| $\GL_2(\Z/56\Z)$-generators: |
$\begin{bmatrix}5&30\\44&35\end{bmatrix}$, $\begin{bmatrix}15&34\\6&39\end{bmatrix}$, $\begin{bmatrix}15&38\\34&31\end{bmatrix}$, $\begin{bmatrix}23&30\\0&17\end{bmatrix}$, $\begin{bmatrix}51&48\\14&37\end{bmatrix}$ |
| Contains $-I$: |
yes |
| Quadratic refinements: |
56.96.1-56.p.1.1, 56.96.1-56.p.1.2, 56.96.1-56.p.1.3, 56.96.1-56.p.1.4, 56.96.1-56.p.1.5, 56.96.1-56.p.1.6, 56.96.1-56.p.1.7, 56.96.1-56.p.1.8, 56.96.1-56.p.1.9, 56.96.1-56.p.1.10, 56.96.1-56.p.1.11, 56.96.1-56.p.1.12, 56.96.1-56.p.1.13, 56.96.1-56.p.1.14, 56.96.1-56.p.1.15, 56.96.1-56.p.1.16, 112.96.1-56.p.1.1, 112.96.1-56.p.1.2, 112.96.1-56.p.1.3, 112.96.1-56.p.1.4, 112.96.1-56.p.1.5, 112.96.1-56.p.1.6, 112.96.1-56.p.1.7, 112.96.1-56.p.1.8, 168.96.1-56.p.1.1, 168.96.1-56.p.1.2, 168.96.1-56.p.1.3, 168.96.1-56.p.1.4, 168.96.1-56.p.1.5, 168.96.1-56.p.1.6, 168.96.1-56.p.1.7, 168.96.1-56.p.1.8, 168.96.1-56.p.1.9, 168.96.1-56.p.1.10, 168.96.1-56.p.1.11, 168.96.1-56.p.1.12, 168.96.1-56.p.1.13, 168.96.1-56.p.1.14, 168.96.1-56.p.1.15, 168.96.1-56.p.1.16, 280.96.1-56.p.1.1, 280.96.1-56.p.1.2, 280.96.1-56.p.1.3, 280.96.1-56.p.1.4, 280.96.1-56.p.1.5, 280.96.1-56.p.1.6, 280.96.1-56.p.1.7, 280.96.1-56.p.1.8, 280.96.1-56.p.1.9, 280.96.1-56.p.1.10, 280.96.1-56.p.1.11, 280.96.1-56.p.1.12, 280.96.1-56.p.1.13, 280.96.1-56.p.1.14, 280.96.1-56.p.1.15, 280.96.1-56.p.1.16 |
| 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 $ | $=$ | $ 3 x^{2} - x y - 2 x z - 4 y^{2} - 2 y z - 2 z^{2} $ |
| $=$ | $7 x y - w^{2}$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 196 x^{4} + 14 x^{3} y + 2 x^{2} y^{2} + 7 x^{2} z^{2} + 2 x y z^{2} - 3 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:
| $\displaystyle X$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle 7z$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{1}{3^8\cdot7}\cdot\frac{251045572771840xz^{11}+147688859867997xz^{9}w^{2}+40272207011064xz^{7}w^{4}+5467146454413xz^{5}w^{6}+379372149828xz^{3}w^{8}+9941802381xzw^{10}+276461163180986y^{2}z^{10}+146253656204334y^{2}z^{8}w^{2}+34451928918036y^{2}z^{6}w^{4}+3755417089410y^{2}z^{4}w^{6}+198939226878y^{2}z^{2}w^{8}+2042693478y^{2}w^{10}+137292729057358yz^{11}+106181420183013yz^{9}w^{2}+32547290610870yz^{7}w^{4}+4925824927317yz^{5}w^{6}+360655798482yz^{3}w^{8}+9941802381yzw^{10}+137751234740224z^{12}+98790852980162z^{10}w^{2}+34065988936686z^{8}w^{4}+6181180386696z^{6}w^{6}+616589973846z^{4}w^{8}+29003516262z^{2}w^{10}+654610410w^{12}}{w^{8}(1792xz^{3}+159xzw^{2}+1862y^{2}z^{2}+42y^{2}w^{2}+1498yz^{3}+159yzw^{2}+1120z^{4}+278z^{2}w^{2}+6w^{4})}$ |
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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.
