$\GL_2(\Z/48\Z)$-generators: |
$\begin{bmatrix}7&5\\20&27\end{bmatrix}$, $\begin{bmatrix}13&36\\28&31\end{bmatrix}$, $\begin{bmatrix}25&36\\0&37\end{bmatrix}$, $\begin{bmatrix}31&14\\44&5\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
48.192.1-48.ch.1.1, 48.192.1-48.ch.1.2, 48.192.1-48.ch.1.3, 48.192.1-48.ch.1.4, 48.192.1-48.ch.1.5, 48.192.1-48.ch.1.6, 48.192.1-48.ch.1.7, 48.192.1-48.ch.1.8, 240.192.1-48.ch.1.1, 240.192.1-48.ch.1.2, 240.192.1-48.ch.1.3, 240.192.1-48.ch.1.4, 240.192.1-48.ch.1.5, 240.192.1-48.ch.1.6, 240.192.1-48.ch.1.7, 240.192.1-48.ch.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 $ | $=$ | $ y^{2} + y z + z^{2} + w^{2} $ |
| $=$ | $3 x^{2} + 2 y^{2} - y z - z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} - x^{2} y^{2} + 12 x^{2} z^{2} + y^{4} - 6 y^{2} z^{2} + 9 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:
$\displaystyle X$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle x$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{3}w$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{2^8}{3}\cdot\frac{(9z^{4}+12z^{2}w^{2}+2w^{4})^{3}(27yz^{3}w^{8}+18yzw^{10}-729z^{12}-2916z^{10}w^{2}-4374z^{8}w^{4}-3024z^{6}w^{6}-918z^{4}w^{8}-81z^{2}w^{10}+w^{12})}{w^{16}(27yz^{7}+54yz^{5}w^{2}+30yz^{3}w^{4}+4yzw^{6}+27z^{8}+63z^{6}w^{2}+45z^{4}w^{4}+10z^{2}w^{6}+w^{8})}$ |
Hi
|
Cover information
Click on a modular curve in the diagram to see information about it.
|
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.