$\GL_2(\Z/48\Z)$-generators: |
$\begin{bmatrix}9&41\\44&41\end{bmatrix}$, $\begin{bmatrix}19&35\\20&9\end{bmatrix}$, $\begin{bmatrix}27&26\\40&3\end{bmatrix}$, $\begin{bmatrix}33&11\\20&13\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
48.96.1-48.co.1.1, 48.96.1-48.co.1.2, 48.96.1-48.co.1.3, 48.96.1-48.co.1.4, 48.96.1-48.co.1.5, 48.96.1-48.co.1.6, 48.96.1-48.co.1.7, 48.96.1-48.co.1.8, 96.96.1-48.co.1.1, 96.96.1-48.co.1.2, 96.96.1-48.co.1.3, 96.96.1-48.co.1.4, 240.96.1-48.co.1.1, 240.96.1-48.co.1.2, 240.96.1-48.co.1.3, 240.96.1-48.co.1.4, 240.96.1-48.co.1.5, 240.96.1-48.co.1.6, 240.96.1-48.co.1.7, 240.96.1-48.co.1.8 |
Cyclic 48-isogeny field degree: |
$16$ |
Cyclic 48-torsion field degree: |
$256$ |
Full 48-torsion field degree: |
$24576$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ x y - 2 y w - 2 z^{2} $ |
| $=$ | $x^{2} - 3 x w + 3 y^{2} + 3 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 12 x^{4} + x^{2} y^{2} + x y 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:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle 2w$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle 2z$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{2^4}{3^2}\cdot\frac{14784xz^{8}w^{3}-5816xz^{4}w^{7}-53568y^{2}z^{8}w^{2}+9060y^{2}z^{4}w^{6}-728y^{2}w^{10}-27648yz^{10}w-32784yz^{6}w^{5}+1468yz^{2}w^{9}-4608z^{12}-60000z^{8}w^{4}+14568z^{4}w^{8}-243w^{12}}{z^{4}(8xz^{4}w^{3}-27y^{2}z^{4}w^{2}+y^{2}w^{6}-30yz^{6}w+6yz^{2}w^{5}-12z^{8}-4z^{4}w^{4})}$ |
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.