$\GL_2(\Z/24\Z)$-generators: |
$\begin{bmatrix}15&16\\8&9\end{bmatrix}$, $\begin{bmatrix}19&8\\8&17\end{bmatrix}$, $\begin{bmatrix}21&20\\16&9\end{bmatrix}$, $\begin{bmatrix}21&20\\16&15\end{bmatrix}$, $\begin{bmatrix}23&0\\0&7\end{bmatrix}$, $\begin{bmatrix}23&2\\8&7\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
24.288.8-24.fp.2.1, 24.288.8-24.fp.2.2, 24.288.8-24.fp.2.3, 24.288.8-24.fp.2.4, 24.288.8-24.fp.2.5, 24.288.8-24.fp.2.6, 24.288.8-24.fp.2.7, 24.288.8-24.fp.2.8, 24.288.8-24.fp.2.9, 24.288.8-24.fp.2.10, 24.288.8-24.fp.2.11, 24.288.8-24.fp.2.12, 24.288.8-24.fp.2.13, 24.288.8-24.fp.2.14, 24.288.8-24.fp.2.15, 24.288.8-24.fp.2.16, 24.288.8-24.fp.2.17, 24.288.8-24.fp.2.18, 24.288.8-24.fp.2.19, 24.288.8-24.fp.2.20, 24.288.8-24.fp.2.21, 24.288.8-24.fp.2.22, 24.288.8-24.fp.2.23, 24.288.8-24.fp.2.24, 48.288.8-24.fp.2.1, 48.288.8-24.fp.2.2, 48.288.8-24.fp.2.3, 48.288.8-24.fp.2.4, 48.288.8-24.fp.2.5, 48.288.8-24.fp.2.6, 48.288.8-24.fp.2.7, 48.288.8-24.fp.2.8, 48.288.8-24.fp.2.9, 48.288.8-24.fp.2.10, 48.288.8-24.fp.2.11, 48.288.8-24.fp.2.12, 48.288.8-24.fp.2.13, 48.288.8-24.fp.2.14, 48.288.8-24.fp.2.15, 48.288.8-24.fp.2.16, 48.288.8-24.fp.2.17, 48.288.8-24.fp.2.18, 48.288.8-24.fp.2.19, 48.288.8-24.fp.2.20, 48.288.8-24.fp.2.21, 48.288.8-24.fp.2.22, 48.288.8-24.fp.2.23, 48.288.8-24.fp.2.24, 48.288.8-24.fp.2.25, 48.288.8-24.fp.2.26, 48.288.8-24.fp.2.27, 48.288.8-24.fp.2.28, 48.288.8-24.fp.2.29, 48.288.8-24.fp.2.30, 48.288.8-24.fp.2.31, 48.288.8-24.fp.2.32, 120.288.8-24.fp.2.1, 120.288.8-24.fp.2.2, 120.288.8-24.fp.2.3, 120.288.8-24.fp.2.4, 120.288.8-24.fp.2.5, 120.288.8-24.fp.2.6, 120.288.8-24.fp.2.7, 120.288.8-24.fp.2.8, 120.288.8-24.fp.2.9, 120.288.8-24.fp.2.10, 120.288.8-24.fp.2.11, 120.288.8-24.fp.2.12, 120.288.8-24.fp.2.13, 120.288.8-24.fp.2.14, 120.288.8-24.fp.2.15, 120.288.8-24.fp.2.16, 120.288.8-24.fp.2.17, 120.288.8-24.fp.2.18, 120.288.8-24.fp.2.19, 120.288.8-24.fp.2.20, 120.288.8-24.fp.2.21, 120.288.8-24.fp.2.22, 120.288.8-24.fp.2.23, 120.288.8-24.fp.2.24, 168.288.8-24.fp.2.1, 168.288.8-24.fp.2.2, 168.288.8-24.fp.2.3, 168.288.8-24.fp.2.4, 168.288.8-24.fp.2.5, 168.288.8-24.fp.2.6, 168.288.8-24.fp.2.7, 168.288.8-24.fp.2.8, 168.288.8-24.fp.2.9, 168.288.8-24.fp.2.10, 168.288.8-24.fp.2.11, 168.288.8-24.fp.2.12, 168.288.8-24.fp.2.13, 168.288.8-24.fp.2.14, 168.288.8-24.fp.2.15, 168.288.8-24.fp.2.16, 168.288.8-24.fp.2.17, 168.288.8-24.fp.2.18, 168.288.8-24.fp.2.19, 168.288.8-24.fp.2.20, 168.288.8-24.fp.2.21, 168.288.8-24.fp.2.22, 168.288.8-24.fp.2.23, 168.288.8-24.fp.2.24, 240.288.8-24.fp.2.1, 240.288.8-24.fp.2.2, 240.288.8-24.fp.2.3, 240.288.8-24.fp.2.4, 240.288.8-24.fp.2.5, 240.288.8-24.fp.2.6, 240.288.8-24.fp.2.7, 240.288.8-24.fp.2.8, 240.288.8-24.fp.2.9, 240.288.8-24.fp.2.10, 240.288.8-24.fp.2.11, 240.288.8-24.fp.2.12, 240.288.8-24.fp.2.13, 240.288.8-24.fp.2.14, 240.288.8-24.fp.2.15, 240.288.8-24.fp.2.16, 240.288.8-24.fp.2.17, 240.288.8-24.fp.2.18, 240.288.8-24.fp.2.19, 240.288.8-24.fp.2.20, 240.288.8-24.fp.2.21, 240.288.8-24.fp.2.22, 240.288.8-24.fp.2.23, 240.288.8-24.fp.2.24, 240.288.8-24.fp.2.25, 240.288.8-24.fp.2.26, 240.288.8-24.fp.2.27, 240.288.8-24.fp.2.28, 240.288.8-24.fp.2.29, 240.288.8-24.fp.2.30, 240.288.8-24.fp.2.31, 240.288.8-24.fp.2.32, 264.288.8-24.fp.2.1, 264.288.8-24.fp.2.2, 264.288.8-24.fp.2.3, 264.288.8-24.fp.2.4, 264.288.8-24.fp.2.5, 264.288.8-24.fp.2.6, 264.288.8-24.fp.2.7, 264.288.8-24.fp.2.8, 264.288.8-24.fp.2.9, 264.288.8-24.fp.2.10, 264.288.8-24.fp.2.11, 264.288.8-24.fp.2.12, 264.288.8-24.fp.2.13, 264.288.8-24.fp.2.14, 264.288.8-24.fp.2.15, 264.288.8-24.fp.2.16, 264.288.8-24.fp.2.17, 264.288.8-24.fp.2.18, 264.288.8-24.fp.2.19, 264.288.8-24.fp.2.20, 264.288.8-24.fp.2.21, 264.288.8-24.fp.2.22, 264.288.8-24.fp.2.23, 264.288.8-24.fp.2.24, 312.288.8-24.fp.2.1, 312.288.8-24.fp.2.2, 312.288.8-24.fp.2.3, 312.288.8-24.fp.2.4, 312.288.8-24.fp.2.5, 312.288.8-24.fp.2.6, 312.288.8-24.fp.2.7, 312.288.8-24.fp.2.8, 312.288.8-24.fp.2.9, 312.288.8-24.fp.2.10, 312.288.8-24.fp.2.11, 312.288.8-24.fp.2.12, 312.288.8-24.fp.2.13, 312.288.8-24.fp.2.14, 312.288.8-24.fp.2.15, 312.288.8-24.fp.2.16, 312.288.8-24.fp.2.17, 312.288.8-24.fp.2.18, 312.288.8-24.fp.2.19, 312.288.8-24.fp.2.20, 312.288.8-24.fp.2.21, 312.288.8-24.fp.2.22, 312.288.8-24.fp.2.23, 312.288.8-24.fp.2.24 |
Cyclic 24-isogeny field degree: |
$4$ |
Cyclic 24-torsion field degree: |
$32$ |
Full 24-torsion field degree: |
$512$ |
Canonical model in $\mathbb{P}^{ 7 }$ defined by 20 equations
$ 0 $ | $=$ | $ z v + w r $ |
| $=$ | $x v + w t + w u$ |
| $=$ | $x r - z t - z u$ |
| $=$ | $2 y v + u r$ |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ - 2 x^{6} z^{2} + 4 x^{4} z^{4} + x^{2} y^{6} - 2 x^{2} z^{6} + y^{6} z^{2} $ |
This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
Canonical model |
$(0:1/4:0:0:-1/2:-1/2:1:1)$, $(0:-1/4:0:0:1/2:1/2:1:1)$, $(0:-1/4:0:0:-1/2:-1/2:-1:1)$, $(0:1/4:0:0:1/2:1/2:-1:1)$ |
Maps between models of this curve
Birational map from canonical model to plane model:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{2}z$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{2}t$ |
Maps to other modular curves
Map
of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve
24.72.4.z.2
:
$\displaystyle X$ |
$=$ |
$\displaystyle -x$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle -w$ |
$\displaystyle W$ |
$=$ |
$\displaystyle r$ |
Equation of the image curve:
$0$ |
$=$ |
$ 4XY-ZW $ |
|
$=$ |
$ 2X^{3}-16Y^{3}-XZ^{2}+YW^{2} $ |
Hi
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Cover information
Click on a modular curve in the diagram to see information about it.
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The following modular covers realize this modular curve as a fiber product over $X(1)$.
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.