$\GL_2(\Z/24\Z)$-generators: |
$\begin{bmatrix}1&17\\12&23\end{bmatrix}$, $\begin{bmatrix}11&20\\0&23\end{bmatrix}$, $\begin{bmatrix}21&22\\4&17\end{bmatrix}$, $\begin{bmatrix}23&6\\12&19\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
24.96.1-24.ed.1.1, 24.96.1-24.ed.1.2, 24.96.1-24.ed.1.3, 24.96.1-24.ed.1.4, 120.96.1-24.ed.1.1, 120.96.1-24.ed.1.2, 120.96.1-24.ed.1.3, 120.96.1-24.ed.1.4, 168.96.1-24.ed.1.1, 168.96.1-24.ed.1.2, 168.96.1-24.ed.1.3, 168.96.1-24.ed.1.4, 264.96.1-24.ed.1.1, 264.96.1-24.ed.1.2, 264.96.1-24.ed.1.3, 264.96.1-24.ed.1.4, 312.96.1-24.ed.1.1, 312.96.1-24.ed.1.2, 312.96.1-24.ed.1.3, 312.96.1-24.ed.1.4 |
Cyclic 24-isogeny field degree: |
$8$ |
Cyclic 24-torsion field degree: |
$64$ |
Full 24-torsion field degree: |
$1536$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ x^{2} + x y + y^{2} + 3 z^{2} $ |
| $=$ | $6 x^{2} + 3 x y + 3 y^{2} - 3 z^{2} - w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 49 x^{4} + 10 x^{2} y^{2} + 42 x^{2} z^{2} + y^{4} + 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:
$\displaystyle X$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{3}w$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{2^4}{3^4}\cdot\frac{(6z^{2}+w^{2})^{3}(18z^{2}+w^{2})^{3}}{z^{8}(12z^{2}+w^{2})^{2}}$ |
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.