$\GL_2(\Z/24\Z)$-generators: |
$\begin{bmatrix}5&3\\4&19\end{bmatrix}$, $\begin{bmatrix}5&4\\14&11\end{bmatrix}$, $\begin{bmatrix}19&18\\0&23\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
48.96.1-24.hc.1.1, 48.96.1-24.hc.1.2, 48.96.1-24.hc.1.3, 48.96.1-24.hc.1.4, 48.96.1-24.hc.1.5, 48.96.1-24.hc.1.6, 240.96.1-24.hc.1.1, 240.96.1-24.hc.1.2, 240.96.1-24.hc.1.3, 240.96.1-24.hc.1.4, 240.96.1-24.hc.1.5, 240.96.1-24.hc.1.6 |
Cyclic 24-isogeny field degree: |
$16$ |
Cyclic 24-torsion field degree: |
$128$ |
Full 24-torsion field degree: |
$1536$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ y^{2} - 10 y z + z^{2} + 2 w^{2} $ |
| $=$ | $12 x^{2} + 3 y z - w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} + 10 x^{2} y^{2} - 12 x^{2} z^{2} + y^{4} - 6 y^{2} z^{2} + 9 z^{4} $ |
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:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle 2x$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{3}w$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle 2^8\cdot3^3\,\frac{z^{3}(70005870yz^{8}-10459206yz^{6}w^{2}+482976yz^{4}w^{4}-7056yz^{2}w^{6}+16yw^{8}-7072029z^{9}-13233294z^{7}w^{2}+1788480z^{5}w^{4}-65800z^{3}w^{6}+592zw^{8})}{280023480yz^{11}-70124940yz^{9}w^{2}+6501060yz^{7}w^{4}-269892yz^{5}w^{6}+4716yz^{3}w^{8}-24yzw^{10}-28288116z^{12}-50075496z^{10}w^{2}+12466629z^{8}w^{4}-1051164z^{6}w^{6}+36810z^{4}w^{8}-468z^{2}w^{10}+w^{12}}$ |
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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.