$\GL_2(\Z/8\Z)$-generators: |
$\begin{bmatrix}3&2\\0&7\end{bmatrix}$, $\begin{bmatrix}3&6\\0&1\end{bmatrix}$, $\begin{bmatrix}5&6\\4&5\end{bmatrix}$, $\begin{bmatrix}7&6\\0&5\end{bmatrix}$, $\begin{bmatrix}7&6\\4&1\end{bmatrix}$ |
$\GL_2(\Z/8\Z)$-subgroup: |
$D_4\times C_2^3$ |
Contains $-I$: |
yes |
Quadratic refinements: |
8.48.1-8.d.1.1, 8.48.1-8.d.1.2, 8.48.1-8.d.1.3, 8.48.1-8.d.1.4, 8.48.1-8.d.1.5, 8.48.1-8.d.1.6, 8.48.1-8.d.1.7, 8.48.1-8.d.1.8, 8.48.1-8.d.1.9, 8.48.1-8.d.1.10, 24.48.1-8.d.1.1, 24.48.1-8.d.1.2, 24.48.1-8.d.1.3, 24.48.1-8.d.1.4, 24.48.1-8.d.1.5, 24.48.1-8.d.1.6, 24.48.1-8.d.1.7, 24.48.1-8.d.1.8, 24.48.1-8.d.1.9, 24.48.1-8.d.1.10, 40.48.1-8.d.1.1, 40.48.1-8.d.1.2, 40.48.1-8.d.1.3, 40.48.1-8.d.1.4, 40.48.1-8.d.1.5, 40.48.1-8.d.1.6, 40.48.1-8.d.1.7, 40.48.1-8.d.1.8, 40.48.1-8.d.1.9, 40.48.1-8.d.1.10, 56.48.1-8.d.1.1, 56.48.1-8.d.1.2, 56.48.1-8.d.1.3, 56.48.1-8.d.1.4, 56.48.1-8.d.1.5, 56.48.1-8.d.1.6, 56.48.1-8.d.1.7, 56.48.1-8.d.1.8, 56.48.1-8.d.1.9, 56.48.1-8.d.1.10, 88.48.1-8.d.1.1, 88.48.1-8.d.1.2, 88.48.1-8.d.1.3, 88.48.1-8.d.1.4, 88.48.1-8.d.1.5, 88.48.1-8.d.1.6, 88.48.1-8.d.1.7, 88.48.1-8.d.1.8, 88.48.1-8.d.1.9, 88.48.1-8.d.1.10, 104.48.1-8.d.1.1, 104.48.1-8.d.1.2, 104.48.1-8.d.1.3, 104.48.1-8.d.1.4, 104.48.1-8.d.1.5, 104.48.1-8.d.1.6, 104.48.1-8.d.1.7, 104.48.1-8.d.1.8, 104.48.1-8.d.1.9, 104.48.1-8.d.1.10, 120.48.1-8.d.1.1, 120.48.1-8.d.1.2, 120.48.1-8.d.1.3, 120.48.1-8.d.1.4, 120.48.1-8.d.1.5, 120.48.1-8.d.1.6, 120.48.1-8.d.1.7, 120.48.1-8.d.1.8, 120.48.1-8.d.1.9, 120.48.1-8.d.1.10, 136.48.1-8.d.1.1, 136.48.1-8.d.1.2, 136.48.1-8.d.1.3, 136.48.1-8.d.1.4, 136.48.1-8.d.1.5, 136.48.1-8.d.1.6, 136.48.1-8.d.1.7, 136.48.1-8.d.1.8, 136.48.1-8.d.1.9, 136.48.1-8.d.1.10, 152.48.1-8.d.1.1, 152.48.1-8.d.1.2, 152.48.1-8.d.1.3, 152.48.1-8.d.1.4, 152.48.1-8.d.1.5, 152.48.1-8.d.1.6, 152.48.1-8.d.1.7, 152.48.1-8.d.1.8, 152.48.1-8.d.1.9, 152.48.1-8.d.1.10, 168.48.1-8.d.1.1, 168.48.1-8.d.1.2, 168.48.1-8.d.1.3, 168.48.1-8.d.1.4, 168.48.1-8.d.1.5, 168.48.1-8.d.1.6, 168.48.1-8.d.1.7, 168.48.1-8.d.1.8, 168.48.1-8.d.1.9, 168.48.1-8.d.1.10, 184.48.1-8.d.1.1, 184.48.1-8.d.1.2, 184.48.1-8.d.1.3, 184.48.1-8.d.1.4, 184.48.1-8.d.1.5, 184.48.1-8.d.1.6, 184.48.1-8.d.1.7, 184.48.1-8.d.1.8, 184.48.1-8.d.1.9, 184.48.1-8.d.1.10, 232.48.1-8.d.1.1, 232.48.1-8.d.1.2, 232.48.1-8.d.1.3, 232.48.1-8.d.1.4, 232.48.1-8.d.1.5, 232.48.1-8.d.1.6, 232.48.1-8.d.1.7, 232.48.1-8.d.1.8, 232.48.1-8.d.1.9, 232.48.1-8.d.1.10, 248.48.1-8.d.1.1, 248.48.1-8.d.1.2, 248.48.1-8.d.1.3, 248.48.1-8.d.1.4, 248.48.1-8.d.1.5, 248.48.1-8.d.1.6, 248.48.1-8.d.1.7, 248.48.1-8.d.1.8, 248.48.1-8.d.1.9, 248.48.1-8.d.1.10, 264.48.1-8.d.1.1, 264.48.1-8.d.1.2, 264.48.1-8.d.1.3, 264.48.1-8.d.1.4, 264.48.1-8.d.1.5, 264.48.1-8.d.1.6, 264.48.1-8.d.1.7, 264.48.1-8.d.1.8, 264.48.1-8.d.1.9, 264.48.1-8.d.1.10, 280.48.1-8.d.1.1, 280.48.1-8.d.1.2, 280.48.1-8.d.1.3, 280.48.1-8.d.1.4, 280.48.1-8.d.1.5, 280.48.1-8.d.1.6, 280.48.1-8.d.1.7, 280.48.1-8.d.1.8, 280.48.1-8.d.1.9, 280.48.1-8.d.1.10, 296.48.1-8.d.1.1, 296.48.1-8.d.1.2, 296.48.1-8.d.1.3, 296.48.1-8.d.1.4, 296.48.1-8.d.1.5, 296.48.1-8.d.1.6, 296.48.1-8.d.1.7, 296.48.1-8.d.1.8, 296.48.1-8.d.1.9, 296.48.1-8.d.1.10, 312.48.1-8.d.1.1, 312.48.1-8.d.1.2, 312.48.1-8.d.1.3, 312.48.1-8.d.1.4, 312.48.1-8.d.1.5, 312.48.1-8.d.1.6, 312.48.1-8.d.1.7, 312.48.1-8.d.1.8, 312.48.1-8.d.1.9, 312.48.1-8.d.1.10, 328.48.1-8.d.1.1, 328.48.1-8.d.1.2, 328.48.1-8.d.1.3, 328.48.1-8.d.1.4, 328.48.1-8.d.1.5, 328.48.1-8.d.1.6, 328.48.1-8.d.1.7, 328.48.1-8.d.1.8, 328.48.1-8.d.1.9, 328.48.1-8.d.1.10 |
Cyclic 8-isogeny field degree: |
$2$ |
Cyclic 8-torsion field degree: |
$8$ |
Full 8-torsion field degree: |
$64$ |
Weierstrass model Weierstrass model
$ y^{2} $ | $=$ | $ x^{3} - 4x $ |
This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
Maps to other modular curves
$j$-invariant map
of degree 24 from the Weierstrass model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle 2^4\,\frac{48x^{2}y^{4}z^{2}+4xy^{6}z+768xy^{2}z^{5}+y^{8}+4096z^{8}}{z^{2}y^{4}x^{2}}$ |
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.