$\GL_2(\Z/24\Z)$-generators: |
$\begin{bmatrix}5&3\\14&17\end{bmatrix}$, $\begin{bmatrix}7&3\\10&23\end{bmatrix}$, $\begin{bmatrix}13&19\\22&11\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
48.48.1-24.l.1.1, 48.48.1-24.l.1.2, 48.48.1-24.l.1.3, 48.48.1-24.l.1.4, 48.48.1-24.l.1.5, 48.48.1-24.l.1.6, 48.48.1-24.l.1.7, 48.48.1-24.l.1.8, 240.48.1-24.l.1.1, 240.48.1-24.l.1.2, 240.48.1-24.l.1.3, 240.48.1-24.l.1.4, 240.48.1-24.l.1.5, 240.48.1-24.l.1.6, 240.48.1-24.l.1.7, 240.48.1-24.l.1.8 |
Cyclic 24-isogeny field degree: |
$16$ |
Cyclic 24-torsion field degree: |
$128$ |
Full 24-torsion field degree: |
$3072$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 4 y^{2} + z^{2} - 2 w^{2} $ |
| $=$ | $8 x^{2} - 3 y z$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 9 x^{4} - 2 y^{2} z^{2} + z^{4} $ |
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
$\displaystyle X$ |
$=$ |
$\displaystyle x$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle \frac{3}{4}w$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{3}{4}z$ |
Maps to other modular curves
$j$-invariant map
of degree 24 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle 2^6\,\frac{(3z^{2}+2w^{2})^{3}}{z^{2}(z^{2}-2w^{2})^{2}}$ |
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.