$\GL_2(\Z/8\Z)$-generators: |
$\begin{bmatrix}1&4\\0&1\end{bmatrix}$, $\begin{bmatrix}5&0\\0&1\end{bmatrix}$, $\begin{bmatrix}5&0\\0&3\end{bmatrix}$, $\begin{bmatrix}7&4\\0&7\end{bmatrix}$ |
$\GL_2(\Z/8\Z)$-subgroup: |
$C_2^4$ |
Contains $-I$: |
yes |
Quadratic refinements: |
8.192.1-8.b.2.1, 8.192.1-8.b.2.2, 8.192.1-8.b.2.3, 8.192.1-8.b.2.4, 8.192.1-8.b.2.5, 8.192.1-8.b.2.6, 16.192.1-8.b.2.1, 16.192.1-8.b.2.2, 16.192.1-8.b.2.3, 16.192.1-8.b.2.4, 24.192.1-8.b.2.1, 24.192.1-8.b.2.2, 24.192.1-8.b.2.3, 24.192.1-8.b.2.4, 24.192.1-8.b.2.5, 24.192.1-8.b.2.6, 40.192.1-8.b.2.1, 40.192.1-8.b.2.2, 40.192.1-8.b.2.3, 40.192.1-8.b.2.4, 40.192.1-8.b.2.5, 40.192.1-8.b.2.6, 48.192.1-8.b.2.1, 48.192.1-8.b.2.2, 48.192.1-8.b.2.3, 48.192.1-8.b.2.4, 56.192.1-8.b.2.1, 56.192.1-8.b.2.2, 56.192.1-8.b.2.3, 56.192.1-8.b.2.4, 56.192.1-8.b.2.5, 56.192.1-8.b.2.6, 80.192.1-8.b.2.1, 80.192.1-8.b.2.2, 80.192.1-8.b.2.3, 80.192.1-8.b.2.4, 88.192.1-8.b.2.1, 88.192.1-8.b.2.2, 88.192.1-8.b.2.3, 88.192.1-8.b.2.4, 88.192.1-8.b.2.5, 88.192.1-8.b.2.6, 104.192.1-8.b.2.1, 104.192.1-8.b.2.2, 104.192.1-8.b.2.3, 104.192.1-8.b.2.4, 104.192.1-8.b.2.5, 104.192.1-8.b.2.6, 112.192.1-8.b.2.1, 112.192.1-8.b.2.2, 112.192.1-8.b.2.3, 112.192.1-8.b.2.4, 120.192.1-8.b.2.1, 120.192.1-8.b.2.2, 120.192.1-8.b.2.3, 120.192.1-8.b.2.4, 120.192.1-8.b.2.5, 120.192.1-8.b.2.6, 136.192.1-8.b.2.1, 136.192.1-8.b.2.2, 136.192.1-8.b.2.3, 136.192.1-8.b.2.4, 136.192.1-8.b.2.5, 136.192.1-8.b.2.6, 152.192.1-8.b.2.1, 152.192.1-8.b.2.2, 152.192.1-8.b.2.3, 152.192.1-8.b.2.4, 152.192.1-8.b.2.5, 152.192.1-8.b.2.6, 168.192.1-8.b.2.1, 168.192.1-8.b.2.2, 168.192.1-8.b.2.3, 168.192.1-8.b.2.4, 168.192.1-8.b.2.5, 168.192.1-8.b.2.6, 176.192.1-8.b.2.1, 176.192.1-8.b.2.2, 176.192.1-8.b.2.3, 176.192.1-8.b.2.4, 184.192.1-8.b.2.1, 184.192.1-8.b.2.2, 184.192.1-8.b.2.3, 184.192.1-8.b.2.4, 184.192.1-8.b.2.5, 184.192.1-8.b.2.6, 208.192.1-8.b.2.1, 208.192.1-8.b.2.2, 208.192.1-8.b.2.3, 208.192.1-8.b.2.4, 232.192.1-8.b.2.1, 232.192.1-8.b.2.2, 232.192.1-8.b.2.3, 232.192.1-8.b.2.4, 232.192.1-8.b.2.5, 232.192.1-8.b.2.6, 240.192.1-8.b.2.1, 240.192.1-8.b.2.2, 240.192.1-8.b.2.3, 240.192.1-8.b.2.4, 248.192.1-8.b.2.1, 248.192.1-8.b.2.2, 248.192.1-8.b.2.3, 248.192.1-8.b.2.4, 248.192.1-8.b.2.5, 248.192.1-8.b.2.6, 264.192.1-8.b.2.1, 264.192.1-8.b.2.2, 264.192.1-8.b.2.3, 264.192.1-8.b.2.4, 264.192.1-8.b.2.5, 264.192.1-8.b.2.6, 272.192.1-8.b.2.1, 272.192.1-8.b.2.2, 272.192.1-8.b.2.3, 272.192.1-8.b.2.4, 280.192.1-8.b.2.1, 280.192.1-8.b.2.2, 280.192.1-8.b.2.3, 280.192.1-8.b.2.4, 280.192.1-8.b.2.5, 280.192.1-8.b.2.6, 296.192.1-8.b.2.1, 296.192.1-8.b.2.2, 296.192.1-8.b.2.3, 296.192.1-8.b.2.4, 296.192.1-8.b.2.5, 296.192.1-8.b.2.6, 304.192.1-8.b.2.1, 304.192.1-8.b.2.2, 304.192.1-8.b.2.3, 304.192.1-8.b.2.4, 312.192.1-8.b.2.1, 312.192.1-8.b.2.2, 312.192.1-8.b.2.3, 312.192.1-8.b.2.4, 312.192.1-8.b.2.5, 312.192.1-8.b.2.6, 328.192.1-8.b.2.1, 328.192.1-8.b.2.2, 328.192.1-8.b.2.3, 328.192.1-8.b.2.4, 328.192.1-8.b.2.5, 328.192.1-8.b.2.6 |
Cyclic 8-isogeny field degree: |
$1$ |
Cyclic 8-torsion field degree: |
$4$ |
Full 8-torsion field degree: |
$16$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ - x w + y z - w^{2} $ |
| $=$ | $2 x^{2} + 2 y^{2} + z^{2} - w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 2 x^{2} y^{2} + x^{2} z^{2} - 4 x y z^{2} + 2 y^{2} z^{2} + 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:
$\displaystyle X$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{49152xy^{22}w-1466368xy^{20}w^{3}+19271680xy^{18}w^{5}-184453120xy^{16}w^{7}+1386215424xy^{14}w^{9}-8600665088xy^{12}w^{11}+45531144704xy^{10}w^{13}-209168105728xy^{8}w^{15}+844995234752xy^{6}w^{17}-3031885394912xy^{4}w^{19}+9737073259728xy^{2}w^{21}+1536xz^{22}w-91648xz^{20}w^{3}+2132480xz^{18}w^{5}-29616640xz^{16}w^{7}+292119552xz^{14}w^{9}-2246878208xz^{12}w^{11}+14287756288xz^{10}w^{13}-78171937792xz^{8}w^{15}+378744772096xz^{6}w^{17}-1660188227072xz^{4}w^{19}+6691527399936xz^{2}w^{21}-19362536159720xw^{23}-4096y^{24}+319488y^{22}w^{2}-5744640y^{20}w^{4}+63764480y^{18}w^{6}-542586624y^{16}w^{8}+3713661952y^{14}w^{10}-21390375680y^{12}w^{12}+106080019200y^{10}w^{14}-459715218416y^{8}w^{16}+1761321068384y^{6}w^{18}-6018128672040y^{4}w^{20}+18471369045464y^{2}w^{22}-64z^{24}+9984z^{22}w^{2}-347520z^{20}w^{4}+6173440z^{18}w^{6}-72378816z^{16}w^{8}+634190336z^{14}w^{10}-4467426560z^{12}w^{12}+26545697280z^{10}w^{14}-137645104064z^{8}w^{16}+638552342272z^{6}w^{18}-2700619079040z^{4}w^{20}+10562968484608z^{2}w^{22}-14116515858985w^{24}}{w^{4}(3072xy^{18}w-91648xy^{16}w^{3}+1061632xy^{14}w^{5}-7266688xy^{12}w^{7}+34910016xy^{10}w^{9}-129157536xy^{8}w^{11}+389799120xy^{6}w^{13}-997721064xy^{4}w^{15}+2227094016xy^{2}w^{17}+96xz^{14}w^{5}-5728xz^{12}w^{7}+133280xz^{10}w^{9}-1851040xz^{8}w^{11}+18253056xz^{6}w^{13}-140166400xz^{4}w^{15}+886853888xz^{2}w^{17}-3379610624xw^{19}-256y^{20}+19968y^{18}w^{2}-347136y^{16}w^{4}+3056768y^{14}w^{6}-17572960y^{12}w^{8}+74607520y^{10}w^{10}-251443200y^{8}w^{12}+705123800y^{6}w^{14}-1700412393y^{4}w^{16}+3612492800y^{2}w^{18}-4z^{16}w^{4}+624z^{14}w^{6}-21720z^{12}w^{8}+385840z^{10}w^{10}-4523492z^{8}w^{12}+39608192z^{6}w^{14}-278215040z^{4}w^{16}+1641357440z^{2}w^{18}-2462761760w^{20})}$ |
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.