$\GL_2(\Z/24\Z)$-generators: |
$\begin{bmatrix}1&8\\8&21\end{bmatrix}$, $\begin{bmatrix}1&18\\12&23\end{bmatrix}$, $\begin{bmatrix}21&2\\16&11\end{bmatrix}$, $\begin{bmatrix}21&22\\16&15\end{bmatrix}$, $\begin{bmatrix}23&6\\12&5\end{bmatrix}$ |
$\GL_2(\Z/24\Z)$-subgroup: |
$C_2^4\times \GL(2,3)$ |
Contains $-I$: |
yes |
Quadratic refinements: |
24.192.1-24.a.1.1, 24.192.1-24.a.1.2, 24.192.1-24.a.1.3, 24.192.1-24.a.1.4, 24.192.1-24.a.1.5, 24.192.1-24.a.1.6, 24.192.1-24.a.1.7, 24.192.1-24.a.1.8, 24.192.1-24.a.1.9, 24.192.1-24.a.1.10, 24.192.1-24.a.1.11, 24.192.1-24.a.1.12, 120.192.1-24.a.1.1, 120.192.1-24.a.1.2, 120.192.1-24.a.1.3, 120.192.1-24.a.1.4, 120.192.1-24.a.1.5, 120.192.1-24.a.1.6, 120.192.1-24.a.1.7, 120.192.1-24.a.1.8, 120.192.1-24.a.1.9, 120.192.1-24.a.1.10, 120.192.1-24.a.1.11, 120.192.1-24.a.1.12, 168.192.1-24.a.1.1, 168.192.1-24.a.1.2, 168.192.1-24.a.1.3, 168.192.1-24.a.1.4, 168.192.1-24.a.1.5, 168.192.1-24.a.1.6, 168.192.1-24.a.1.7, 168.192.1-24.a.1.8, 168.192.1-24.a.1.9, 168.192.1-24.a.1.10, 168.192.1-24.a.1.11, 168.192.1-24.a.1.12, 264.192.1-24.a.1.1, 264.192.1-24.a.1.2, 264.192.1-24.a.1.3, 264.192.1-24.a.1.4, 264.192.1-24.a.1.5, 264.192.1-24.a.1.6, 264.192.1-24.a.1.7, 264.192.1-24.a.1.8, 264.192.1-24.a.1.9, 264.192.1-24.a.1.10, 264.192.1-24.a.1.11, 264.192.1-24.a.1.12, 312.192.1-24.a.1.1, 312.192.1-24.a.1.2, 312.192.1-24.a.1.3, 312.192.1-24.a.1.4, 312.192.1-24.a.1.5, 312.192.1-24.a.1.6, 312.192.1-24.a.1.7, 312.192.1-24.a.1.8, 312.192.1-24.a.1.9, 312.192.1-24.a.1.10, 312.192.1-24.a.1.11, 312.192.1-24.a.1.12 |
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} + 2 z^{2} $ |
| $=$ | $3 x^{2} - 3 y^{2} - w^{2}$ |
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
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{2^4}{3^4}\cdot\frac{(36z^{4}-36z^{3}w+18z^{2}w^{2}-6zw^{3}+w^{4})^{3}(36z^{4}+36z^{3}w+18z^{2}w^{2}+6zw^{3}+w^{4})^{3}}{w^{8}z^{8}(6z^{2}-w^{2})^{2}(6z^{2}+w^{2})^{2}}$ |
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.