$\GL_2(\Z/24\Z)$-generators: |
$\begin{bmatrix}1&9\\0&17\end{bmatrix}$, $\begin{bmatrix}3&17\\20&21\end{bmatrix}$, $\begin{bmatrix}5&15\\12&19\end{bmatrix}$, $\begin{bmatrix}13&4\\4&11\end{bmatrix}$, $\begin{bmatrix}21&13\\16&9\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
24.144.4-24.fo.1.1, 24.144.4-24.fo.1.2, 24.144.4-24.fo.1.3, 24.144.4-24.fo.1.4, 24.144.4-24.fo.1.5, 24.144.4-24.fo.1.6, 24.144.4-24.fo.1.7, 24.144.4-24.fo.1.8, 24.144.4-24.fo.1.9, 24.144.4-24.fo.1.10, 24.144.4-24.fo.1.11, 24.144.4-24.fo.1.12, 24.144.4-24.fo.1.13, 24.144.4-24.fo.1.14, 24.144.4-24.fo.1.15, 24.144.4-24.fo.1.16, 48.144.4-24.fo.1.1, 48.144.4-24.fo.1.2, 48.144.4-24.fo.1.3, 48.144.4-24.fo.1.4, 48.144.4-24.fo.1.5, 48.144.4-24.fo.1.6, 48.144.4-24.fo.1.7, 48.144.4-24.fo.1.8, 120.144.4-24.fo.1.1, 120.144.4-24.fo.1.2, 120.144.4-24.fo.1.3, 120.144.4-24.fo.1.4, 120.144.4-24.fo.1.5, 120.144.4-24.fo.1.6, 120.144.4-24.fo.1.7, 120.144.4-24.fo.1.8, 120.144.4-24.fo.1.9, 120.144.4-24.fo.1.10, 120.144.4-24.fo.1.11, 120.144.4-24.fo.1.12, 120.144.4-24.fo.1.13, 120.144.4-24.fo.1.14, 120.144.4-24.fo.1.15, 120.144.4-24.fo.1.16, 168.144.4-24.fo.1.1, 168.144.4-24.fo.1.2, 168.144.4-24.fo.1.3, 168.144.4-24.fo.1.4, 168.144.4-24.fo.1.5, 168.144.4-24.fo.1.6, 168.144.4-24.fo.1.7, 168.144.4-24.fo.1.8, 168.144.4-24.fo.1.9, 168.144.4-24.fo.1.10, 168.144.4-24.fo.1.11, 168.144.4-24.fo.1.12, 168.144.4-24.fo.1.13, 168.144.4-24.fo.1.14, 168.144.4-24.fo.1.15, 168.144.4-24.fo.1.16, 240.144.4-24.fo.1.1, 240.144.4-24.fo.1.2, 240.144.4-24.fo.1.3, 240.144.4-24.fo.1.4, 240.144.4-24.fo.1.5, 240.144.4-24.fo.1.6, 240.144.4-24.fo.1.7, 240.144.4-24.fo.1.8, 264.144.4-24.fo.1.1, 264.144.4-24.fo.1.2, 264.144.4-24.fo.1.3, 264.144.4-24.fo.1.4, 264.144.4-24.fo.1.5, 264.144.4-24.fo.1.6, 264.144.4-24.fo.1.7, 264.144.4-24.fo.1.8, 264.144.4-24.fo.1.9, 264.144.4-24.fo.1.10, 264.144.4-24.fo.1.11, 264.144.4-24.fo.1.12, 264.144.4-24.fo.1.13, 264.144.4-24.fo.1.14, 264.144.4-24.fo.1.15, 264.144.4-24.fo.1.16, 312.144.4-24.fo.1.1, 312.144.4-24.fo.1.2, 312.144.4-24.fo.1.3, 312.144.4-24.fo.1.4, 312.144.4-24.fo.1.5, 312.144.4-24.fo.1.6, 312.144.4-24.fo.1.7, 312.144.4-24.fo.1.8, 312.144.4-24.fo.1.9, 312.144.4-24.fo.1.10, 312.144.4-24.fo.1.11, 312.144.4-24.fo.1.12, 312.144.4-24.fo.1.13, 312.144.4-24.fo.1.14, 312.144.4-24.fo.1.15, 312.144.4-24.fo.1.16 |
Cyclic 24-isogeny field degree: |
$8$ |
Cyclic 24-torsion field degree: |
$64$ |
Full 24-torsion field degree: |
$1024$ |
Canonical model in $\mathbb{P}^{ 3 }$
$ 0 $ | $=$ | $ 4 y^{2} + z^{2} + z w + w^{2} $ |
| $=$ | $3 x^{3} + y z^{2} + 2 y z w$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{6} + 36 y^{4} z^{2} + 12 y^{2} z^{4} $ |
This modular curve has no real points, and therefore no rational points.
Maps between models of this curve
Birational map from canonical model to plane model:
$\displaystyle X$ |
$=$ |
$\displaystyle x$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle \frac{2}{3}y$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{2}z$ |
Maps to other modular curves
$j$-invariant map
of degree 72 from the canonical model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{2^8}{3}\cdot\frac{(z^{2}+4zw+w^{2})^{3}(2z^{2}+2zw-w^{2})^{3}}{z^{2}(z+2w)^{2}(z^{2}+zw+w^{2})^{4}}$ |
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.