$\GL_2(\Z/48\Z)$-generators: |
$\begin{bmatrix}1&32\\0&37\end{bmatrix}$, $\begin{bmatrix}37&1\\20&13\end{bmatrix}$, $\begin{bmatrix}39&26\\20&25\end{bmatrix}$, $\begin{bmatrix}43&30\\8&43\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
48.192.1-48.cv.1.1, 48.192.1-48.cv.1.2, 48.192.1-48.cv.1.3, 48.192.1-48.cv.1.4, 48.192.1-48.cv.1.5, 48.192.1-48.cv.1.6, 48.192.1-48.cv.1.7, 48.192.1-48.cv.1.8, 240.192.1-48.cv.1.1, 240.192.1-48.cv.1.2, 240.192.1-48.cv.1.3, 240.192.1-48.cv.1.4, 240.192.1-48.cv.1.5, 240.192.1-48.cv.1.6, 240.192.1-48.cv.1.7, 240.192.1-48.cv.1.8 |
Cyclic 48-isogeny field degree: |
$8$ |
Cyclic 48-torsion field degree: |
$128$ |
Full 48-torsion field degree: |
$12288$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 2 x^{2} - 2 x y + 2 y^{2} - w^{2} $ |
| $=$ | $3 y^{2} + 3 z^{2} + 2 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} - x^{2} y^{2} + 84 x^{2} z^{2} + y^{4} - 60 y^{2} z^{2} + 1764 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:
$\displaystyle X$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle x$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{6}w$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{2^8}{3}\cdot\frac{(81z^{8}+216z^{6}w^{2}+180z^{4}w^{4}+48z^{2}w^{6}+w^{8})^{3}}{w^{16}z^{2}(3z^{2}+2w^{2})^{2}(3z^{2}+4w^{2})}$ |
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.