$\GL_2(\Z/48\Z)$-generators: |
$\begin{bmatrix}3&31\\4&15\end{bmatrix}$, $\begin{bmatrix}21&5\\44&33\end{bmatrix}$, $\begin{bmatrix}29&33\\4&17\end{bmatrix}$, $\begin{bmatrix}37&36\\20&43\end{bmatrix}$, $\begin{bmatrix}41&39\\32&35\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
48.96.1-48.by.2.1, 48.96.1-48.by.2.2, 48.96.1-48.by.2.3, 48.96.1-48.by.2.4, 48.96.1-48.by.2.5, 48.96.1-48.by.2.6, 48.96.1-48.by.2.7, 48.96.1-48.by.2.8, 48.96.1-48.by.2.9, 48.96.1-48.by.2.10, 48.96.1-48.by.2.11, 48.96.1-48.by.2.12, 48.96.1-48.by.2.13, 48.96.1-48.by.2.14, 48.96.1-48.by.2.15, 48.96.1-48.by.2.16, 240.96.1-48.by.2.1, 240.96.1-48.by.2.2, 240.96.1-48.by.2.3, 240.96.1-48.by.2.4, 240.96.1-48.by.2.5, 240.96.1-48.by.2.6, 240.96.1-48.by.2.7, 240.96.1-48.by.2.8, 240.96.1-48.by.2.9, 240.96.1-48.by.2.10, 240.96.1-48.by.2.11, 240.96.1-48.by.2.12, 240.96.1-48.by.2.13, 240.96.1-48.by.2.14, 240.96.1-48.by.2.15, 240.96.1-48.by.2.16 |
Cyclic 48-isogeny field degree: |
$8$ |
Cyclic 48-torsion field degree: |
$128$ |
Full 48-torsion field degree: |
$24576$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 8 x y + 2 y^{2} + z^{2} $ |
| $=$ | $24 x^{2} + 2 x y + 2 y^{2} + z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} + 3 x^{2} y^{2} + 3 x^{2} z^{2} + 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:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{3}w$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{2}z$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{1}{3^2}\cdot\frac{382204494y^{2}z^{10}+1529167896y^{2}z^{8}w^{2}+650897856y^{2}z^{6}w^{4}+49082112y^{2}z^{4}w^{6}-41472y^{2}z^{2}w^{8}-387072y^{2}w^{10}+95550759z^{12}+254975040z^{10}w^{2}-14059008z^{8}w^{4}+5177088z^{6}w^{6}+2730240z^{4}w^{8}-589824z^{2}w^{10}-131072w^{12}}{w^{2}z^{2}(54y^{2}z^{6}+504y^{2}z^{4}w^{2}+1056y^{2}z^{2}w^{4}+128y^{2}w^{6}+27z^{8}+288z^{6}w^{2}+816z^{4}w^{4}+512z^{2}w^{6})}$ |
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.