$\GL_2(\Z/48\Z)$-generators: |
$\begin{bmatrix}11&39\\28&43\end{bmatrix}$, $\begin{bmatrix}21&28\\28&27\end{bmatrix}$, $\begin{bmatrix}25&5\\28&9\end{bmatrix}$, $\begin{bmatrix}27&13\\16&29\end{bmatrix}$, $\begin{bmatrix}35&31\\32&41\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
48.96.1-48.cd.2.1, 48.96.1-48.cd.2.2, 48.96.1-48.cd.2.3, 48.96.1-48.cd.2.4, 48.96.1-48.cd.2.5, 48.96.1-48.cd.2.6, 48.96.1-48.cd.2.7, 48.96.1-48.cd.2.8, 48.96.1-48.cd.2.9, 48.96.1-48.cd.2.10, 48.96.1-48.cd.2.11, 48.96.1-48.cd.2.12, 48.96.1-48.cd.2.13, 48.96.1-48.cd.2.14, 48.96.1-48.cd.2.15, 48.96.1-48.cd.2.16, 240.96.1-48.cd.2.1, 240.96.1-48.cd.2.2, 240.96.1-48.cd.2.3, 240.96.1-48.cd.2.4, 240.96.1-48.cd.2.5, 240.96.1-48.cd.2.6, 240.96.1-48.cd.2.7, 240.96.1-48.cd.2.8, 240.96.1-48.cd.2.9, 240.96.1-48.cd.2.10, 240.96.1-48.cd.2.11, 240.96.1-48.cd.2.12, 240.96.1-48.cd.2.13, 240.96.1-48.cd.2.14, 240.96.1-48.cd.2.15, 240.96.1-48.cd.2.16 |
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 $ | $=$ | $ 8 x^{2} + 2 x z + y^{2} $ |
| $=$ | $8 x^{2} - 22 x z + y^{2} + 6 z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 9 x^{4} + x^{2} y^{2} + 9 x^{2} z^{2} + 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:
$\displaystyle X$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle 3y$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{2}w$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle -2^5\,\frac{1492992y^{12}+746496y^{10}w^{2}-404352y^{8}w^{4}-76032y^{6}w^{6}+54864y^{4}w^{8}-9576y^{2}w^{10}-1469664z^{12}-1819584z^{10}w^{2}+1318032z^{8}w^{4}-1963440z^{6}w^{6}-368874z^{4}w^{8}-7560z^{2}w^{10}+725w^{12}}{w^{2}(20736y^{8}w^{2}+3456y^{6}w^{4}+24y^{2}w^{8}+15552z^{10}+1296z^{8}w^{2}-3456z^{6}w^{4}-1296z^{4}w^{6}-168z^{2}w^{8}-7w^{10})}$ |
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.