$\GL_2(\Z/24\Z)$-generators: |
$\begin{bmatrix}1&10\\6&19\end{bmatrix}$, $\begin{bmatrix}2&5\\21&13\end{bmatrix}$, $\begin{bmatrix}13&5\\9&20\end{bmatrix}$, $\begin{bmatrix}16&7\\3&17\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
24.128.1-24.a.1.1, 24.128.1-24.a.1.2, 24.128.1-24.a.1.3, 24.128.1-24.a.1.4, 24.128.1-24.a.1.5, 24.128.1-24.a.1.6, 24.128.1-24.a.1.7, 24.128.1-24.a.1.8, 120.128.1-24.a.1.1, 120.128.1-24.a.1.2, 120.128.1-24.a.1.3, 120.128.1-24.a.1.4, 120.128.1-24.a.1.5, 120.128.1-24.a.1.6, 120.128.1-24.a.1.7, 120.128.1-24.a.1.8, 168.128.1-24.a.1.1, 168.128.1-24.a.1.2, 168.128.1-24.a.1.3, 168.128.1-24.a.1.4, 168.128.1-24.a.1.5, 168.128.1-24.a.1.6, 168.128.1-24.a.1.7, 168.128.1-24.a.1.8, 264.128.1-24.a.1.1, 264.128.1-24.a.1.2, 264.128.1-24.a.1.3, 264.128.1-24.a.1.4, 264.128.1-24.a.1.5, 264.128.1-24.a.1.6, 264.128.1-24.a.1.7, 264.128.1-24.a.1.8, 312.128.1-24.a.1.1, 312.128.1-24.a.1.2, 312.128.1-24.a.1.3, 312.128.1-24.a.1.4, 312.128.1-24.a.1.5, 312.128.1-24.a.1.6, 312.128.1-24.a.1.7, 312.128.1-24.a.1.8 |
Cyclic 24-isogeny field degree: |
$12$ |
Cyclic 24-torsion field degree: |
$96$ |
Full 24-torsion field degree: |
$1152$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 3 x^{2} + 6 x y + z^{2} + w^{2} $ |
| $=$ | $9 x^{2} - 6 x y + 6 y^{2} - 2 z^{2} + 2 z w - 2 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 75 x^{4} - 60 x^{3} z - 36 x^{2} y^{2} + 162 x^{2} z^{2} + 40 x y^{2} z - 60 x z^{3} + 12 y^{4} + \cdots + 75 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:
$\displaystyle X$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle 3y$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Maps to other modular curves
$j$-invariant map
of degree 64 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle -\frac{3^2}{2^2}\cdot\frac{(z^{2}+3w^{2})(17496y^{2}z^{11}w-25272y^{2}z^{9}w^{3}-379728y^{2}z^{7}w^{5}+365520y^{2}z^{5}w^{7}+2003832y^{2}z^{3}w^{9}-2117016y^{2}zw^{11}-729z^{14}-2916z^{13}w+9963z^{12}w^{2}+1296z^{11}w^{3}+21123z^{10}w^{4}+67500z^{9}w^{5}-163241z^{8}w^{6}+2368z^{7}w^{7}-160875z^{6}w^{8}-394892z^{5}w^{9}+617761z^{4}w^{10}+18864z^{3}w^{11}+197505z^{2}w^{12}+352836zw^{13}-107811w^{14})}{w^{4}(1458y^{2}z^{9}w+2376y^{2}z^{7}w^{3}+3372y^{2}z^{5}w^{5}+2376y^{2}z^{3}w^{7}+1458y^{2}zw^{9}-81z^{12}-243z^{11}w+432z^{10}w^{2}-639z^{9}w^{3}+1049z^{8}w^{4}-958z^{7}w^{5}+1584z^{6}w^{6}-958z^{5}w^{7}+1049z^{4}w^{8}-639z^{3}w^{9}+432z^{2}w^{10}-243zw^{11}-81w^{12})}$ |
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.