$\GL_2(\Z/24\Z)$-generators: |
$\begin{bmatrix}1&5\\20&11\end{bmatrix}$, $\begin{bmatrix}17&19\\8&3\end{bmatrix}$, $\begin{bmatrix}19&19\\20&13\end{bmatrix}$, $\begin{bmatrix}21&5\\20&3\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
24.96.1-24.dz.1.1, 24.96.1-24.dz.1.2, 24.96.1-24.dz.1.3, 24.96.1-24.dz.1.4, 120.96.1-24.dz.1.1, 120.96.1-24.dz.1.2, 120.96.1-24.dz.1.3, 120.96.1-24.dz.1.4, 168.96.1-24.dz.1.1, 168.96.1-24.dz.1.2, 168.96.1-24.dz.1.3, 168.96.1-24.dz.1.4, 264.96.1-24.dz.1.1, 264.96.1-24.dz.1.2, 264.96.1-24.dz.1.3, 264.96.1-24.dz.1.4, 312.96.1-24.dz.1.1, 312.96.1-24.dz.1.2, 312.96.1-24.dz.1.3, 312.96.1-24.dz.1.4 |
Cyclic 24-isogeny field degree: |
$8$ |
Cyclic 24-torsion field degree: |
$64$ |
Full 24-torsion field degree: |
$1536$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 3 x^{2} - z w $ |
| $=$ | $24 y^{2} - z^{2} - w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} - 6 y^{2} z^{2} + 9 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:
$\displaystyle X$ |
$=$ |
$\displaystyle x$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle 2y$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{3}w$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle -2^4\,\frac{(z^{2}-4zw+w^{2})^{3}(z^{2}+4zw+w^{2})^{3}}{w^{2}z^{2}(z^{2}+w^{2})^{4}}$ |
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.