$\GL_2(\Z/48\Z)$-generators: |
$\begin{bmatrix}17&17\\44&45\end{bmatrix}$, $\begin{bmatrix}25&20\\24&37\end{bmatrix}$, $\begin{bmatrix}27&22\\8&35\end{bmatrix}$, $\begin{bmatrix}47&33\\8&17\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
48.192.1-48.ed.1.1, 48.192.1-48.ed.1.2, 48.192.1-48.ed.1.3, 48.192.1-48.ed.1.4, 48.192.1-48.ed.1.5, 48.192.1-48.ed.1.6, 48.192.1-48.ed.1.7, 48.192.1-48.ed.1.8, 96.192.1-48.ed.1.1, 96.192.1-48.ed.1.2, 96.192.1-48.ed.1.3, 96.192.1-48.ed.1.4, 240.192.1-48.ed.1.1, 240.192.1-48.ed.1.2, 240.192.1-48.ed.1.3, 240.192.1-48.ed.1.4, 240.192.1-48.ed.1.5, 240.192.1-48.ed.1.6, 240.192.1-48.ed.1.7, 240.192.1-48.ed.1.8 |
Cyclic 48-isogeny field degree: |
$4$ |
Cyclic 48-torsion field degree: |
$64$ |
Full 48-torsion field degree: |
$12288$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ x^{2} + y^{2} - y z + 2 z^{2} + w^{2} $ |
| $=$ | $3 x^{2} - y^{2} + y z - z^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} - 10 x^{2} y^{2} - x^{2} z^{2} + 49 y^{4} + 14 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:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle x$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle -2^4\,\frac{(z^{8}-28z^{6}w^{2}-10z^{4}w^{4}+4z^{2}w^{6}+w^{8})^{3}}{w^{2}z^{4}(z^{2}+w^{2})^{8}(2z^{2}+w^{2})}$ |
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.