$\GL_2(\Z/24\Z)$-generators: |
$\begin{bmatrix}3&22\\16&3\end{bmatrix}$, $\begin{bmatrix}3&22\\16&21\end{bmatrix}$, $\begin{bmatrix}13&10\\16&21\end{bmatrix}$, $\begin{bmatrix}13&22\\8&7\end{bmatrix}$, $\begin{bmatrix}19&16\\12&5\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
24.96.1-24.bd.2.1, 24.96.1-24.bd.2.2, 24.96.1-24.bd.2.3, 24.96.1-24.bd.2.4, 24.96.1-24.bd.2.5, 24.96.1-24.bd.2.6, 24.96.1-24.bd.2.7, 24.96.1-24.bd.2.8, 24.96.1-24.bd.2.9, 24.96.1-24.bd.2.10, 24.96.1-24.bd.2.11, 24.96.1-24.bd.2.12, 24.96.1-24.bd.2.13, 24.96.1-24.bd.2.14, 24.96.1-24.bd.2.15, 24.96.1-24.bd.2.16, 120.96.1-24.bd.2.1, 120.96.1-24.bd.2.2, 120.96.1-24.bd.2.3, 120.96.1-24.bd.2.4, 120.96.1-24.bd.2.5, 120.96.1-24.bd.2.6, 120.96.1-24.bd.2.7, 120.96.1-24.bd.2.8, 120.96.1-24.bd.2.9, 120.96.1-24.bd.2.10, 120.96.1-24.bd.2.11, 120.96.1-24.bd.2.12, 120.96.1-24.bd.2.13, 120.96.1-24.bd.2.14, 120.96.1-24.bd.2.15, 120.96.1-24.bd.2.16, 168.96.1-24.bd.2.1, 168.96.1-24.bd.2.2, 168.96.1-24.bd.2.3, 168.96.1-24.bd.2.4, 168.96.1-24.bd.2.5, 168.96.1-24.bd.2.6, 168.96.1-24.bd.2.7, 168.96.1-24.bd.2.8, 168.96.1-24.bd.2.9, 168.96.1-24.bd.2.10, 168.96.1-24.bd.2.11, 168.96.1-24.bd.2.12, 168.96.1-24.bd.2.13, 168.96.1-24.bd.2.14, 168.96.1-24.bd.2.15, 168.96.1-24.bd.2.16, 264.96.1-24.bd.2.1, 264.96.1-24.bd.2.2, 264.96.1-24.bd.2.3, 264.96.1-24.bd.2.4, 264.96.1-24.bd.2.5, 264.96.1-24.bd.2.6, 264.96.1-24.bd.2.7, 264.96.1-24.bd.2.8, 264.96.1-24.bd.2.9, 264.96.1-24.bd.2.10, 264.96.1-24.bd.2.11, 264.96.1-24.bd.2.12, 264.96.1-24.bd.2.13, 264.96.1-24.bd.2.14, 264.96.1-24.bd.2.15, 264.96.1-24.bd.2.16, 312.96.1-24.bd.2.1, 312.96.1-24.bd.2.2, 312.96.1-24.bd.2.3, 312.96.1-24.bd.2.4, 312.96.1-24.bd.2.5, 312.96.1-24.bd.2.6, 312.96.1-24.bd.2.7, 312.96.1-24.bd.2.8, 312.96.1-24.bd.2.9, 312.96.1-24.bd.2.10, 312.96.1-24.bd.2.11, 312.96.1-24.bd.2.12, 312.96.1-24.bd.2.13, 312.96.1-24.bd.2.14, 312.96.1-24.bd.2.15, 312.96.1-24.bd.2.16 |
Cyclic 24-isogeny field degree: |
$8$ |
Cyclic 24-torsion field degree: |
$64$ |
Full 24-torsion field degree: |
$1536$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 3 x y - 3 x z - y^{2} - 2 y z - z^{2} $ |
| $=$ | $x^{2} + 3 y^{2} - 2 y z + 3 z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 7 x^{4} - 17 x^{3} z + x^{2} y^{2} + 24 x^{2} z^{2} - 2 x y^{2} z - 17 x z^{3} + y^{2} z^{2} + 7 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:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle \frac{3}{2}w$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle z$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{1}{7^4}\cdot\frac{838835850240xz^{11}+2228511849984xz^{9}w^{2}+1675270150848xz^{7}w^{4}+1425184954368xz^{5}w^{6}+3792037530372xz^{3}w^{8}+2057821117440y^{2}z^{10}+4819671666432y^{2}z^{8}w^{2}+6556624084416y^{2}z^{6}w^{4}-573460026160y^{2}z^{4}w^{6}-5246448360888y^{2}z^{2}w^{8}-2659434619443y^{2}w^{10}-1218985267200yz^{11}-826703771904yz^{9}w^{2}-3961138597248yz^{7}w^{4}+415628021872yz^{5}w^{6}+7460661097848yz^{3}w^{8}+1220785667241yzw^{10}+351221302272z^{12}-1072058135040z^{10}w^{2}+1608759445440z^{8}w^{4}-468182443456z^{6}w^{6}-8056812437184z^{4}w^{8}-5503462767222z^{2}w^{10}-867763964928w^{12}}{w^{4}(626688xz^{7}-172032xz^{5}w^{2}+65268xz^{3}w^{4}+1380352y^{2}z^{6}-1390592y^{2}z^{4}w^{2}-122304y^{2}z^{2}w^{4}+9261y^{2}w^{6}-753664yz^{7}+591872yz^{5}w^{2}+453096yz^{3}w^{4}+30429yzw^{6}+753664z^{8}-183296z^{6}w^{2}-227304z^{4}w^{4}-39690z^{2}w^{6})}$ |
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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.