$\GL_2(\Z/56\Z)$-generators: |
$\begin{bmatrix}1&34\\38&17\end{bmatrix}$, $\begin{bmatrix}13&22\\10&55\end{bmatrix}$, $\begin{bmatrix}15&54\\38&27\end{bmatrix}$, $\begin{bmatrix}35&34\\6&35\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
56.96.1-56.j.1.1, 56.96.1-56.j.1.2, 56.96.1-56.j.1.3, 56.96.1-56.j.1.4, 168.96.1-56.j.1.1, 168.96.1-56.j.1.2, 168.96.1-56.j.1.3, 168.96.1-56.j.1.4, 280.96.1-56.j.1.1, 280.96.1-56.j.1.2, 280.96.1-56.j.1.3, 280.96.1-56.j.1.4 |
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 $ | $=$ | $ 4 x^{2} - x y + 2 x z + 4 y^{2} - 2 y z + 2 z^{2} $ |
| $=$ | $7 x^{2} - 7 y^{2} - w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} - 4 x^{3} y + 32 x^{2} y^{2} + 21 x^{2} z^{2} - 56 x y^{3} - 42 x y z^{2} + 252 y^{4} + \cdots + 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:
$\displaystyle X$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{2}y$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{7}w$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle 2^2\cdot3^3\,\frac{233027228703360xz^{11}+847051341963904xz^{9}w^{2}+992128434546816xz^{7}w^{4}+384759509762880xz^{5}w^{6}+40526381976696xz^{3}w^{8}+639081185904xzw^{10}+1013537489095040y^{2}z^{10}+3296116902390480y^{2}z^{8}w^{2}+3735644513160672y^{2}z^{6}w^{4}+1442902814740800y^{2}z^{4}w^{6}+146914657906968y^{2}z^{2}w^{8}+2379981459555y^{2}w^{10}-233027228703360yz^{11}-746377232967296yz^{9}w^{2}-789193374371136yz^{7}w^{4}-258969385664640yz^{5}w^{6}-15765504949656yz^{3}w^{8}+399805190064yzw^{10}+213040248570688z^{12}+822788493405760z^{10}w^{2}+1143018783803232z^{8}w^{4}+630617486946048z^{6}w^{6}+141956649588852z^{4}w^{8}+11252553878412z^{2}w^{10}+161917849557w^{12}}{14564201793960xz^{11}-6812862687176xz^{9}w^{2}-3657369948372xz^{7}w^{4}+1407674735964xz^{5}w^{6}+9341551800xz^{3}w^{8}-357128352xzw^{10}+63346093068440y^{2}z^{10}-41201461279881y^{2}z^{8}w^{2}-12137764970898y^{2}z^{6}w^{4}+7583369655519y^{2}z^{4}w^{6}-251864770248y^{2}z^{2}w^{8}-33748629264y^{2}w^{10}-14564201793960yz^{11}+13104994499464yz^{9}w^{2}+1120681696176yz^{7}w^{4}-1678016550672yz^{5}w^{6}+106463929824yz^{3}w^{8}-1428513408yzw^{10}+13315015535668z^{12}-4855191047420z^{10}w^{2}-5232606587019z^{8}w^{4}+753924496878z^{6}w^{6}+473502915081z^{4}w^{8}-14552980344z^{2}w^{10}-3596792688w^{12}}$ |
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.