$\GL_2(\Z/24\Z)$-generators: |
$\begin{bmatrix}11&10\\4&9\end{bmatrix}$, $\begin{bmatrix}17&0\\0&7\end{bmatrix}$, $\begin{bmatrix}17&10\\8&3\end{bmatrix}$, $\begin{bmatrix}21&8\\4&3\end{bmatrix}$ |
$\GL_2(\Z/24\Z)$-subgroup: |
$C_2\times D_4\times \GL(2,3)$ |
Contains $-I$: |
yes |
Quadratic refinements: |
24.192.1-24.bd.2.1, 24.192.1-24.bd.2.2, 24.192.1-24.bd.2.3, 24.192.1-24.bd.2.4, 24.192.1-24.bd.2.5, 24.192.1-24.bd.2.6, 24.192.1-24.bd.2.7, 24.192.1-24.bd.2.8, 120.192.1-24.bd.2.1, 120.192.1-24.bd.2.2, 120.192.1-24.bd.2.3, 120.192.1-24.bd.2.4, 120.192.1-24.bd.2.5, 120.192.1-24.bd.2.6, 120.192.1-24.bd.2.7, 120.192.1-24.bd.2.8, 168.192.1-24.bd.2.1, 168.192.1-24.bd.2.2, 168.192.1-24.bd.2.3, 168.192.1-24.bd.2.4, 168.192.1-24.bd.2.5, 168.192.1-24.bd.2.6, 168.192.1-24.bd.2.7, 168.192.1-24.bd.2.8, 264.192.1-24.bd.2.1, 264.192.1-24.bd.2.2, 264.192.1-24.bd.2.3, 264.192.1-24.bd.2.4, 264.192.1-24.bd.2.5, 264.192.1-24.bd.2.6, 264.192.1-24.bd.2.7, 264.192.1-24.bd.2.8, 312.192.1-24.bd.2.1, 312.192.1-24.bd.2.2, 312.192.1-24.bd.2.3, 312.192.1-24.bd.2.4, 312.192.1-24.bd.2.5, 312.192.1-24.bd.2.6, 312.192.1-24.bd.2.7, 312.192.1-24.bd.2.8 |
Cyclic 24-isogeny field degree: |
$8$ |
Cyclic 24-torsion field degree: |
$32$ |
Full 24-torsion field degree: |
$768$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ x^{2} - 2 y^{2} - 2 z^{2} $ |
| $=$ | $2 x^{2} + 2 y^{2} - z^{2} - 2 w^{2}$ |
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known 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
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{2^4}{3^4}\cdot\frac{(9z^{4}-18z^{3}w+18z^{2}w^{2}-12zw^{3}+4w^{4})^{3}(9z^{4}+18z^{3}w+18z^{2}w^{2}+12zw^{3}+4w^{4})^{3}}{w^{8}z^{8}(3z^{2}-2w^{2})^{2}(3z^{2}+2w^{2})^{2}}$ |
Hi
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Cover information
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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.