$\GL_2(\Z/48\Z)$-generators: |
$\begin{bmatrix}19&9\\44&43\end{bmatrix}$, $\begin{bmatrix}19&26\\8&17\end{bmatrix}$, $\begin{bmatrix}19&39\\12&1\end{bmatrix}$, $\begin{bmatrix}47&47\\20&21\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
48.96.1-48.e.1.1, 48.96.1-48.e.1.2, 48.96.1-48.e.1.3, 48.96.1-48.e.1.4, 48.96.1-48.e.1.5, 48.96.1-48.e.1.6, 48.96.1-48.e.1.7, 48.96.1-48.e.1.8, 240.96.1-48.e.1.1, 240.96.1-48.e.1.2, 240.96.1-48.e.1.3, 240.96.1-48.e.1.4, 240.96.1-48.e.1.5, 240.96.1-48.e.1.6, 240.96.1-48.e.1.7, 240.96.1-48.e.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 $ | $=$ | $ 6 x^{2} + z w $ |
| $=$ | $y^{2} + 8 z^{2} + 2 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 9 x^{4} + 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:
$\displaystyle X$ |
$=$ |
$\displaystyle x$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{4}y$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{2}w$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle 2^8\,\frac{(z^{4}+4z^{2}w^{2}+w^{4})^{3}}{w^{2}z^{8}(4z^{2}+w^{2})}$ |
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.