$\GL_2(\Z/8\Z)$-generators: |
$\begin{bmatrix}1&3\\4&7\end{bmatrix}$, $\begin{bmatrix}3&3\\2&1\end{bmatrix}$, $\begin{bmatrix}7&3\\2&5\end{bmatrix}$ |
$\GL_2(\Z/8\Z)$-subgroup: |
$C_2^2.D_4$ |
Contains $-I$: |
yes |
Quadratic refinements: |
8.96.1-8.bo.1.1, 16.96.1-8.bo.1.1, 16.96.1-8.bo.1.2, 24.96.1-8.bo.1.1, 40.96.1-8.bo.1.1, 48.96.1-8.bo.1.1, 48.96.1-8.bo.1.2, 56.96.1-8.bo.1.1, 80.96.1-8.bo.1.1, 80.96.1-8.bo.1.2, 88.96.1-8.bo.1.1, 104.96.1-8.bo.1.1, 112.96.1-8.bo.1.1, 112.96.1-8.bo.1.2, 120.96.1-8.bo.1.1, 136.96.1-8.bo.1.1, 152.96.1-8.bo.1.1, 168.96.1-8.bo.1.1, 176.96.1-8.bo.1.1, 176.96.1-8.bo.1.2, 184.96.1-8.bo.1.1, 208.96.1-8.bo.1.1, 208.96.1-8.bo.1.2, 232.96.1-8.bo.1.1, 240.96.1-8.bo.1.1, 240.96.1-8.bo.1.2, 248.96.1-8.bo.1.1, 264.96.1-8.bo.1.1, 272.96.1-8.bo.1.1, 272.96.1-8.bo.1.2, 280.96.1-8.bo.1.1, 296.96.1-8.bo.1.1, 304.96.1-8.bo.1.1, 304.96.1-8.bo.1.2, 312.96.1-8.bo.1.1, 328.96.1-8.bo.1.1 |
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 $ | $=$ | $ 4 x^{2} - y^{2} - y z - z^{2} - w^{2} $ |
| $=$ | $3 y^{2} + 2 y z + 3 z^{2} + 2 w^{2}$ |
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:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle 2x$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle 2^8\cdot3^3\,\frac{z^{3}(230yz^{8}+258yz^{6}w^{2}-2160yz^{4}w^{4}-3024yz^{2}w^{6}-1296yw^{8}-33z^{9}-1426z^{7}w^{2}-3744z^{5}w^{4}-2520z^{3}w^{6}-432zw^{8})}{920yz^{11}+2220yz^{9}w^{2}-2700yz^{7}w^{4}-3780yz^{5}w^{6}-1620yz^{3}w^{8}-132z^{12}+3368z^{10}w^{2}+4113z^{8}w^{4}+396z^{6}w^{6}-1998z^{4}w^{8}-972z^{2}w^{10}-243w^{12}}$ |
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.