| $\GL_2(\Z/24\Z)$-generators: |
$\begin{bmatrix}1&3\\12&7\end{bmatrix}$, $\begin{bmatrix}1&15\\0&5\end{bmatrix}$, $\begin{bmatrix}1&21\\18&5\end{bmatrix}$, $\begin{bmatrix}5&10\\12&19\end{bmatrix}$ |
| Contains $-I$: |
yes |
| Quadratic refinements: |
24.96.1-24.eq.1.1, 24.96.1-24.eq.1.2, 24.96.1-24.eq.1.3, 24.96.1-24.eq.1.4, 24.96.1-24.eq.1.5, 24.96.1-24.eq.1.6, 24.96.1-24.eq.1.7, 24.96.1-24.eq.1.8, 120.96.1-24.eq.1.1, 120.96.1-24.eq.1.2, 120.96.1-24.eq.1.3, 120.96.1-24.eq.1.4, 120.96.1-24.eq.1.5, 120.96.1-24.eq.1.6, 120.96.1-24.eq.1.7, 120.96.1-24.eq.1.8, 168.96.1-24.eq.1.1, 168.96.1-24.eq.1.2, 168.96.1-24.eq.1.3, 168.96.1-24.eq.1.4, 168.96.1-24.eq.1.5, 168.96.1-24.eq.1.6, 168.96.1-24.eq.1.7, 168.96.1-24.eq.1.8, 264.96.1-24.eq.1.1, 264.96.1-24.eq.1.2, 264.96.1-24.eq.1.3, 264.96.1-24.eq.1.4, 264.96.1-24.eq.1.5, 264.96.1-24.eq.1.6, 264.96.1-24.eq.1.7, 264.96.1-24.eq.1.8, 312.96.1-24.eq.1.1, 312.96.1-24.eq.1.2, 312.96.1-24.eq.1.3, 312.96.1-24.eq.1.4, 312.96.1-24.eq.1.5, 312.96.1-24.eq.1.6, 312.96.1-24.eq.1.7, 312.96.1-24.eq.1.8 |
| Cyclic 24-isogeny field degree: |
$4$ |
| Cyclic 24-torsion field degree: |
$32$ |
| Full 24-torsion field degree: |
$1536$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 6 x y + z^{2} $ |
| $=$ | $2 x^{2} + 2 x y + 18 y^{2} - 3 z^{2} - w^{2}$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 3 x^{4} - 6 x^{2} y^{2} - 20 x^{2} z^{2} + 12 z^{4} $ |
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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:
| $\displaystyle X$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{6}w$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{6}z$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -\frac{1}{3}\cdot\frac{(4z^{2}+3w^{2})(559104y^{2}z^{8}-78336y^{2}z^{6}w^{2}-196992y^{2}z^{4}w^{4}-1179360y^{2}z^{2}w^{6}-353808y^{2}w^{8}-10240z^{10}+9216z^{8}w^{2}+46656z^{6}w^{4}+212112z^{4}w^{6}+131220z^{2}w^{8}+19683w^{10})}{w^{2}z^{4}(48y^{2}z^{4}-36y^{2}z^{2}w^{2}-54y^{2}w^{4}-8z^{6}+3z^{4}w^{2})}$ |
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.
