$\GL_2(\Z/24\Z)$-generators: |
$\begin{bmatrix}1&12\\16&11\end{bmatrix}$, $\begin{bmatrix}5&9\\4&23\end{bmatrix}$, $\begin{bmatrix}7&18\\12&1\end{bmatrix}$, $\begin{bmatrix}11&18\\20&19\end{bmatrix}$, $\begin{bmatrix}23&6\\8&13\end{bmatrix}$ |
$\GL_2(\Z/24\Z)$-subgroup: |
Group 768.1035917 |
Contains $-I$: |
yes |
Quadratic refinements: |
24.192.1-24.dl.3.1, 24.192.1-24.dl.3.2, 24.192.1-24.dl.3.3, 24.192.1-24.dl.3.4, 24.192.1-24.dl.3.5, 24.192.1-24.dl.3.6, 24.192.1-24.dl.3.7, 24.192.1-24.dl.3.8, 24.192.1-24.dl.3.9, 24.192.1-24.dl.3.10, 24.192.1-24.dl.3.11, 24.192.1-24.dl.3.12, 24.192.1-24.dl.3.13, 24.192.1-24.dl.3.14, 24.192.1-24.dl.3.15, 24.192.1-24.dl.3.16, 120.192.1-24.dl.3.1, 120.192.1-24.dl.3.2, 120.192.1-24.dl.3.3, 120.192.1-24.dl.3.4, 120.192.1-24.dl.3.5, 120.192.1-24.dl.3.6, 120.192.1-24.dl.3.7, 120.192.1-24.dl.3.8, 120.192.1-24.dl.3.9, 120.192.1-24.dl.3.10, 120.192.1-24.dl.3.11, 120.192.1-24.dl.3.12, 120.192.1-24.dl.3.13, 120.192.1-24.dl.3.14, 120.192.1-24.dl.3.15, 120.192.1-24.dl.3.16, 168.192.1-24.dl.3.1, 168.192.1-24.dl.3.2, 168.192.1-24.dl.3.3, 168.192.1-24.dl.3.4, 168.192.1-24.dl.3.5, 168.192.1-24.dl.3.6, 168.192.1-24.dl.3.7, 168.192.1-24.dl.3.8, 168.192.1-24.dl.3.9, 168.192.1-24.dl.3.10, 168.192.1-24.dl.3.11, 168.192.1-24.dl.3.12, 168.192.1-24.dl.3.13, 168.192.1-24.dl.3.14, 168.192.1-24.dl.3.15, 168.192.1-24.dl.3.16, 264.192.1-24.dl.3.1, 264.192.1-24.dl.3.2, 264.192.1-24.dl.3.3, 264.192.1-24.dl.3.4, 264.192.1-24.dl.3.5, 264.192.1-24.dl.3.6, 264.192.1-24.dl.3.7, 264.192.1-24.dl.3.8, 264.192.1-24.dl.3.9, 264.192.1-24.dl.3.10, 264.192.1-24.dl.3.11, 264.192.1-24.dl.3.12, 264.192.1-24.dl.3.13, 264.192.1-24.dl.3.14, 264.192.1-24.dl.3.15, 264.192.1-24.dl.3.16, 312.192.1-24.dl.3.1, 312.192.1-24.dl.3.2, 312.192.1-24.dl.3.3, 312.192.1-24.dl.3.4, 312.192.1-24.dl.3.5, 312.192.1-24.dl.3.6, 312.192.1-24.dl.3.7, 312.192.1-24.dl.3.8, 312.192.1-24.dl.3.9, 312.192.1-24.dl.3.10, 312.192.1-24.dl.3.11, 312.192.1-24.dl.3.12, 312.192.1-24.dl.3.13, 312.192.1-24.dl.3.14, 312.192.1-24.dl.3.15, 312.192.1-24.dl.3.16 |
Cyclic 24-isogeny field degree: |
$2$ |
Cyclic 24-torsion field degree: |
$16$ |
Full 24-torsion field degree: |
$768$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 4 x^{2} + 2 x y + z^{2} $ |
| $=$ | $10 x^{2} - 16 x y - 6 y^{2} - 2 z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 3 x^{4} + 6 x^{2} y^{2} - 4 x^{2} z^{2} - 4 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:
$\displaystyle X$ |
$=$ |
$\displaystyle x$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{6}w$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{2}z$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{1}{3^4}\cdot\frac{(6z^{2}-w^{2})^{3}(3668281344y^{2}z^{16}+228427776y^{2}z^{14}w^{2}-274337280y^{2}z^{12}w^{4}+721861632y^{2}z^{10}w^{6}-262175616y^{2}z^{8}w^{8}+44343936y^{2}z^{6}w^{10}-4167072y^{2}z^{4}w^{12}+209664y^{2}z^{2}w^{14}-4368y^{2}w^{16}+634894848z^{18}-569389824z^{16}w^{2}+1776473856z^{14}w^{4}-3322187136z^{12}w^{6}+1275575040z^{10}w^{8}-237536064z^{8}w^{10}+25713072z^{6}w^{12}-1673784z^{4}w^{14}+61278z^{2}w^{16}-973w^{18})}{w^{2}z^{8}(12z^{2}-w^{2})(23328y^{2}z^{10}+3888y^{2}z^{8}w^{2}-7128y^{2}z^{6}w^{4}+1836y^{2}z^{4}w^{6}-180y^{2}z^{2}w^{8}+6y^{2}w^{10}-104976z^{12}-14580z^{10}w^{2}-2025z^{8}w^{4}+864z^{6}w^{6}-189z^{4}w^{8}+24z^{2}w^{10}-w^{12})}$ |
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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.