$\GL_2(\Z/24\Z)$-generators: |
$\begin{bmatrix}7&9\\16&1\end{bmatrix}$, $\begin{bmatrix}13&20\\0&13\end{bmatrix}$, $\begin{bmatrix}13&20\\12&1\end{bmatrix}$, $\begin{bmatrix}17&9\\2&19\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
24.96.1-24.hq.1.1, 24.96.1-24.hq.1.2, 48.96.1-24.hq.1.1, 48.96.1-24.hq.1.2, 48.96.1-24.hq.1.3, 48.96.1-24.hq.1.4, 120.96.1-24.hq.1.1, 120.96.1-24.hq.1.2, 168.96.1-24.hq.1.1, 168.96.1-24.hq.1.2, 240.96.1-24.hq.1.1, 240.96.1-24.hq.1.2, 240.96.1-24.hq.1.3, 240.96.1-24.hq.1.4, 264.96.1-24.hq.1.1, 264.96.1-24.hq.1.2, 312.96.1-24.hq.1.1, 312.96.1-24.hq.1.2 |
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 $ | $=$ | $ x^{2} + x z - 6 y^{2} + z^{2} $ |
| $=$ | $4 x^{2} + 7 x z + 6 y^{2} + 7 z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} + 4 x^{2} y^{2} + 3 x^{2} z^{2} + 100 y^{4} + 60 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
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle 2^4\cdot3^3\,\frac{139128192xz^{11}-43545600xz^{9}w^{2}-106747200xz^{7}w^{4}-17976000xz^{5}w^{6}-425000xz^{3}w^{8}-150000xzw^{10}+59579712z^{12}-246856896z^{10}w^{2}-118793520z^{8}w^{4}+2760800z^{6}w^{6}+4767500z^{4}w^{8}+262500z^{2}w^{10}-3125w^{12}}{34782048xz^{11}+38831400xz^{9}w^{2}+2567700xz^{7}w^{4}-2349000xz^{5}w^{6}-450000xz^{3}w^{8}-37500xzw^{10}+14894928z^{12}+55071576z^{10}w^{2}+25777845z^{8}w^{4}+4099950z^{6}w^{6}+354375z^{4}w^{8}+18750z^{2}w^{10}+3125w^{12}}$ |
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.