$\GL_2(\Z/48\Z)$-generators: |
$\begin{bmatrix}5&3\\28&43\end{bmatrix}$, $\begin{bmatrix}23&19\\36&7\end{bmatrix}$, $\begin{bmatrix}35&20\\40&9\end{bmatrix}$, $\begin{bmatrix}41&5\\12&1\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
48.96.1-48.ba.1.1, 48.96.1-48.ba.1.2, 48.96.1-48.ba.1.3, 48.96.1-48.ba.1.4, 48.96.1-48.ba.1.5, 48.96.1-48.ba.1.6, 48.96.1-48.ba.1.7, 48.96.1-48.ba.1.8, 240.96.1-48.ba.1.1, 240.96.1-48.ba.1.2, 240.96.1-48.ba.1.3, 240.96.1-48.ba.1.4, 240.96.1-48.ba.1.5, 240.96.1-48.ba.1.6, 240.96.1-48.ba.1.7, 240.96.1-48.ba.1.8 |
Cyclic 48-isogeny field degree: |
$16$ |
Cyclic 48-torsion field degree: |
$256$ |
Full 48-torsion field degree: |
$24576$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ x^{2} + x w + z^{2} + w^{2} $ |
| $=$ | $ - x z + 12 y^{2} - 2 z w$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} - 3 x^{2} y z + 3 y^{2} z^{2} + 9 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:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{2}w$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{6}z$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{2^8}{3}\cdot\frac{(2z^{2}+3w^{2})^{3}(9xz^{4}w-z^{6}+27z^{4}w^{2}+54z^{2}w^{4}+27w^{6})}{z^{8}(2xz^{2}w+3xw^{3}-z^{4}-z^{2}w^{2})}$ |
Hi
|
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.