$\GL_2(\Z/56\Z)$-generators: |
$\begin{bmatrix}9&32\\52&21\end{bmatrix}$, $\begin{bmatrix}29&30\\30&9\end{bmatrix}$, $\begin{bmatrix}48&1\\21&20\end{bmatrix}$, $\begin{bmatrix}50&43\\7&26\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
56.96.1-56.dl.1.1, 56.96.1-56.dl.1.2, 56.96.1-56.dl.1.3, 56.96.1-56.dl.1.4, 168.96.1-56.dl.1.1, 168.96.1-56.dl.1.2, 168.96.1-56.dl.1.3, 168.96.1-56.dl.1.4, 280.96.1-56.dl.1.1, 280.96.1-56.dl.1.2, 280.96.1-56.dl.1.3, 280.96.1-56.dl.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 $ | $=$ | $ 2 x^{2} + x z + 7 y^{2} + z^{2} $ |
| $=$ | $13 x^{2} + 3 x z - 7 y^{2} + 3 z^{2} - w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} + 26 x^{2} y^{2} + 21 x^{2} z^{2} + 225 y^{4} + 420 y^{2} z^{2} + 196 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:
$\displaystyle X$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{7}w$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{2^4\cdot3^3}{7^2}\cdot\frac{88291339136xz^{11}-10923044092800xz^{9}w^{2}+16875038577600xz^{7}w^{4}-7455623616000xz^{5}w^{6}+969527475000xz^{3}w^{8}-46309725000xzw^{10}-5705799555904z^{12}+12387922851136z^{10}w^{2}-8471103132240z^{8}w^{4}+2329605784800z^{6}w^{6}-338267947500z^{4}w^{8}+4443862500z^{2}w^{10}+1010728125w^{12}}{450466016xz^{11}+39289895400xz^{9}w^{2}+8470772100xz^{7}w^{4}-4733788500xz^{5}w^{6}-629775000xz^{3}w^{8}+72900000xzw^{10}-29111222224z^{12}-5029507784z^{10}w^{2}+8211744135z^{8}w^{4}-419526450z^{6}w^{6}-629150625z^{4}w^{8}+54675000z^{2}w^{10}+12150000w^{12}}$ |
Hi
|
Cover information
Click on a modular curve in the diagram to see information about it.
|
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.