$\GL_2(\Z/56\Z)$-generators: |
$\begin{bmatrix}1&18\\4&19\end{bmatrix}$, $\begin{bmatrix}7&44\\38&53\end{bmatrix}$, $\begin{bmatrix}9&42\\6&41\end{bmatrix}$, $\begin{bmatrix}31&8\\16&15\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
56.192.1-56.t.2.1, 56.192.1-56.t.2.2, 56.192.1-56.t.2.3, 56.192.1-56.t.2.4, 56.192.1-56.t.2.5, 56.192.1-56.t.2.6, 56.192.1-56.t.2.7, 56.192.1-56.t.2.8, 112.192.1-56.t.2.1, 112.192.1-56.t.2.2, 112.192.1-56.t.2.3, 112.192.1-56.t.2.4, 168.192.1-56.t.2.1, 168.192.1-56.t.2.2, 168.192.1-56.t.2.3, 168.192.1-56.t.2.4, 168.192.1-56.t.2.5, 168.192.1-56.t.2.6, 168.192.1-56.t.2.7, 168.192.1-56.t.2.8, 280.192.1-56.t.2.1, 280.192.1-56.t.2.2, 280.192.1-56.t.2.3, 280.192.1-56.t.2.4, 280.192.1-56.t.2.5, 280.192.1-56.t.2.6, 280.192.1-56.t.2.7, 280.192.1-56.t.2.8 |
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} $ |
| $=$ | $14 x^{2} - 3 y^{2} - 2 y z + 2 z^{2}$ |
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} $ |
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:
$\displaystyle X$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle 2x$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{7}w$ |
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.