$\GL_2(\Z/16\Z)$-generators: |
$\begin{bmatrix}5&10\\8&7\end{bmatrix}$, $\begin{bmatrix}5&14\\0&3\end{bmatrix}$, $\begin{bmatrix}7&8\\8&11\end{bmatrix}$, $\begin{bmatrix}13&0\\0&11\end{bmatrix}$ |
$\GL_2(\Z/16\Z)$-subgroup: |
$C_2\times C_8:C_4^2$ |
Contains $-I$: |
yes |
Quadratic refinements: |
16.192.1-16.a.1.1, 16.192.1-16.a.1.2, 16.192.1-16.a.1.3, 16.192.1-16.a.1.4, 16.192.1-16.a.1.5, 16.192.1-16.a.1.6, 48.192.1-16.a.1.1, 48.192.1-16.a.1.2, 48.192.1-16.a.1.3, 48.192.1-16.a.1.4, 48.192.1-16.a.1.5, 48.192.1-16.a.1.6, 80.192.1-16.a.1.1, 80.192.1-16.a.1.2, 80.192.1-16.a.1.3, 80.192.1-16.a.1.4, 80.192.1-16.a.1.5, 80.192.1-16.a.1.6, 112.192.1-16.a.1.1, 112.192.1-16.a.1.2, 112.192.1-16.a.1.3, 112.192.1-16.a.1.4, 112.192.1-16.a.1.5, 112.192.1-16.a.1.6, 176.192.1-16.a.1.1, 176.192.1-16.a.1.2, 176.192.1-16.a.1.3, 176.192.1-16.a.1.4, 176.192.1-16.a.1.5, 176.192.1-16.a.1.6, 208.192.1-16.a.1.1, 208.192.1-16.a.1.2, 208.192.1-16.a.1.3, 208.192.1-16.a.1.4, 208.192.1-16.a.1.5, 208.192.1-16.a.1.6, 240.192.1-16.a.1.1, 240.192.1-16.a.1.2, 240.192.1-16.a.1.3, 240.192.1-16.a.1.4, 240.192.1-16.a.1.5, 240.192.1-16.a.1.6, 272.192.1-16.a.1.1, 272.192.1-16.a.1.2, 272.192.1-16.a.1.3, 272.192.1-16.a.1.4, 272.192.1-16.a.1.5, 272.192.1-16.a.1.6, 304.192.1-16.a.1.1, 304.192.1-16.a.1.2, 304.192.1-16.a.1.3, 304.192.1-16.a.1.4, 304.192.1-16.a.1.5, 304.192.1-16.a.1.6 |
Cyclic 16-isogeny field degree: |
$2$ |
Cyclic 16-torsion field degree: |
$16$ |
Full 16-torsion field degree: |
$256$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ x^{2} + y z $ |
| $=$ | $y^{2} + z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} + 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:
$\displaystyle X$ |
$=$ |
$\displaystyle x$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle w$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle z$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle 2^8\,\frac{(z^{8}+2z^{6}w^{2}+5z^{4}w^{4}+4z^{2}w^{6}+w^{8})^{3}}{w^{4}z^{8}(z^{2}+w^{2})^{4}(2z^{2}+w^{2})^{2}}$ |
Hi
|
Cover information
Click on a modular curve in the diagram to see information about it.
|
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.