$\GL_2(\Z/24\Z)$-generators: |
$\begin{bmatrix}1&3\\18&19\end{bmatrix}$, $\begin{bmatrix}7&1\\6&11\end{bmatrix}$, $\begin{bmatrix}7&5\\18&17\end{bmatrix}$, $\begin{bmatrix}17&0\\12&19\end{bmatrix}$, $\begin{bmatrix}17&16\\12&11\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
24.96.1-24.ct.1.1, 24.96.1-24.ct.1.2, 24.96.1-24.ct.1.3, 24.96.1-24.ct.1.4, 24.96.1-24.ct.1.5, 24.96.1-24.ct.1.6, 24.96.1-24.ct.1.7, 24.96.1-24.ct.1.8, 24.96.1-24.ct.1.9, 24.96.1-24.ct.1.10, 24.96.1-24.ct.1.11, 24.96.1-24.ct.1.12, 120.96.1-24.ct.1.1, 120.96.1-24.ct.1.2, 120.96.1-24.ct.1.3, 120.96.1-24.ct.1.4, 120.96.1-24.ct.1.5, 120.96.1-24.ct.1.6, 120.96.1-24.ct.1.7, 120.96.1-24.ct.1.8, 120.96.1-24.ct.1.9, 120.96.1-24.ct.1.10, 120.96.1-24.ct.1.11, 120.96.1-24.ct.1.12, 168.96.1-24.ct.1.1, 168.96.1-24.ct.1.2, 168.96.1-24.ct.1.3, 168.96.1-24.ct.1.4, 168.96.1-24.ct.1.5, 168.96.1-24.ct.1.6, 168.96.1-24.ct.1.7, 168.96.1-24.ct.1.8, 168.96.1-24.ct.1.9, 168.96.1-24.ct.1.10, 168.96.1-24.ct.1.11, 168.96.1-24.ct.1.12, 264.96.1-24.ct.1.1, 264.96.1-24.ct.1.2, 264.96.1-24.ct.1.3, 264.96.1-24.ct.1.4, 264.96.1-24.ct.1.5, 264.96.1-24.ct.1.6, 264.96.1-24.ct.1.7, 264.96.1-24.ct.1.8, 264.96.1-24.ct.1.9, 264.96.1-24.ct.1.10, 264.96.1-24.ct.1.11, 264.96.1-24.ct.1.12, 312.96.1-24.ct.1.1, 312.96.1-24.ct.1.2, 312.96.1-24.ct.1.3, 312.96.1-24.ct.1.4, 312.96.1-24.ct.1.5, 312.96.1-24.ct.1.6, 312.96.1-24.ct.1.7, 312.96.1-24.ct.1.8, 312.96.1-24.ct.1.9, 312.96.1-24.ct.1.10, 312.96.1-24.ct.1.11, 312.96.1-24.ct.1.12 |
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 $ | $=$ | $ x^{2} - 2 y z $ |
| $=$ | $8 x^{2} + 2 y^{2} + 4 y z + 18 z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 9 x^{4} + 20 x^{2} z^{2} + 2 y^{2} z^{2} + 4 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:
$\displaystyle X$ |
$=$ |
$\displaystyle x$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle w$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle 3z$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle -\frac{1}{2^3}\cdot\frac{23855104yz^{11}-16056320yz^{9}w^{2}+3760128yz^{7}w^{4}-342016yz^{5}w^{6}+8384yz^{3}w^{8}-48yzw^{10}+23592960z^{12}-14614528z^{10}w^{2}+2764800z^{8}w^{4}-114688z^{6}w^{6}-11264z^{4}w^{8}+336z^{2}w^{10}-w^{12}}{w^{2}z^{6}(2592yz^{3}-36yzw^{2}+2592z^{4}+126z^{2}w^{2}-w^{4})}$ |
Hi
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Cover information
Click on a modular curve in the diagram to see information about it.
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The following modular covers realize this modular curve as a fiber product over $X(1)$.
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.