| $\GL_2(\Z/24\Z)$-generators: |
$\begin{bmatrix}5&0\\18&7\end{bmatrix}$, $\begin{bmatrix}7&4\\18&5\end{bmatrix}$, $\begin{bmatrix}11&16\\0&13\end{bmatrix}$, $\begin{bmatrix}17&2\\18&13\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: |
$C_2^4:D_{12}$ |
| Contains $-I$: |
yes |
| Quadratic refinements: |
24.384.5-24.bz.1.1, 24.384.5-24.bz.1.2, 24.384.5-24.bz.1.3, 24.384.5-24.bz.1.4, 24.384.5-24.bz.1.5, 24.384.5-24.bz.1.6, 24.384.5-24.bz.1.7, 24.384.5-24.bz.1.8, 120.384.5-24.bz.1.1, 120.384.5-24.bz.1.2, 120.384.5-24.bz.1.3, 120.384.5-24.bz.1.4, 120.384.5-24.bz.1.5, 120.384.5-24.bz.1.6, 120.384.5-24.bz.1.7, 120.384.5-24.bz.1.8, 168.384.5-24.bz.1.1, 168.384.5-24.bz.1.2, 168.384.5-24.bz.1.3, 168.384.5-24.bz.1.4, 168.384.5-24.bz.1.5, 168.384.5-24.bz.1.6, 168.384.5-24.bz.1.7, 168.384.5-24.bz.1.8, 264.384.5-24.bz.1.1, 264.384.5-24.bz.1.2, 264.384.5-24.bz.1.3, 264.384.5-24.bz.1.4, 264.384.5-24.bz.1.5, 264.384.5-24.bz.1.6, 264.384.5-24.bz.1.7, 264.384.5-24.bz.1.8, 312.384.5-24.bz.1.1, 312.384.5-24.bz.1.2, 312.384.5-24.bz.1.3, 312.384.5-24.bz.1.4, 312.384.5-24.bz.1.5, 312.384.5-24.bz.1.6, 312.384.5-24.bz.1.7, 312.384.5-24.bz.1.8 |
| Cyclic 24-isogeny field degree: |
$4$ |
| Cyclic 24-torsion field degree: |
$16$ |
| Full 24-torsion field degree: |
$384$ |
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ 2 x y - y^{2} - w^{2} $ |
| $=$ | $2 x^{2} - 2 x y - y^{2} - w^{2} - t^{2}$ |
| $=$ | $3 z^{2} - 2 w^{2} - t^{2}$ |
![Copy content]()
![Toggle raw display]()
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 144 x^{8} - 360 x^{6} y^{2} - 480 x^{6} z^{2} + 9 x^{4} y^{4} + 24 x^{4} y^{2} z^{2} + 376 x^{4} z^{4} + \cdots + z^{8} $ |
![Copy content]()
![Toggle raw display]()
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 canonical model to plane model:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle 4z+4w$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle t$ |
Maps to other modular curves
$j$-invariant map
of degree 192 from the canonical model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -2^4\,\frac{(4w^{4}+2w^{2}t^{2}+t^{4})(186368y^{2}w^{18}+419328y^{2}w^{16}t^{2}+364032y^{2}w^{14}t^{4}+147840y^{2}w^{12}t^{6}+17856y^{2}w^{10}t^{8}-8928y^{2}w^{8}t^{10}-18480y^{2}w^{6}t^{12}-11376y^{2}w^{4}t^{14}-3276y^{2}w^{2}t^{16}-364y^{2}t^{18}-62464w^{20}-109568w^{18}t^{2}-67840w^{16}t^{4}-20096w^{14}t^{6}-18688w^{12}t^{8}-22784w^{10}t^{10}-27616w^{8}t^{12}-21872w^{6}t^{14}-10024w^{4}t^{16}-2430w^{2}t^{18}-243t^{20})}{t^{4}w^{4}(2w^{2}+t^{2})^{2}(64y^{2}w^{10}+80y^{2}w^{8}t^{2}+16y^{2}w^{6}t^{4}-8y^{2}w^{4}t^{6}-10y^{2}w^{2}t^{8}-2y^{2}t^{10}+64w^{12}+112w^{10}t^{2}+60w^{8}t^{4}+4w^{6}t^{6}+w^{4}t^{8})}$ |
Hi
|
|
Cover information
Click on a modular curve in the diagram to see information about it.
|
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.
