$\GL_2(\Z/24\Z)$-generators: |
$\begin{bmatrix}8&5\\9&19\end{bmatrix}$, $\begin{bmatrix}13&18\\6&11\end{bmatrix}$, $\begin{bmatrix}17&17\\6&11\end{bmatrix}$, $\begin{bmatrix}19&12\\12&23\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
24.64.1-24.d.1.1, 24.64.1-24.d.1.2, 24.64.1-24.d.1.3, 24.64.1-24.d.1.4, 24.64.1-24.d.1.5, 24.64.1-24.d.1.6, 24.64.1-24.d.1.7, 24.64.1-24.d.1.8, 120.64.1-24.d.1.1, 120.64.1-24.d.1.2, 120.64.1-24.d.1.3, 120.64.1-24.d.1.4, 120.64.1-24.d.1.5, 120.64.1-24.d.1.6, 120.64.1-24.d.1.7, 120.64.1-24.d.1.8, 168.64.1-24.d.1.1, 168.64.1-24.d.1.2, 168.64.1-24.d.1.3, 168.64.1-24.d.1.4, 168.64.1-24.d.1.5, 168.64.1-24.d.1.6, 168.64.1-24.d.1.7, 168.64.1-24.d.1.8, 264.64.1-24.d.1.1, 264.64.1-24.d.1.2, 264.64.1-24.d.1.3, 264.64.1-24.d.1.4, 264.64.1-24.d.1.5, 264.64.1-24.d.1.6, 264.64.1-24.d.1.7, 264.64.1-24.d.1.8, 312.64.1-24.d.1.1, 312.64.1-24.d.1.2, 312.64.1-24.d.1.3, 312.64.1-24.d.1.4, 312.64.1-24.d.1.5, 312.64.1-24.d.1.6, 312.64.1-24.d.1.7, 312.64.1-24.d.1.8 |
Cyclic 24-isogeny field degree: |
$12$ |
Cyclic 24-torsion field degree: |
$96$ |
Full 24-torsion field degree: |
$2304$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 3 x^{2} - 6 x z + 2 y w $ |
| $=$ | $9 x^{2} + 6 x z - 9 y^{2} + 6 z^{2} - w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} - 6 x^{3} y + 9 x^{2} y^{2} - 4 x^{2} z^{2} - 4 x y z^{2} - 6 y^{2} z^{2} + 4 z^{4} $ |
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:
$\displaystyle X$ |
$=$ |
$\displaystyle x$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle \frac{2}{3}z$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{3}w$ |
Maps to other modular curves
$j$-invariant map
of degree 32 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle 2\cdot3^2\,\frac{z(3949236xyz^{4}w+944640xyz^{2}w^{3}+2304xyw^{5}-4434876xz^{6}-7676100xz^{4}w^{2}-222720xz^{2}w^{4}+768xw^{6}+1478316yz^{5}w+2229408yz^{3}w^{3}+65280yzw^{5}+9z^{7}-713340z^{5}w^{2}+49952z^{3}w^{4}+11520zw^{6})}{43200xyz^{5}w+16896xyz^{3}w^{3}+411xyzw^{5}-46656xz^{7}-97152xz^{5}w^{2}-9271xz^{3}w^{4}+17xzw^{6}+15552yz^{6}w+28736yz^{4}w^{3}+2573yz^{2}w^{5}+8yw^{7}-7872z^{6}w^{2}-434z^{4}w^{4}+279z^{2}w^{6}+2w^{8}}$ |
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.