$\GL_2(\Z/48\Z)$-generators: |
$\begin{bmatrix}11&6\\8&11\end{bmatrix}$, $\begin{bmatrix}29&8\\28&15\end{bmatrix}$, $\begin{bmatrix}43&3\\16&41\end{bmatrix}$, $\begin{bmatrix}43&12\\44&29\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
48.192.1-48.dp.2.1, 48.192.1-48.dp.2.2, 48.192.1-48.dp.2.3, 48.192.1-48.dp.2.4, 48.192.1-48.dp.2.5, 48.192.1-48.dp.2.6, 48.192.1-48.dp.2.7, 48.192.1-48.dp.2.8, 96.192.1-48.dp.2.1, 96.192.1-48.dp.2.2, 96.192.1-48.dp.2.3, 96.192.1-48.dp.2.4, 240.192.1-48.dp.2.1, 240.192.1-48.dp.2.2, 240.192.1-48.dp.2.3, 240.192.1-48.dp.2.4, 240.192.1-48.dp.2.5, 240.192.1-48.dp.2.6, 240.192.1-48.dp.2.7, 240.192.1-48.dp.2.8 |
Cyclic 48-isogeny field degree: |
$4$ |
Cyclic 48-torsion field degree: |
$64$ |
Full 48-torsion field degree: |
$12288$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ x^{2} + 2 x z + 2 z^{2} - 2 w^{2} $ |
| $=$ | $x z + 3 y^{2} + z^{2} + 3 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 9 x^{4} - 2 x^{2} z^{2} + y^{4} + 2 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:
$\displaystyle X$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle 3y$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle 3w$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle 2^8\,\frac{12xz^{21}w^{2}-120xz^{19}w^{4}+380xz^{17}w^{6}-160xz^{15}w^{8}-1680xz^{13}w^{10}+4256xz^{11}w^{12}-4576xz^{9}w^{14}+2304xz^{7}w^{16}-304xz^{5}w^{18}-160xz^{3}w^{20}+48xzw^{22}-z^{24}+24z^{22}w^{2}-126z^{20}w^{4}+1754z^{16}w^{8}-5760z^{14}w^{10}+8112z^{12}w^{12}-4576z^{10}w^{14}-1140z^{8}w^{16}+2848z^{6}w^{18}-1336z^{4}w^{20}+192z^{2}w^{22}+8w^{24}}{w^{16}(z-w)^{2}(z+w)^{2}(4xzw^{2}-z^{4}+6z^{2}w^{2}+3w^{4})}$ |
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.