$\GL_2(\Z/48\Z)$-generators: |
$\begin{bmatrix}15&1\\44&15\end{bmatrix}$, $\begin{bmatrix}17&32\\40&29\end{bmatrix}$, $\begin{bmatrix}17&35\\8&27\end{bmatrix}$, $\begin{bmatrix}43&27\\4&43\end{bmatrix}$, $\begin{bmatrix}47&4\\16&7\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
48.192.1-48.cs.1.1, 48.192.1-48.cs.1.2, 48.192.1-48.cs.1.3, 48.192.1-48.cs.1.4, 48.192.1-48.cs.1.5, 48.192.1-48.cs.1.6, 48.192.1-48.cs.1.7, 48.192.1-48.cs.1.8, 48.192.1-48.cs.1.9, 48.192.1-48.cs.1.10, 48.192.1-48.cs.1.11, 48.192.1-48.cs.1.12, 96.192.1-48.cs.1.1, 96.192.1-48.cs.1.2, 96.192.1-48.cs.1.3, 96.192.1-48.cs.1.4, 96.192.1-48.cs.1.5, 96.192.1-48.cs.1.6, 96.192.1-48.cs.1.7, 96.192.1-48.cs.1.8, 240.192.1-48.cs.1.1, 240.192.1-48.cs.1.2, 240.192.1-48.cs.1.3, 240.192.1-48.cs.1.4, 240.192.1-48.cs.1.5, 240.192.1-48.cs.1.6, 240.192.1-48.cs.1.7, 240.192.1-48.cs.1.8, 240.192.1-48.cs.1.9, 240.192.1-48.cs.1.10, 240.192.1-48.cs.1.11, 240.192.1-48.cs.1.12 |
Cyclic 48-isogeny field degree: |
$8$ |
Cyclic 48-torsion field degree: |
$64$ |
Full 48-torsion field degree: |
$12288$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 6 x^{2} - y w - w^{2} $ |
| $=$ | $y^{2} + 8 y w - 6 z^{2} + 8 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} + 36 x^{2} z^{2} - 6 y^{2} z^{2} + 36 z^{4} $ |
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
$\displaystyle X$ |
$=$ |
$\displaystyle x$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{6}w$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{2^8}{3}\cdot\frac{4251528yz^{22}w+268791048yz^{20}w^{3}+4944841992yz^{18}w^{5}+41071886244yz^{16}w^{7}+183263146704yz^{14}w^{9}+474733116432yz^{12}w^{11}+730998404784yz^{10}w^{13}+654785649288yz^{8}w^{15}+316037513160yz^{6}w^{17}+70831627848yz^{4}w^{19}+7203397032yz^{2}w^{21}+271669860yw^{23}-531441z^{24}-85030560z^{22}w^{2}-2165917320z^{20}w^{4}-20333405052z^{18}w^{6}-88587061587z^{16}w^{8}-182459485440z^{14}w^{10}-110159341632z^{12}w^{12}+219728793960z^{10}w^{14}+445479541101z^{8}w^{16}+296544997728z^{6}w^{18}+75398085576z^{4}w^{20}+8151155172z^{2}w^{22}+318281039w^{24}}{w^{8}z^{2}(233280yz^{12}w+10023264yz^{10}w^{3}+120000528yz^{8}w^{5}+605560968yz^{6}w^{7}+1468002888yz^{4}w^{9}+1693055208yz^{2}w^{11}+745778864yw^{13}-34992z^{14}-3709152z^{12}w^{2}-59087880z^{10}w^{4}-309950712z^{8}w^{6}-566507547z^{6}w^{8}+72432000z^{4}w^{10}+1192519620z^{2}w^{12}+873734288w^{14})}$ |
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.