$\GL_2(\Z/24\Z)$-generators: |
$\begin{bmatrix}5&2\\6&13\end{bmatrix}$, $\begin{bmatrix}7&13\\6&13\end{bmatrix}$, $\begin{bmatrix}17&22\\0&17\end{bmatrix}$, $\begin{bmatrix}23&16\\0&1\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
24.96.1-24.ig.1.1, 24.96.1-24.ig.1.2, 24.96.1-24.ig.1.3, 24.96.1-24.ig.1.4, 24.96.1-24.ig.1.5, 24.96.1-24.ig.1.6, 24.96.1-24.ig.1.7, 24.96.1-24.ig.1.8, 120.96.1-24.ig.1.1, 120.96.1-24.ig.1.2, 120.96.1-24.ig.1.3, 120.96.1-24.ig.1.4, 120.96.1-24.ig.1.5, 120.96.1-24.ig.1.6, 120.96.1-24.ig.1.7, 120.96.1-24.ig.1.8, 168.96.1-24.ig.1.1, 168.96.1-24.ig.1.2, 168.96.1-24.ig.1.3, 168.96.1-24.ig.1.4, 168.96.1-24.ig.1.5, 168.96.1-24.ig.1.6, 168.96.1-24.ig.1.7, 168.96.1-24.ig.1.8, 264.96.1-24.ig.1.1, 264.96.1-24.ig.1.2, 264.96.1-24.ig.1.3, 264.96.1-24.ig.1.4, 264.96.1-24.ig.1.5, 264.96.1-24.ig.1.6, 264.96.1-24.ig.1.7, 264.96.1-24.ig.1.8, 312.96.1-24.ig.1.1, 312.96.1-24.ig.1.2, 312.96.1-24.ig.1.3, 312.96.1-24.ig.1.4, 312.96.1-24.ig.1.5, 312.96.1-24.ig.1.6, 312.96.1-24.ig.1.7, 312.96.1-24.ig.1.8 |
Cyclic 24-isogeny field degree: |
$4$ |
Cyclic 24-torsion field degree: |
$32$ |
Full 24-torsion field degree: |
$1536$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ x^{2} - 2 y z $ |
| $=$ | $24 x^{2} - 6 y^{2} + 12 y z - 54 z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 9 x^{4} - 20 x^{2} 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:
$\displaystyle X$ |
$=$ |
$\displaystyle x$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{3}w$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle 3z$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{1}{2^3\cdot3^3}\cdot\frac{17390370816yz^{11}+3901685760yz^{9}w^{2}+304570368yz^{7}w^{4}+9234432yz^{5}w^{6}+75456yz^{3}w^{8}+144yzw^{10}-17199267840z^{12}-3551330304z^{10}w^{2}-223948800z^{8}w^{4}-3096576z^{6}w^{6}+101376z^{4}w^{8}+1008z^{2}w^{10}+w^{12}}{w^{2}z^{6}(23328yz^{3}+108yzw^{2}-23328z^{4}+378z^{2}w^{2}+w^{4})}$ |
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.