$\GL_2(\Z/24\Z)$-generators: |
$\begin{bmatrix}3&4\\2&21\end{bmatrix}$, $\begin{bmatrix}3&20\\22&17\end{bmatrix}$, $\begin{bmatrix}7&20\\8&23\end{bmatrix}$, $\begin{bmatrix}9&20\\16&17\end{bmatrix}$, $\begin{bmatrix}21&4\\16&5\end{bmatrix}$ |
$\GL_2(\Z/24\Z)$-subgroup: |
$C_2^4\times \GL(2,3)$ |
Contains $-I$: |
yes |
Quadratic refinements: |
24.192.1-24.l.1.1, 24.192.1-24.l.1.2, 24.192.1-24.l.1.3, 24.192.1-24.l.1.4, 24.192.1-24.l.1.5, 24.192.1-24.l.1.6, 24.192.1-24.l.1.7, 24.192.1-24.l.1.8, 24.192.1-24.l.1.9, 24.192.1-24.l.1.10, 24.192.1-24.l.1.11, 24.192.1-24.l.1.12, 24.192.1-24.l.1.13, 24.192.1-24.l.1.14, 24.192.1-24.l.1.15, 24.192.1-24.l.1.16, 48.192.1-24.l.1.1, 48.192.1-24.l.1.2, 48.192.1-24.l.1.3, 48.192.1-24.l.1.4, 48.192.1-24.l.1.5, 48.192.1-24.l.1.6, 48.192.1-24.l.1.7, 48.192.1-24.l.1.8, 120.192.1-24.l.1.1, 120.192.1-24.l.1.2, 120.192.1-24.l.1.3, 120.192.1-24.l.1.4, 120.192.1-24.l.1.5, 120.192.1-24.l.1.6, 120.192.1-24.l.1.7, 120.192.1-24.l.1.8, 120.192.1-24.l.1.9, 120.192.1-24.l.1.10, 120.192.1-24.l.1.11, 120.192.1-24.l.1.12, 120.192.1-24.l.1.13, 120.192.1-24.l.1.14, 120.192.1-24.l.1.15, 120.192.1-24.l.1.16, 168.192.1-24.l.1.1, 168.192.1-24.l.1.2, 168.192.1-24.l.1.3, 168.192.1-24.l.1.4, 168.192.1-24.l.1.5, 168.192.1-24.l.1.6, 168.192.1-24.l.1.7, 168.192.1-24.l.1.8, 168.192.1-24.l.1.9, 168.192.1-24.l.1.10, 168.192.1-24.l.1.11, 168.192.1-24.l.1.12, 168.192.1-24.l.1.13, 168.192.1-24.l.1.14, 168.192.1-24.l.1.15, 168.192.1-24.l.1.16, 240.192.1-24.l.1.1, 240.192.1-24.l.1.2, 240.192.1-24.l.1.3, 240.192.1-24.l.1.4, 240.192.1-24.l.1.5, 240.192.1-24.l.1.6, 240.192.1-24.l.1.7, 240.192.1-24.l.1.8, 264.192.1-24.l.1.1, 264.192.1-24.l.1.2, 264.192.1-24.l.1.3, 264.192.1-24.l.1.4, 264.192.1-24.l.1.5, 264.192.1-24.l.1.6, 264.192.1-24.l.1.7, 264.192.1-24.l.1.8, 264.192.1-24.l.1.9, 264.192.1-24.l.1.10, 264.192.1-24.l.1.11, 264.192.1-24.l.1.12, 264.192.1-24.l.1.13, 264.192.1-24.l.1.14, 264.192.1-24.l.1.15, 264.192.1-24.l.1.16, 312.192.1-24.l.1.1, 312.192.1-24.l.1.2, 312.192.1-24.l.1.3, 312.192.1-24.l.1.4, 312.192.1-24.l.1.5, 312.192.1-24.l.1.6, 312.192.1-24.l.1.7, 312.192.1-24.l.1.8, 312.192.1-24.l.1.9, 312.192.1-24.l.1.10, 312.192.1-24.l.1.11, 312.192.1-24.l.1.12, 312.192.1-24.l.1.13, 312.192.1-24.l.1.14, 312.192.1-24.l.1.15, 312.192.1-24.l.1.16 |
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 $ | $=$ | $ 3 x y - z w $ |
| $=$ | $6 x^{2} - 3 y^{2} + 2 z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 3 x^{4} - 18 x^{2} y^{2} - x^{2} z^{2} - 6 y^{2} z^{2} $ |
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:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{3}z$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{(2z^{2}-2zw+w^{2})^{3}(2z^{2}+2zw+w^{2})^{3}(4z^{4}+8z^{2}w^{2}+w^{4})^{3}}{w^{8}z^{8}(2z^{2}+w^{2})^{4}}$ |
Hi
|
Cover information
Click on a modular curve in the diagram to see information about it.
|
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.