$\GL_2(\Z/48\Z)$-generators: |
$\begin{bmatrix}15&4\\32&31\end{bmatrix}$, $\begin{bmatrix}27&40\\40&27\end{bmatrix}$, $\begin{bmatrix}39&17\\40&9\end{bmatrix}$, $\begin{bmatrix}41&13\\44&9\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
48.192.1-48.bb.2.1, 48.192.1-48.bb.2.2, 48.192.1-48.bb.2.3, 48.192.1-48.bb.2.4, 48.192.1-48.bb.2.5, 48.192.1-48.bb.2.6, 48.192.1-48.bb.2.7, 48.192.1-48.bb.2.8, 240.192.1-48.bb.2.1, 240.192.1-48.bb.2.2, 240.192.1-48.bb.2.3, 240.192.1-48.bb.2.4, 240.192.1-48.bb.2.5, 240.192.1-48.bb.2.6, 240.192.1-48.bb.2.7, 240.192.1-48.bb.2.8 |
Cyclic 48-isogeny field degree: |
$8$ |
Cyclic 48-torsion field degree: |
$128$ |
Full 48-torsion field degree: |
$12288$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 3 x^{2} - 2 y^{2} - z^{2} $ |
| $=$ | $3 x^{2} + 6 y^{2} + z^{2} + w^{2}$ |
This modular curve has no real points, and therefore no rational points.
Maps to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle -2\,\frac{(4z^{4}-16z^{3}w-28z^{2}w^{2}-8zw^{3}+w^{4})^{3}(4z^{4}+16z^{3}w-28z^{2}w^{2}+8zw^{3}+w^{4})^{3}}{w^{2}z^{2}(2z^{2}-w^{2})^{2}(2z^{2}+w^{2})^{8}}$ |
Hi
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Cover information
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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.