$\GL_2(\Z/24\Z)$-generators: |
$\begin{bmatrix}1&14\\20&17\end{bmatrix}$, $\begin{bmatrix}5&10\\0&13\end{bmatrix}$, $\begin{bmatrix}7&12\\20&19\end{bmatrix}$, $\begin{bmatrix}7&20\\8&23\end{bmatrix}$, $\begin{bmatrix}13&4\\16&15\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
24.96.1-24.bh.2.1, 24.96.1-24.bh.2.2, 24.96.1-24.bh.2.3, 24.96.1-24.bh.2.4, 24.96.1-24.bh.2.5, 24.96.1-24.bh.2.6, 24.96.1-24.bh.2.7, 24.96.1-24.bh.2.8, 24.96.1-24.bh.2.9, 24.96.1-24.bh.2.10, 24.96.1-24.bh.2.11, 24.96.1-24.bh.2.12, 24.96.1-24.bh.2.13, 24.96.1-24.bh.2.14, 24.96.1-24.bh.2.15, 24.96.1-24.bh.2.16, 120.96.1-24.bh.2.1, 120.96.1-24.bh.2.2, 120.96.1-24.bh.2.3, 120.96.1-24.bh.2.4, 120.96.1-24.bh.2.5, 120.96.1-24.bh.2.6, 120.96.1-24.bh.2.7, 120.96.1-24.bh.2.8, 120.96.1-24.bh.2.9, 120.96.1-24.bh.2.10, 120.96.1-24.bh.2.11, 120.96.1-24.bh.2.12, 120.96.1-24.bh.2.13, 120.96.1-24.bh.2.14, 120.96.1-24.bh.2.15, 120.96.1-24.bh.2.16, 168.96.1-24.bh.2.1, 168.96.1-24.bh.2.2, 168.96.1-24.bh.2.3, 168.96.1-24.bh.2.4, 168.96.1-24.bh.2.5, 168.96.1-24.bh.2.6, 168.96.1-24.bh.2.7, 168.96.1-24.bh.2.8, 168.96.1-24.bh.2.9, 168.96.1-24.bh.2.10, 168.96.1-24.bh.2.11, 168.96.1-24.bh.2.12, 168.96.1-24.bh.2.13, 168.96.1-24.bh.2.14, 168.96.1-24.bh.2.15, 168.96.1-24.bh.2.16, 264.96.1-24.bh.2.1, 264.96.1-24.bh.2.2, 264.96.1-24.bh.2.3, 264.96.1-24.bh.2.4, 264.96.1-24.bh.2.5, 264.96.1-24.bh.2.6, 264.96.1-24.bh.2.7, 264.96.1-24.bh.2.8, 264.96.1-24.bh.2.9, 264.96.1-24.bh.2.10, 264.96.1-24.bh.2.11, 264.96.1-24.bh.2.12, 264.96.1-24.bh.2.13, 264.96.1-24.bh.2.14, 264.96.1-24.bh.2.15, 264.96.1-24.bh.2.16, 312.96.1-24.bh.2.1, 312.96.1-24.bh.2.2, 312.96.1-24.bh.2.3, 312.96.1-24.bh.2.4, 312.96.1-24.bh.2.5, 312.96.1-24.bh.2.6, 312.96.1-24.bh.2.7, 312.96.1-24.bh.2.8, 312.96.1-24.bh.2.9, 312.96.1-24.bh.2.10, 312.96.1-24.bh.2.11, 312.96.1-24.bh.2.12, 312.96.1-24.bh.2.13, 312.96.1-24.bh.2.14, 312.96.1-24.bh.2.15, 312.96.1-24.bh.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} $ |
| $=$ | $2 x^{2} + 6 y^{2} + 4 y z + 6 z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 7 x^{4} + 17 x^{3} z + 2 x^{2} y^{2} + 24 x^{2} z^{2} + 4 x y^{2} z + 17 x z^{3} + 2 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:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle \frac{3}{4}w$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle z$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{1}{7^4}\cdot\frac{26842747207680xz^{11}+35656189599744xz^{9}w^{2}+13402161206784xz^{7}w^{4}+5700739817472xz^{5}w^{6}+7584075060744xz^{3}w^{8}-65850275758080y^{2}z^{10}-77114746662912y^{2}z^{8}w^{2}-52452992675328y^{2}z^{6}w^{4}+2293840104640y^{2}z^{4}w^{6}+10492896721776y^{2}z^{2}w^{8}+2659434619443y^{2}w^{10}-39007528550400yz^{11}-13227260350464yz^{9}w^{2}-31689108777984yz^{7}w^{4}+1662512087488yz^{5}w^{6}+14921322195696yz^{3}w^{8}+1220785667241yzw^{10}-11239081672704z^{12}+17152930160640z^{10}w^{2}-12870075563520z^{8}w^{4}+1872729773824z^{6}w^{6}+16113624874368z^{4}w^{8}+5503462767222z^{2}w^{10}+433881982464w^{12}}{w^{4}(5013504xz^{7}-688128xz^{5}w^{2}+130536xz^{3}w^{4}-11042816y^{2}z^{6}+5562368y^{2}z^{4}w^{2}+244608y^{2}z^{2}w^{4}-9261y^{2}w^{6}-6029312yz^{7}+2367488yz^{5}w^{2}+906192yz^{3}w^{4}+30429yzw^{6}-6029312z^{8}+733184z^{6}w^{2}+454608z^{4}w^{4}+39690z^{2}w^{6})}$ |
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.