$\GL_2(\Z/16\Z)$-generators: |
$\begin{bmatrix}1&15\\12&1\end{bmatrix}$, $\begin{bmatrix}11&2\\0&9\end{bmatrix}$, $\begin{bmatrix}13&14\\0&7\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
16.96.1-16.c.1.1, 16.96.1-16.c.1.2, 16.96.1-16.c.1.3, 16.96.1-16.c.1.4, 32.96.1-16.c.1.1, 32.96.1-16.c.1.2, 48.96.1-16.c.1.1, 48.96.1-16.c.1.2, 48.96.1-16.c.1.3, 48.96.1-16.c.1.4, 80.96.1-16.c.1.1, 80.96.1-16.c.1.2, 80.96.1-16.c.1.3, 80.96.1-16.c.1.4, 96.96.1-16.c.1.1, 96.96.1-16.c.1.2, 112.96.1-16.c.1.1, 112.96.1-16.c.1.2, 112.96.1-16.c.1.3, 112.96.1-16.c.1.4, 160.96.1-16.c.1.1, 160.96.1-16.c.1.2, 176.96.1-16.c.1.1, 176.96.1-16.c.1.2, 176.96.1-16.c.1.3, 176.96.1-16.c.1.4, 208.96.1-16.c.1.1, 208.96.1-16.c.1.2, 208.96.1-16.c.1.3, 208.96.1-16.c.1.4, 224.96.1-16.c.1.1, 224.96.1-16.c.1.2, 240.96.1-16.c.1.1, 240.96.1-16.c.1.2, 240.96.1-16.c.1.3, 240.96.1-16.c.1.4, 272.96.1-16.c.1.1, 272.96.1-16.c.1.2, 272.96.1-16.c.1.3, 272.96.1-16.c.1.4, 304.96.1-16.c.1.1, 304.96.1-16.c.1.2, 304.96.1-16.c.1.3, 304.96.1-16.c.1.4 |
Cyclic 16-isogeny field degree: |
$2$ |
Cyclic 16-torsion field degree: |
$16$ |
Full 16-torsion field degree: |
$512$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ - 2 y w + z^{2} $ |
| $=$ | $x^{2} + 32 y^{2} + 2 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} + 2 y^{2} z^{2} + 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:
$\displaystyle X$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{2}x$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{2}w$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{(16y^{4}+16y^{2}w^{2}+w^{4})^{3}}{w^{2}y^{8}(16y^{2}+w^{2})}$ |
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.