$\GL_2(\Z/24\Z)$-generators: |
$\begin{bmatrix}5&16\\2&3\end{bmatrix}$, $\begin{bmatrix}11&20\\8&19\end{bmatrix}$, $\begin{bmatrix}17&14\\2&7\end{bmatrix}$, $\begin{bmatrix}19&4\\8&7\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
24.96.1-24.bn.1.1, 24.96.1-24.bn.1.2, 24.96.1-24.bn.1.3, 24.96.1-24.bn.1.4, 120.96.1-24.bn.1.1, 120.96.1-24.bn.1.2, 120.96.1-24.bn.1.3, 120.96.1-24.bn.1.4, 168.96.1-24.bn.1.1, 168.96.1-24.bn.1.2, 168.96.1-24.bn.1.3, 168.96.1-24.bn.1.4, 264.96.1-24.bn.1.1, 264.96.1-24.bn.1.2, 264.96.1-24.bn.1.3, 264.96.1-24.bn.1.4, 312.96.1-24.bn.1.1, 312.96.1-24.bn.1.2, 312.96.1-24.bn.1.3, 312.96.1-24.bn.1.4 |
Cyclic 24-isogeny field degree: |
$8$ |
Cyclic 24-torsion field degree: |
$64$ |
Full 24-torsion field degree: |
$1536$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ x y + x z + y^{2} + z^{2} $ |
| $=$ | $6 x^{2} - 18 x y + 6 x z + 6 y^{2} + 6 z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 5 x^{4} + 4 x^{3} z + 6 x^{2} y^{2} + 6 x^{2} z^{2} + 12 x y^{2} z + 4 x z^{3} + 6 y^{2} z^{2} + 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:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{6}w$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle z$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle -\frac{2^3}{3\cdot5^4}\cdot\frac{7287496998912xz^{11}-152842567680xz^{9}w^{2}-298374451200xz^{7}w^{4}-1831680000xz^{5}w^{6}+1724400000xz^{3}w^{8}+36445170917376y^{2}z^{10}+6832028436480y^{2}z^{8}w^{2}-672391756800y^{2}z^{6}w^{4}-87829920000y^{2}z^{4}w^{6}+4419900000y^{2}z^{2}w^{8}+119325000y^{2}w^{10}+6075096920064yz^{9}w^{2}+911302410240yz^{7}w^{4}-37680422400yz^{5}w^{6}-4556160000yz^{3}w^{8}+94950000yzw^{10}+7285170917376z^{12}+1060577224704z^{10}w^{2}+26766339840z^{8}w^{4}+9265017600z^{6}w^{6}+15390000z^{4}w^{8}-24375000z^{2}w^{10}+9765625w^{12}}{w^{4}(839808xz^{7}-20160xz^{5}w^{2}-2400xz^{3}w^{4}+4217184y^{2}z^{6}+437400y^{2}z^{4}w^{2}+6600y^{2}z^{2}w^{4}+125y^{2}w^{6}+421776yz^{5}w^{2}+29280yz^{3}w^{4}+450yzw^{6}+842184z^{8}+50616z^{6}w^{2}+11280z^{4}w^{4}+325z^{2}w^{6})}$ |
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.