$\GL_2(\Z/8\Z)$-generators: |
$\begin{bmatrix}3&7\\2&1\end{bmatrix}$, $\begin{bmatrix}5&1\\6&7\end{bmatrix}$, $\begin{bmatrix}7&4\\6&5\end{bmatrix}$ |
$\GL_2(\Z/8\Z)$-subgroup: |
$C_2^3:C_4$ |
Contains $-I$: |
yes |
Quadratic refinements: |
8.96.1-8.r.1.1, 8.96.1-8.r.1.2, 16.96.1-8.r.1.1, 16.96.1-8.r.1.2, 24.96.1-8.r.1.1, 24.96.1-8.r.1.2, 40.96.1-8.r.1.1, 40.96.1-8.r.1.2, 48.96.1-8.r.1.1, 48.96.1-8.r.1.2, 56.96.1-8.r.1.1, 56.96.1-8.r.1.2, 80.96.1-8.r.1.1, 80.96.1-8.r.1.2, 88.96.1-8.r.1.1, 88.96.1-8.r.1.2, 104.96.1-8.r.1.1, 104.96.1-8.r.1.2, 112.96.1-8.r.1.1, 112.96.1-8.r.1.2, 120.96.1-8.r.1.1, 120.96.1-8.r.1.2, 136.96.1-8.r.1.1, 136.96.1-8.r.1.2, 152.96.1-8.r.1.1, 152.96.1-8.r.1.2, 168.96.1-8.r.1.1, 168.96.1-8.r.1.2, 176.96.1-8.r.1.1, 176.96.1-8.r.1.2, 184.96.1-8.r.1.1, 184.96.1-8.r.1.2, 208.96.1-8.r.1.1, 208.96.1-8.r.1.2, 232.96.1-8.r.1.1, 232.96.1-8.r.1.2, 240.96.1-8.r.1.1, 240.96.1-8.r.1.2, 248.96.1-8.r.1.1, 248.96.1-8.r.1.2, 264.96.1-8.r.1.1, 264.96.1-8.r.1.2, 272.96.1-8.r.1.1, 272.96.1-8.r.1.2, 280.96.1-8.r.1.1, 280.96.1-8.r.1.2, 296.96.1-8.r.1.1, 296.96.1-8.r.1.2, 304.96.1-8.r.1.1, 304.96.1-8.r.1.2, 312.96.1-8.r.1.1, 312.96.1-8.r.1.2, 328.96.1-8.r.1.1, 328.96.1-8.r.1.2 |
Cyclic 8-isogeny field degree: |
$4$ |
Cyclic 8-torsion field degree: |
$16$ |
Full 8-torsion field degree: |
$32$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ x^{2} + y^{2} - y z + z^{2} + w^{2} $ |
| $=$ | $2 x^{2} - y^{2} - z^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} - 2 x^{2} y^{2} + 9 y^{4} + 6 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:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle x$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle 2^4\cdot3^3\,\frac{14720yz^{11}+48192yz^{9}w^{2}-544320yz^{7}w^{4}-59616yz^{5}w^{6}+911736yz^{3}w^{8}-218700yzw^{10}+2112z^{12}-150656z^{10}w^{2}+32976z^{8}w^{4}+921600z^{6}w^{6}-43092z^{4}w^{8}-398520z^{2}w^{10}+30375w^{12}}{3680yz^{11}+7296yz^{9}w^{2}+9504yz^{7}w^{4}+6264yz^{5}w^{6}+2754yz^{3}w^{8}+1458yzw^{10}+528z^{12}-1376z^{10}w^{2}-6120z^{8}w^{4}-5904z^{6}w^{6}-3915z^{4}w^{8}-1458z^{2}w^{10}+243w^{12}}$ |
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.