$\GL_2(\Z/48\Z)$-generators: |
$\begin{bmatrix}5&25\\4&33\end{bmatrix}$, $\begin{bmatrix}15&20\\40&27\end{bmatrix}$, $\begin{bmatrix}33&1\\32&15\end{bmatrix}$, $\begin{bmatrix}35&37\\12&11\end{bmatrix}$, $\begin{bmatrix}41&0\\12&35\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
48.96.1-48.bl.1.1, 48.96.1-48.bl.1.2, 48.96.1-48.bl.1.3, 48.96.1-48.bl.1.4, 48.96.1-48.bl.1.5, 48.96.1-48.bl.1.6, 48.96.1-48.bl.1.7, 48.96.1-48.bl.1.8, 48.96.1-48.bl.1.9, 48.96.1-48.bl.1.10, 48.96.1-48.bl.1.11, 48.96.1-48.bl.1.12, 48.96.1-48.bl.1.13, 48.96.1-48.bl.1.14, 48.96.1-48.bl.1.15, 48.96.1-48.bl.1.16, 96.96.1-48.bl.1.1, 96.96.1-48.bl.1.2, 96.96.1-48.bl.1.3, 96.96.1-48.bl.1.4, 96.96.1-48.bl.1.5, 96.96.1-48.bl.1.6, 96.96.1-48.bl.1.7, 96.96.1-48.bl.1.8, 240.96.1-48.bl.1.1, 240.96.1-48.bl.1.2, 240.96.1-48.bl.1.3, 240.96.1-48.bl.1.4, 240.96.1-48.bl.1.5, 240.96.1-48.bl.1.6, 240.96.1-48.bl.1.7, 240.96.1-48.bl.1.8, 240.96.1-48.bl.1.9, 240.96.1-48.bl.1.10, 240.96.1-48.bl.1.11, 240.96.1-48.bl.1.12, 240.96.1-48.bl.1.13, 240.96.1-48.bl.1.14, 240.96.1-48.bl.1.15, 240.96.1-48.bl.1.16 |
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 $ | $=$ | $ 3 x w + z^{2} + z w + w^{2} $ |
| $=$ | $12 x^{2} + y^{2} + z^{2} + z w + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 4 x^{4} + 8 x^{3} z + 15 x^{2} z^{2} + 11 x z^{3} + 3 y^{2} z^{2} + 7 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:
$\displaystyle X$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{2}{3}\cdot\frac{18432xy^{10}w+472320xy^{8}w^{3}+4078080xy^{6}w^{5}+12581568xy^{4}w^{7}+7879032xy^{2}w^{9}+2985255xw^{11}-1024y^{12}-25344y^{10}w^{2}-213696y^{8}w^{4}-616896y^{6}w^{6}+28188y^{4}w^{8}-995085y^{2}w^{10}-746496w^{12}}{w^{2}y^{2}(480xy^{6}w-1152xy^{4}w^{3}+594xy^{2}w^{5}-81xw^{7}-32y^{8}+216y^{6}w^{2}-162y^{4}w^{4}+27y^{2}w^{6})}$ |
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.