$\GL_2(\Z/48\Z)$-generators: |
$\begin{bmatrix}19&24\\12&41\end{bmatrix}$, $\begin{bmatrix}19&37\\0&5\end{bmatrix}$, $\begin{bmatrix}19&40\\40&27\end{bmatrix}$, $\begin{bmatrix}35&11\\16&21\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
48.192.1-48.bb.1.1, 48.192.1-48.bb.1.2, 48.192.1-48.bb.1.3, 48.192.1-48.bb.1.4, 48.192.1-48.bb.1.5, 48.192.1-48.bb.1.6, 48.192.1-48.bb.1.7, 48.192.1-48.bb.1.8, 240.192.1-48.bb.1.1, 240.192.1-48.bb.1.2, 240.192.1-48.bb.1.3, 240.192.1-48.bb.1.4, 240.192.1-48.bb.1.5, 240.192.1-48.bb.1.6, 240.192.1-48.bb.1.7, 240.192.1-48.bb.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} + z^{2} $ |
| $=$ | $2 x^{2} + 2 x y - 4 y^{2} + 5 z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 49 x^{4} + 60 x^{2} y^{2} + 14 x^{2} z^{2} + 36 y^{4} + 6 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:
$\displaystyle X$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle x$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{(16z^{8}+128z^{6}w^{2}+80z^{4}w^{4}+16z^{2}w^{6}+w^{8})^{3}}{w^{2}z^{16}(4z^{2}+w^{2})^{2}(8z^{2}+w^{2})}$ |
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.