$\GL_2(\Z/16\Z)$-generators: |
$\begin{bmatrix}1&5\\8&15\end{bmatrix}$, $\begin{bmatrix}1&10\\0&9\end{bmatrix}$, $\begin{bmatrix}7&6\\0&7\end{bmatrix}$, $\begin{bmatrix}9&7\\0&11\end{bmatrix}$ |
$\GL_2(\Z/16\Z)$-subgroup: |
$C_2\times C_4^2.Q_8$ |
Contains $-I$: |
yes |
Quadratic refinements: |
16.192.1-16.n.2.1, 16.192.1-16.n.2.2, 16.192.1-16.n.2.3, 16.192.1-16.n.2.4, 16.192.1-16.n.2.5, 16.192.1-16.n.2.6, 48.192.1-16.n.2.1, 48.192.1-16.n.2.2, 48.192.1-16.n.2.3, 48.192.1-16.n.2.4, 48.192.1-16.n.2.5, 48.192.1-16.n.2.6, 80.192.1-16.n.2.1, 80.192.1-16.n.2.2, 80.192.1-16.n.2.3, 80.192.1-16.n.2.4, 80.192.1-16.n.2.5, 80.192.1-16.n.2.6, 112.192.1-16.n.2.1, 112.192.1-16.n.2.2, 112.192.1-16.n.2.3, 112.192.1-16.n.2.4, 112.192.1-16.n.2.5, 112.192.1-16.n.2.6, 176.192.1-16.n.2.1, 176.192.1-16.n.2.2, 176.192.1-16.n.2.3, 176.192.1-16.n.2.4, 176.192.1-16.n.2.5, 176.192.1-16.n.2.6, 208.192.1-16.n.2.1, 208.192.1-16.n.2.2, 208.192.1-16.n.2.3, 208.192.1-16.n.2.4, 208.192.1-16.n.2.5, 208.192.1-16.n.2.6, 240.192.1-16.n.2.1, 240.192.1-16.n.2.2, 240.192.1-16.n.2.3, 240.192.1-16.n.2.4, 240.192.1-16.n.2.5, 240.192.1-16.n.2.6, 272.192.1-16.n.2.1, 272.192.1-16.n.2.2, 272.192.1-16.n.2.3, 272.192.1-16.n.2.4, 272.192.1-16.n.2.5, 272.192.1-16.n.2.6, 304.192.1-16.n.2.1, 304.192.1-16.n.2.2, 304.192.1-16.n.2.3, 304.192.1-16.n.2.4, 304.192.1-16.n.2.5, 304.192.1-16.n.2.6 |
Cyclic 16-isogeny field degree: |
$2$ |
Cyclic 16-torsion field degree: |
$8$ |
Full 16-torsion field degree: |
$256$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ y^{2} + 2 z w $ |
| $=$ | $x^{2} + 2 y^{2} + z^{2} - 2 z w + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 4 x^{4} + 12 x^{2} z^{2} + 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:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle 2x$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle 2w$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{(z^{8}-8z^{7}w+12z^{6}w^{2}+8z^{5}w^{3}-10z^{4}w^{4}+8z^{3}w^{5}+12z^{2}w^{6}-8zw^{7}+w^{8})^{3}}{w^{8}z^{8}(z-w)^{4}(z+w)^{2}(z^{2}-6zw+w^{2})}$ |
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.