$\GL_2(\Z/48\Z)$-generators: |
$\begin{bmatrix}1&37\\32&39\end{bmatrix}$, $\begin{bmatrix}9&16\\8&21\end{bmatrix}$, $\begin{bmatrix}11&34\\8&27\end{bmatrix}$, $\begin{bmatrix}23&20\\16&11\end{bmatrix}$, $\begin{bmatrix}47&24\\24&35\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
48.192.1-48.bj.2.1, 48.192.1-48.bj.2.2, 48.192.1-48.bj.2.3, 48.192.1-48.bj.2.4, 48.192.1-48.bj.2.5, 48.192.1-48.bj.2.6, 48.192.1-48.bj.2.7, 48.192.1-48.bj.2.8, 48.192.1-48.bj.2.9, 48.192.1-48.bj.2.10, 48.192.1-48.bj.2.11, 48.192.1-48.bj.2.12, 240.192.1-48.bj.2.1, 240.192.1-48.bj.2.2, 240.192.1-48.bj.2.3, 240.192.1-48.bj.2.4, 240.192.1-48.bj.2.5, 240.192.1-48.bj.2.6, 240.192.1-48.bj.2.7, 240.192.1-48.bj.2.8, 240.192.1-48.bj.2.9, 240.192.1-48.bj.2.10, 240.192.1-48.bj.2.11, 240.192.1-48.bj.2.12 |
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} + 2 y^{2} + z^{2} + w^{2} $ |
| $=$ | $y^{2} + 3 z w$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 9 x^{4} + 18 x^{2} z^{2} + 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:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle 3x$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle 3w$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{(z^{8}-8z^{7}w+12z^{6}w^{2}+8z^{5}w^{3}-10z^{4}w^{4}+8z^{3}w^{5}+12z^{2}w^{6}-8zw^{7}+w^{8})^{3}}{w^{8}z^{8}(z-w)^{4}(z+w)^{2}(z^{2}-6zw+w^{2})}$ |
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.