + 225 y^{4} - 210 y^{2} z^{2} + 49 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:
| $\displaystyle X$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle 2x$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{2}{7}w$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{2^4\cdot3^3}{7^2}\cdot\frac{88291339136yz^{11}-10923044092800yz^{9}w^{2}+16875038577600yz^{7}w^{4}-7455623616000yz^{5}w^{6}+969527475000yz^{3}w^{8}-46309725000yzw^{10}+5705799555904z^{12}-12387922851136z^{10}w^{2}+8471103132240z^{8}w^{4}-2329605784800z^{6}w^{6}+338267947500z^{4}w^{8}-4443862500z^{2}w^{10}-1010728125w^{12}}{450466016yz^{11}+39289895400yz^{9}w^{2}+8470772100yz^{7}w^{4}-4733788500yz^{5}w^{6}-629775000yz^{3}w^{8}+72900000yzw^{10}+29111222224z^{12}+5029507784z^{10}w^{2}-8211744135z^{8}w^{4}+419526450z^{6}w^{6}+629150625z^{4}w^{8}-54675000z^{2}w^{10}-12150000w^{12}}$ |
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.
