| $\GL_2(\Z/56\Z)$-generators: |
$\begin{bmatrix}31&34\\24&29\end{bmatrix}$, $\begin{bmatrix}41&0\\14&27\end{bmatrix}$, $\begin{bmatrix}43&50\\6&19\end{bmatrix}$, $\begin{bmatrix}45&22\\36&51\end{bmatrix}$ |
| Contains $-I$: |
yes |
| Quadratic refinements: |
56.192.1-56.i.1.1, 56.192.1-56.i.1.2, 56.192.1-56.i.1.3, 56.192.1-56.i.1.4, 56.192.1-56.i.1.5, 56.192.1-56.i.1.6, 56.192.1-56.i.1.7, 56.192.1-56.i.1.8, 112.192.1-56.i.1.1, 112.192.1-56.i.1.2, 112.192.1-56.i.1.3, 112.192.1-56.i.1.4, 168.192.1-56.i.1.1, 168.192.1-56.i.1.2, 168.192.1-56.i.1.3, 168.192.1-56.i.1.4, 168.192.1-56.i.1.5, 168.192.1-56.i.1.6, 168.192.1-56.i.1.7, 168.192.1-56.i.1.8, 280.192.1-56.i.1.1, 280.192.1-56.i.1.2, 280.192.1-56.i.1.3, 280.192.1-56.i.1.4, 280.192.1-56.i.1.5, 280.192.1-56.i.1.6, 280.192.1-56.i.1.7, 280.192.1-56.i.1.8 |
| Cyclic 56-isogeny field degree: |
$16$ |
| Cyclic 56-torsion field degree: |
$384$ |
| Full 56-torsion field degree: |
$32256$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 2 y^{2} + y z + z^{2} - w^{2} $ |
| $=$ | $14 x^{2} + 3 y^{2} - 2 y z - 2 z^{2}$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 2 x^{4} - 3 x^{2} y^{2} - 112 x^{2} z^{2} + 2 y^{4} + 84 y^{2} z^{2} + 882 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:
| $\displaystyle X$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle 2x$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{7}w$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -\frac{1}{2^4\cdot7^2}\cdot\frac{29274322430115yz^{23}-184010026703580yz^{21}w^{2}+501845527373400yz^{19}w^{4}-778372654701600yz^{17}w^{6}+750369003943104yz^{15}w^{8}-454500231365376yz^{13}w^{10}+160383273103872yz^{11}w^{12}-22473470638080yz^{9}w^{14}-4032000145152yz^{7}w^{16}+1543916473344yz^{5}w^{18}-5313337344yz^{3}w^{20}-19224059904yzw^{22}-13384524723367z^{24}+95961644164533z^{22}w^{2}-300431070677934z^{20}w^{4}+539371153594840z^{18}w^{6}-607568288240760z^{16}w^{8}+433541233312320z^{14}w^{10}-179600049469568z^{12}w^{12}+25608227301888z^{10}w^{14}+11123540577408z^{8}w^{16}-4765186935040z^{6}w^{18}-35068432896z^{4}w^{20}+198026078208z^{2}w^{22}+14723188736w^{24}}{w^{8}(10941357yz^{15}-43765428yz^{13}w^{2}+69667416yz^{11}w^{4}-56142240yz^{9}w^{6}+24211488yz^{7}w^{8}-5507712yz^{5}w^{10}+622848yz^{3}w^{12}-27648yzw^{14}-10470761z^{16}+49429387z^{14}w^{2}-94700242z^{12}w^{4}+94385368z^{10}w^{6}-52418632z^{8}w^{8}+16408672z^{6}w^{10}-2892992z^{4}w^{12}+269568z^{2}w^{14}-10368w^{16})}$ |
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.
