$\GL_2(\Z/8\Z)$-generators: |
$\begin{bmatrix}1&4\\4&5\end{bmatrix}$, $\begin{bmatrix}3&0\\0&3\end{bmatrix}$, $\begin{bmatrix}3&5\\3&6\end{bmatrix}$ |
$\GL_2(\Z/8\Z)$-subgroup: |
$C_2^2\times C_{12}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
8.64.1-8.a.1.1, 8.64.1-8.a.1.2, 24.64.1-8.a.1.1, 24.64.1-8.a.1.2, 40.64.1-8.a.1.1, 40.64.1-8.a.1.2, 56.64.1-8.a.1.1, 56.64.1-8.a.1.2, 88.64.1-8.a.1.1, 88.64.1-8.a.1.2, 104.64.1-8.a.1.1, 104.64.1-8.a.1.2, 120.64.1-8.a.1.1, 120.64.1-8.a.1.2, 136.64.1-8.a.1.1, 136.64.1-8.a.1.2, 152.64.1-8.a.1.1, 152.64.1-8.a.1.2, 168.64.1-8.a.1.1, 168.64.1-8.a.1.2, 184.64.1-8.a.1.1, 184.64.1-8.a.1.2, 232.64.1-8.a.1.1, 232.64.1-8.a.1.2, 248.64.1-8.a.1.1, 248.64.1-8.a.1.2, 264.64.1-8.a.1.1, 264.64.1-8.a.1.2, 280.64.1-8.a.1.1, 280.64.1-8.a.1.2, 296.64.1-8.a.1.1, 296.64.1-8.a.1.2, 312.64.1-8.a.1.1, 312.64.1-8.a.1.2, 328.64.1-8.a.1.1, 328.64.1-8.a.1.2 |
Cyclic 8-isogeny field degree: |
$12$ |
Cyclic 8-torsion field degree: |
$48$ |
Full 8-torsion field degree: |
$48$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 2 x y - 2 x z - 2 x w - y^{2} + z w $ |
| $=$ | $4 x^{2} + 2 x y + 2 x z + 2 x w + y^{2} + z^{2} - z w + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} - 2 x^{3} y + 2 x^{3} z + x^{2} y^{2} + 3 x^{2} z^{2} - 2 x y^{3} - 4 x y^{2} z + 2 x z^{3} + \cdots + 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 32 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle 2^{10}\cdot3^2\,\frac{7468xz^{7}+6272xz^{6}w-21084xz^{5}w^{2}+40320xz^{4}w^{3}+40320xz^{3}w^{4}-21084xz^{2}w^{5}+6272xzw^{6}+7468xw^{7}+3395y^{2}z^{6}-2870y^{2}z^{5}w-1589y^{2}z^{4}w^{2}+21556y^{2}z^{3}w^{3}-1589y^{2}z^{2}w^{4}-2870y^{2}zw^{5}+3395y^{2}w^{6}+38yz^{7}-1324yz^{6}w+2698yz^{5}w^{2}+2308yz^{4}w^{3}+2308yz^{3}w^{4}+2698yz^{2}w^{5}-1324yzw^{6}+38yw^{7}+1336z^{8}-5280z^{7}w+4223z^{6}w^{2}+11850z^{5}w^{3}-20145z^{4}w^{4}+11850z^{3}w^{5}+4223z^{2}w^{6}-5280zw^{7}+1336w^{8}}{4528xz^{7}-67440xz^{6}w-139664xz^{5}w^{2}+378448xz^{4}w^{3}+378448xz^{3}w^{4}-139664xz^{2}w^{5}-67440xzw^{6}+4528xw^{7}+368y^{2}z^{6}-39120y^{2}z^{5}w+6096y^{2}z^{4}w^{2}+168928y^{2}z^{3}w^{3}+6096y^{2}z^{2}w^{4}-39120y^{2}zw^{5}+368y^{2}w^{6}-1072yz^{7}-5936yz^{6}w+15856yz^{5}w^{2}+10992yz^{4}w^{3}+10992yz^{3}w^{4}+15856yz^{2}w^{5}-5936yzw^{6}-1072yw^{7}-425z^{8}-19448z^{7}w+46876z^{6}w^{2}+60248z^{5}w^{3}-152566z^{4}w^{4}+60248z^{3}w^{5}+46876z^{2}w^{6}-19448zw^{7}-425w^{8}}$ |
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.