$\GL_2(\Z/8\Z)$-generators: |
$\begin{bmatrix}1&4\\0&1\end{bmatrix}$, $\begin{bmatrix}3&0\\0&3\end{bmatrix}$, $\begin{bmatrix}3&2\\2&1\end{bmatrix}$, $\begin{bmatrix}5&0\\0&1\end{bmatrix}$ |
$\GL_2(\Z/8\Z)$-subgroup: |
$C_2^3\times C_4$ |
Contains $-I$: |
yes |
Quadratic refinements: |
8.96.1-8.b.1.1, 8.96.1-8.b.1.2, 8.96.1-8.b.1.3, 24.96.1-8.b.1.1, 24.96.1-8.b.1.2, 24.96.1-8.b.1.3, 40.96.1-8.b.1.1, 40.96.1-8.b.1.2, 40.96.1-8.b.1.3, 56.96.1-8.b.1.1, 56.96.1-8.b.1.2, 56.96.1-8.b.1.3, 88.96.1-8.b.1.1, 88.96.1-8.b.1.2, 88.96.1-8.b.1.3, 104.96.1-8.b.1.1, 104.96.1-8.b.1.2, 104.96.1-8.b.1.3, 120.96.1-8.b.1.1, 120.96.1-8.b.1.2, 120.96.1-8.b.1.3, 136.96.1-8.b.1.1, 136.96.1-8.b.1.2, 136.96.1-8.b.1.3, 152.96.1-8.b.1.1, 152.96.1-8.b.1.2, 152.96.1-8.b.1.3, 168.96.1-8.b.1.1, 168.96.1-8.b.1.2, 168.96.1-8.b.1.3, 184.96.1-8.b.1.1, 184.96.1-8.b.1.2, 184.96.1-8.b.1.3, 232.96.1-8.b.1.1, 232.96.1-8.b.1.2, 232.96.1-8.b.1.3, 248.96.1-8.b.1.1, 248.96.1-8.b.1.2, 248.96.1-8.b.1.3, 264.96.1-8.b.1.1, 264.96.1-8.b.1.2, 264.96.1-8.b.1.3, 280.96.1-8.b.1.1, 280.96.1-8.b.1.2, 280.96.1-8.b.1.3, 296.96.1-8.b.1.1, 296.96.1-8.b.1.2, 296.96.1-8.b.1.3, 312.96.1-8.b.1.1, 312.96.1-8.b.1.2, 312.96.1-8.b.1.3, 328.96.1-8.b.1.1, 328.96.1-8.b.1.2, 328.96.1-8.b.1.3 |
Cyclic 8-isogeny field degree: |
$2$ |
Cyclic 8-torsion field degree: |
$8$ |
Full 8-torsion field degree: |
$32$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ x y - x z - y^{2} - z^{2} $ |
| $=$ | $x^{2} + 3 x y + x z + y^{2} + z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 5 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:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle w$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle z$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle -\frac{2^4}{5^4}\cdot\frac{156196352xz^{11}-19655680xz^{9}w^{2}-230227200xz^{7}w^{4}-8480000xz^{5}w^{6}+47900000xz^{3}w^{8}+781146496y^{2}z^{10}+878604480y^{2}z^{8}w^{2}-518820800y^{2}z^{6}w^{4}-406620000y^{2}z^{4}w^{6}+122775000y^{2}z^{2}w^{8}+19887500y^{2}w^{10}-781262464yz^{9}w^{2}-703165440yz^{7}w^{4}+174446400yz^{5}w^{6}+126560000yz^{3}w^{8}-15825000yzw^{10}+156146496z^{12}+136391104z^{10}w^{2}+20653040z^{8}w^{4}+42893600z^{6}w^{6}+427500z^{4}w^{8}-4062500z^{2}w^{10}+9765625w^{12}}{w^{4}(3888xz^{7}-560xz^{5}w^{2}-400xz^{3}w^{4}+19524y^{2}z^{6}+12150y^{2}z^{4}w^{2}+1100y^{2}z^{2}w^{4}+125y^{2}w^{6}-11716yz^{5}w^{2}-4880yz^{3}w^{4}-450yzw^{6}+3899z^{8}+1406z^{6}w^{2}+1880z^{4}w^{4}+325z^{2}w^{6})}$ |
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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.