$\GL_2(\Z/8\Z)$-generators: |
$\begin{bmatrix}1&0\\0&3\end{bmatrix}$, $\begin{bmatrix}3&2\\0&5\end{bmatrix}$, $\begin{bmatrix}7&4\\0&7\end{bmatrix}$ |
$\GL_2(\Z/8\Z)$-subgroup: |
$C_2\times D_4$ |
Contains $-I$: |
yes |
Quadratic refinements: |
8.192.1-8.i.2.1, 8.192.1-8.i.2.2, 8.192.1-8.i.2.3, 8.192.1-8.i.2.4, 16.192.1-8.i.2.1, 16.192.1-8.i.2.2, 24.192.1-8.i.2.1, 24.192.1-8.i.2.2, 24.192.1-8.i.2.3, 24.192.1-8.i.2.4, 40.192.1-8.i.2.1, 40.192.1-8.i.2.2, 40.192.1-8.i.2.3, 40.192.1-8.i.2.4, 48.192.1-8.i.2.1, 48.192.1-8.i.2.2, 56.192.1-8.i.2.1, 56.192.1-8.i.2.2, 56.192.1-8.i.2.3, 56.192.1-8.i.2.4, 80.192.1-8.i.2.1, 80.192.1-8.i.2.2, 88.192.1-8.i.2.1, 88.192.1-8.i.2.2, 88.192.1-8.i.2.3, 88.192.1-8.i.2.4, 104.192.1-8.i.2.1, 104.192.1-8.i.2.2, 104.192.1-8.i.2.3, 104.192.1-8.i.2.4, 112.192.1-8.i.2.1, 112.192.1-8.i.2.2, 120.192.1-8.i.2.1, 120.192.1-8.i.2.2, 120.192.1-8.i.2.3, 120.192.1-8.i.2.4, 136.192.1-8.i.2.1, 136.192.1-8.i.2.2, 136.192.1-8.i.2.3, 136.192.1-8.i.2.4, 152.192.1-8.i.2.1, 152.192.1-8.i.2.2, 152.192.1-8.i.2.3, 152.192.1-8.i.2.4, 168.192.1-8.i.2.1, 168.192.1-8.i.2.2, 168.192.1-8.i.2.3, 168.192.1-8.i.2.4, 176.192.1-8.i.2.1, 176.192.1-8.i.2.2, 184.192.1-8.i.2.1, 184.192.1-8.i.2.2, 184.192.1-8.i.2.3, 184.192.1-8.i.2.4, 208.192.1-8.i.2.1, 208.192.1-8.i.2.2, 232.192.1-8.i.2.1, 232.192.1-8.i.2.2, 232.192.1-8.i.2.3, 232.192.1-8.i.2.4, 240.192.1-8.i.2.1, 240.192.1-8.i.2.2, 248.192.1-8.i.2.1, 248.192.1-8.i.2.2, 248.192.1-8.i.2.3, 248.192.1-8.i.2.4, 264.192.1-8.i.2.1, 264.192.1-8.i.2.2, 264.192.1-8.i.2.3, 264.192.1-8.i.2.4, 272.192.1-8.i.2.1, 272.192.1-8.i.2.2, 280.192.1-8.i.2.1, 280.192.1-8.i.2.2, 280.192.1-8.i.2.3, 280.192.1-8.i.2.4, 296.192.1-8.i.2.1, 296.192.1-8.i.2.2, 296.192.1-8.i.2.3, 296.192.1-8.i.2.4, 304.192.1-8.i.2.1, 304.192.1-8.i.2.2, 312.192.1-8.i.2.1, 312.192.1-8.i.2.2, 312.192.1-8.i.2.3, 312.192.1-8.i.2.4, 328.192.1-8.i.2.1, 328.192.1-8.i.2.2, 328.192.1-8.i.2.3, 328.192.1-8.i.2.4 |
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^{2} - x y - x z + 2 y z $ |
| $=$ | $x y + x z - 2 y^{2} + 2 y z - 2 z^{2} - 2 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 2 x^{4} - 4 x^{3} z + x^{2} y^{2} + 6 x^{2} z^{2} - 2 x y^{2} z - 4 x z^{3} + 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:
$\displaystyle X$ |
$=$ |
$\displaystyle x$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle 2w$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle 2z$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle 2^2\,\frac{887040xz^{23}+2204544xz^{21}w^{2}-1654656xz^{19}w^{4}-14230144xz^{17}w^{6}-20976256xz^{15}w^{8}-10792512xz^{13}w^{10}+4253248xz^{11}w^{12}+8391808xz^{9}w^{14}+4190448xz^{7}w^{16}+883944xz^{5}w^{18}+65544xz^{3}w^{20}-887040y^{2}z^{22}+7329312y^{2}z^{18}w^{4}+16330064y^{2}z^{16}w^{6}+9930624y^{2}z^{14}w^{8}-7591584y^{2}z^{12}w^{10}-14721088y^{2}z^{10}w^{12}-8008992y^{2}z^{8}w^{14}-1092016y^{2}z^{6}w^{16}+294672y^{2}z^{4}w^{18}+73734y^{2}z^{2}w^{20}+4095y^{2}w^{22}+1774080yz^{23}+2674176yz^{21}w^{2}-8805120yz^{19}w^{4}-31397792yz^{17}w^{6}-29950208yz^{15}w^{8}+919488yz^{13}w^{10}+23152000yz^{11}w^{12}+16801088yz^{9}w^{14}+3746528yz^{7}w^{16}-260928yz^{5}w^{18}-131088yz^{3}w^{20}-8190yzw^{22}-1406656z^{24}-4004736z^{22}w^{2}+2665536z^{20}w^{4}+28243792z^{18}w^{6}+47294832z^{16}w^{8}+27314976z^{14}w^{10}-11352864z^{12}w^{12}-25908576z^{10}w^{14}-14396444z^{8}w^{16}-2792344z^{6}w^{18}+114252z^{4}w^{20}+69639z^{2}w^{22}+4094w^{24}}{w^{4}z^{4}(3264xz^{15}-48352xz^{13}w^{2}-75424xz^{11}w^{4}-14128xz^{9}w^{6}+10744xz^{7}w^{8}+3196xz^{5}w^{10}+64xz^{3}w^{12}-4xzw^{14}+7880y^{2}z^{14}+66980y^{2}z^{12}w^{2}+31968y^{2}z^{10}w^{4}-37128y^{2}z^{8}w^{6}-13330y^{2}z^{6}w^{8}+543y^{2}z^{4}w^{10}+28y^{2}z^{2}w^{12}-y^{2}w^{14}-117128yz^{13}w^{2}-109632yz^{11}w^{4}+47280yz^{9}w^{6}+30368yz^{7}w^{8}-406yz^{5}w^{10}-112yz^{3}w^{12}+6yzw^{14}-3264z^{16}+82180z^{14}w^{2}+173468z^{12}w^{4}+61256z^{10}w^{6}-46048z^{8}w^{8}-19053z^{6}w^{10}+276z^{4}w^{12}+47z^{2}w^{14}-2w^{16})}$ |
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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.