$\GL_2(\Z/48\Z)$-generators: |
$\begin{bmatrix}9&29\\44&25\end{bmatrix}$, $\begin{bmatrix}29&25\\8&39\end{bmatrix}$, $\begin{bmatrix}33&25\\32&15\end{bmatrix}$, $\begin{bmatrix}39&19\\32&25\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
48.192.1-48.dd.1.1, 48.192.1-48.dd.1.2, 48.192.1-48.dd.1.3, 48.192.1-48.dd.1.4, 48.192.1-48.dd.1.5, 48.192.1-48.dd.1.6, 48.192.1-48.dd.1.7, 48.192.1-48.dd.1.8, 96.192.1-48.dd.1.1, 96.192.1-48.dd.1.2, 96.192.1-48.dd.1.3, 96.192.1-48.dd.1.4, 240.192.1-48.dd.1.1, 240.192.1-48.dd.1.2, 240.192.1-48.dd.1.3, 240.192.1-48.dd.1.4, 240.192.1-48.dd.1.5, 240.192.1-48.dd.1.6, 240.192.1-48.dd.1.7, 240.192.1-48.dd.1.8 |
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} + z^{2} - z w + w^{2} $ |
| $=$ | $8 y^{2} + z^{2} + 2 z w - 2 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 36 x^{4} - 12 x^{2} y^{2} - 6 x^{2} z^{2} + y^{4} + 4 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:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle \frac{3}{2}x$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{3}{2}w$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{2^2}{3}\cdot\frac{(193z^{8}-1144z^{7}w+2896z^{6}w^{2}-4240z^{5}w^{3}+3976z^{4}w^{4}-2272z^{3}w^{5}+832z^{2}w^{6}-64zw^{7}+16w^{8})^{3}}{z^{2}(z-2w)^{2}(z^{2}-zw+w^{2})^{2}(z^{2}+2zw-2w^{2})^{8}}$ |
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.