$\GL_2(\Z/16\Z)$-generators: |
$\begin{bmatrix}7&9\\0&13\end{bmatrix}$, $\begin{bmatrix}13&7\\0&9\end{bmatrix}$, $\begin{bmatrix}15&4\\0&15\end{bmatrix}$ |
$\GL_2(\Z/16\Z)$-subgroup: |
$C_2^4.\SD_{16}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
16.192.1-16.p.1.1, 16.192.1-16.p.1.2, 16.192.1-16.p.1.3, 16.192.1-16.p.1.4, 32.192.1-16.p.1.1, 32.192.1-16.p.1.2, 48.192.1-16.p.1.1, 48.192.1-16.p.1.2, 48.192.1-16.p.1.3, 48.192.1-16.p.1.4, 80.192.1-16.p.1.1, 80.192.1-16.p.1.2, 80.192.1-16.p.1.3, 80.192.1-16.p.1.4, 96.192.1-16.p.1.1, 96.192.1-16.p.1.2, 112.192.1-16.p.1.1, 112.192.1-16.p.1.2, 112.192.1-16.p.1.3, 112.192.1-16.p.1.4, 160.192.1-16.p.1.1, 160.192.1-16.p.1.2, 176.192.1-16.p.1.1, 176.192.1-16.p.1.2, 176.192.1-16.p.1.3, 176.192.1-16.p.1.4, 208.192.1-16.p.1.1, 208.192.1-16.p.1.2, 208.192.1-16.p.1.3, 208.192.1-16.p.1.4, 224.192.1-16.p.1.1, 224.192.1-16.p.1.2, 240.192.1-16.p.1.1, 240.192.1-16.p.1.2, 240.192.1-16.p.1.3, 240.192.1-16.p.1.4, 272.192.1-16.p.1.1, 272.192.1-16.p.1.2, 272.192.1-16.p.1.3, 272.192.1-16.p.1.4, 304.192.1-16.p.1.1, 304.192.1-16.p.1.2, 304.192.1-16.p.1.3, 304.192.1-16.p.1.4 |
Cyclic 16-isogeny field degree: |
$1$ |
Cyclic 16-torsion field degree: |
$8$ |
Full 16-torsion field degree: |
$256$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ x y - y^{2} - 3 z^{2} - w^{2} $ |
| $=$ | $x^{2} - 2 x y + 2 y^{2} - 2 z^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 9 x^{4} - 2 x^{2} y^{2} + 6 x^{2} z^{2} + y^{4} + 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 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle 2^8\,\frac{(z^{8}+16z^{6}w^{2}+20z^{4}w^{4}+8z^{2}w^{6}+w^{8})^{3}}{w^{2}z^{16}(2z^{2}+w^{2})^{2}(4z^{2}+w^{2})}$ |
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.