$\GL_2(\Z/56\Z)$-generators: |
$\begin{bmatrix}17&40\\6&55\end{bmatrix}$, $\begin{bmatrix}17&52\\36&13\end{bmatrix}$, $\begin{bmatrix}39&46\\32&45\end{bmatrix}$, $\begin{bmatrix}47&12\\44&19\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
56.192.1-56.bw.1.1, 56.192.1-56.bw.1.2, 56.192.1-56.bw.1.3, 56.192.1-56.bw.1.4, 56.192.1-56.bw.1.5, 56.192.1-56.bw.1.6, 56.192.1-56.bw.1.7, 56.192.1-56.bw.1.8, 168.192.1-56.bw.1.1, 168.192.1-56.bw.1.2, 168.192.1-56.bw.1.3, 168.192.1-56.bw.1.4, 168.192.1-56.bw.1.5, 168.192.1-56.bw.1.6, 168.192.1-56.bw.1.7, 168.192.1-56.bw.1.8, 280.192.1-56.bw.1.1, 280.192.1-56.bw.1.2, 280.192.1-56.bw.1.3, 280.192.1-56.bw.1.4, 280.192.1-56.bw.1.5, 280.192.1-56.bw.1.6, 280.192.1-56.bw.1.7, 280.192.1-56.bw.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 x^{2} - y^{2} - z^{2} $ |
| $=$ | $4 x^{2} + 2 y^{2} + 2 y w + 5 z^{2} + 2 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 9 x^{4} + 12 x^{2} y^{2} + 16 x^{2} z^{2} + 4 y^{4} + 6 y^{2} z^{2} + 4 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:
$\displaystyle X$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle 2x$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle -\frac{2681928yz^{22}w-211788yz^{20}w^{3}-17582868yz^{18}w^{5}-13119534yz^{16}w^{7}+20892960yz^{14}w^{9}+42601104yz^{12}w^{11}+34492752yz^{10}w^{13}+16270488yz^{8}w^{15}+4822200yz^{6}w^{17}+888300yz^{4}w^{19}+93060yz^{2}w^{21}+4230yw^{23}+7189057z^{24}-27626406z^{22}w^{2}-1397817z^{20}w^{4}+54268255z^{18}w^{6}+36194361z^{16}w^{8}-23807256z^{14}w^{10}-47705476z^{12}w^{12}-32902140z^{10}w^{14}-13174095z^{8}w^{16}-3341530z^{6}w^{18}-531783z^{4}w^{20}-48531z^{2}w^{22}-1934w^{24}}{z^{8}(3024yz^{14}w+19464yz^{12}w^{3}+49176yz^{10}w^{5}+61764yz^{8}w^{7}+40920yz^{6}w^{9}+14508yz^{4}w^{11}+2604yz^{2}w^{13}+186yw^{15}-3969z^{16}-29484z^{14}w^{2}-90406z^{12}w^{4}-146506z^{10}w^{6}-133721z^{8}w^{8}-68794z^{6}w^{10}-19721z^{4}w^{12}-2941z^{2}w^{14}-178w^{16})}$ |
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.