$\GL_2(\Z/48\Z)$-generators: |
$\begin{bmatrix}1&22\\16&33\end{bmatrix}$, $\begin{bmatrix}3&26\\32&7\end{bmatrix}$, $\begin{bmatrix}5&44\\0&17\end{bmatrix}$, $\begin{bmatrix}11&30\\36&5\end{bmatrix}$, $\begin{bmatrix}35&8\\8&3\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
48.192.1-48.o.1.1, 48.192.1-48.o.1.2, 48.192.1-48.o.1.3, 48.192.1-48.o.1.4, 48.192.1-48.o.1.5, 48.192.1-48.o.1.6, 48.192.1-48.o.1.7, 48.192.1-48.o.1.8, 48.192.1-48.o.1.9, 48.192.1-48.o.1.10, 48.192.1-48.o.1.11, 48.192.1-48.o.1.12, 240.192.1-48.o.1.1, 240.192.1-48.o.1.2, 240.192.1-48.o.1.3, 240.192.1-48.o.1.4, 240.192.1-48.o.1.5, 240.192.1-48.o.1.6, 240.192.1-48.o.1.7, 240.192.1-48.o.1.8, 240.192.1-48.o.1.9, 240.192.1-48.o.1.10, 240.192.1-48.o.1.11, 240.192.1-48.o.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 z + 2 y^{2} + z^{2} $ |
| $=$ | $6 x^{2} + 8 x z - 8 y^{2} + 8 z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} + 6 y^{2} z^{2} + 4 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:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{6}w$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{2}z$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{2^4}{3^4}\cdot\frac{(1296z^{8}+432z^{6}w^{2}+180z^{4}w^{4}+24z^{2}w^{6}+w^{8})^{3}}{w^{4}z^{8}(6z^{2}+w^{2})^{4}(12z^{2}+w^{2})^{2}}$ |
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.