$\GL_2(\Z/48\Z)$-generators: |
$\begin{bmatrix}5&46\\16&45\end{bmatrix}$, $\begin{bmatrix}23&3\\24&13\end{bmatrix}$, $\begin{bmatrix}27&2\\32&43\end{bmatrix}$, $\begin{bmatrix}29&27\\16&7\end{bmatrix}$, $\begin{bmatrix}35&33\\24&5\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
48.192.1-48.bi.1.1, 48.192.1-48.bi.1.2, 48.192.1-48.bi.1.3, 48.192.1-48.bi.1.4, 48.192.1-48.bi.1.5, 48.192.1-48.bi.1.6, 48.192.1-48.bi.1.7, 48.192.1-48.bi.1.8, 48.192.1-48.bi.1.9, 48.192.1-48.bi.1.10, 48.192.1-48.bi.1.11, 48.192.1-48.bi.1.12, 240.192.1-48.bi.1.1, 240.192.1-48.bi.1.2, 240.192.1-48.bi.1.3, 240.192.1-48.bi.1.4, 240.192.1-48.bi.1.5, 240.192.1-48.bi.1.6, 240.192.1-48.bi.1.7, 240.192.1-48.bi.1.8, 240.192.1-48.bi.1.9, 240.192.1-48.bi.1.10, 240.192.1-48.bi.1.11, 240.192.1-48.bi.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 $ | $=$ | $ x^{2} + y^{2} + z^{2} + w^{2} $ |
| $=$ | $6 y z + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 36 x^{4} + x^{2} y^{2} + 36 x^{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:
$\displaystyle X$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle 6x$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle 2\,\frac{544195584y^{24}+2176782336y^{22}w^{2}+3446572032y^{20}w^{4}+2680667136y^{18}w^{6}+975017088y^{16}w^{8}+58786560y^{14}w^{10}-66204864y^{12}w^{12}-16936128y^{10}w^{14}+471420y^{8}w^{16}+876960y^{6}w^{18}+131976y^{4}w^{20}-14184y^{2}w^{22}+544195584z^{24}+2176782336z^{22}w^{2}+3446572032z^{20}w^{4}+2680667136z^{18}w^{6}+975017088z^{16}w^{8}+58786560z^{14}w^{10}-66204864z^{12}w^{12}-16936128z^{10}w^{14}+471420z^{8}w^{16}+876960z^{6}w^{18}+131976z^{4}w^{20}-14184z^{2}w^{22}-5473w^{24}}{w^{16}(648y^{8}+864y^{6}w^{2}+216y^{4}w^{4}-24y^{2}w^{6}+648z^{8}+864z^{6}w^{2}+216z^{4}w^{4}-24z^{2}w^{6}-13w^{8})}$ |
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.