$\GL_2(\Z/24\Z)$-generators: |
$\begin{bmatrix}17&6\\4&23\end{bmatrix}$, $\begin{bmatrix}17&14\\4&21\end{bmatrix}$, $\begin{bmatrix}17&22\\12&17\end{bmatrix}$, $\begin{bmatrix}23&7\\18&19\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
24.48.1-24.s.1.1, 24.48.1-24.s.1.2, 24.48.1-24.s.1.3, 24.48.1-24.s.1.4, 48.48.1-24.s.1.1, 48.48.1-24.s.1.2, 48.48.1-24.s.1.3, 48.48.1-24.s.1.4, 120.48.1-24.s.1.1, 120.48.1-24.s.1.2, 120.48.1-24.s.1.3, 120.48.1-24.s.1.4, 168.48.1-24.s.1.1, 168.48.1-24.s.1.2, 168.48.1-24.s.1.3, 168.48.1-24.s.1.4, 240.48.1-24.s.1.1, 240.48.1-24.s.1.2, 240.48.1-24.s.1.3, 240.48.1-24.s.1.4, 264.48.1-24.s.1.1, 264.48.1-24.s.1.2, 264.48.1-24.s.1.3, 264.48.1-24.s.1.4, 312.48.1-24.s.1.1, 312.48.1-24.s.1.2, 312.48.1-24.s.1.3, 312.48.1-24.s.1.4 |
Cyclic 24-isogeny field degree: |
$16$ |
Cyclic 24-torsion field degree: |
$128$ |
Full 24-torsion field degree: |
$3072$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ y^{2} + y z + z^{2} + w^{2} $ |
| $=$ | $12 x^{2} - y w + z w$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} + 3 x^{2} y z + 3 y^{2} z^{2} + 9 z^{4} $ |
This modular curve has no real points, and therefore no rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
$\displaystyle X$ |
$=$ |
$\displaystyle x$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{2}z$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{6}w$ |
Maps to other modular curves
$j$-invariant map
of degree 24 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle -2^8\cdot3^3\,\frac{z^{3}(yw^{2}-z^{3}-zw^{2})}{w^{4}(3yz+w^{2})}$ |
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.