$\GL_2(\Z/24\Z)$-generators: |
$\begin{bmatrix}1&10\\12&7\end{bmatrix}$, $\begin{bmatrix}5&4\\12&23\end{bmatrix}$, $\begin{bmatrix}9&10\\4&13\end{bmatrix}$, $\begin{bmatrix}19&0\\4&11\end{bmatrix}$ |
$\GL_2(\Z/24\Z)$-subgroup: |
$C_2\times D_4\times \GL(2,3)$ |
Contains $-I$: |
yes |
Quadratic refinements: |
24.192.1-24.bj.2.1, 24.192.1-24.bj.2.2, 24.192.1-24.bj.2.3, 24.192.1-24.bj.2.4, 24.192.1-24.bj.2.5, 24.192.1-24.bj.2.6, 24.192.1-24.bj.2.7, 24.192.1-24.bj.2.8, 120.192.1-24.bj.2.1, 120.192.1-24.bj.2.2, 120.192.1-24.bj.2.3, 120.192.1-24.bj.2.4, 120.192.1-24.bj.2.5, 120.192.1-24.bj.2.6, 120.192.1-24.bj.2.7, 120.192.1-24.bj.2.8, 168.192.1-24.bj.2.1, 168.192.1-24.bj.2.2, 168.192.1-24.bj.2.3, 168.192.1-24.bj.2.4, 168.192.1-24.bj.2.5, 168.192.1-24.bj.2.6, 168.192.1-24.bj.2.7, 168.192.1-24.bj.2.8, 264.192.1-24.bj.2.1, 264.192.1-24.bj.2.2, 264.192.1-24.bj.2.3, 264.192.1-24.bj.2.4, 264.192.1-24.bj.2.5, 264.192.1-24.bj.2.6, 264.192.1-24.bj.2.7, 264.192.1-24.bj.2.8, 312.192.1-24.bj.2.1, 312.192.1-24.bj.2.2, 312.192.1-24.bj.2.3, 312.192.1-24.bj.2.4, 312.192.1-24.bj.2.5, 312.192.1-24.bj.2.6, 312.192.1-24.bj.2.7, 312.192.1-24.bj.2.8 |
Cyclic 24-isogeny field degree: |
$8$ |
Cyclic 24-torsion field degree: |
$64$ |
Full 24-torsion field degree: |
$768$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ x^{2} + y^{2} - z^{2} + w^{2} $ |
| $=$ | $3 y^{2} + 3 z^{2} - w^{2}$ |
This modular curve has no real points, and therefore no rational points.
Maps to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{2^8}{3^2}\cdot\frac{(81z^{8}-108z^{6}w^{2}+45z^{4}w^{4}-6z^{2}w^{6}+w^{8})^{3}}{w^{8}z^{4}(3z^{2}-2w^{2})^{2}(3z^{2}-w^{2})^{4}}$ |
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.