$\GL_2(\Z/24\Z)$-generators: |
$\begin{bmatrix}3&16\\10&17\end{bmatrix}$, $\begin{bmatrix}7&21\\4&5\end{bmatrix}$, $\begin{bmatrix}11&9\\2&13\end{bmatrix}$, $\begin{bmatrix}13&3\\18&7\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
24.96.1-24.co.1.1, 24.96.1-24.co.1.2, 24.96.1-24.co.1.3, 24.96.1-24.co.1.4, 120.96.1-24.co.1.1, 120.96.1-24.co.1.2, 120.96.1-24.co.1.3, 120.96.1-24.co.1.4, 168.96.1-24.co.1.1, 168.96.1-24.co.1.2, 168.96.1-24.co.1.3, 168.96.1-24.co.1.4, 264.96.1-24.co.1.1, 264.96.1-24.co.1.2, 264.96.1-24.co.1.3, 264.96.1-24.co.1.4, 312.96.1-24.co.1.1, 312.96.1-24.co.1.2, 312.96.1-24.co.1.3, 312.96.1-24.co.1.4 |
Cyclic 24-isogeny field degree: |
$16$ |
Cyclic 24-torsion field degree: |
$128$ |
Full 24-torsion field degree: |
$1536$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ x^{2} - y^{2} + y z - z^{2} $ |
| $=$ | $4 x^{2} + 3 y^{2} + 2 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 9 x^{4} - 30 x^{2} y^{2} - 6 x^{2} z^{2} + 49 y^{4} + 28 y^{2} z^{2} + 4 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:
$\displaystyle X$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle x$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{2^4}{3^4}\cdot\frac{30195296640yz^{11}-109698660288yz^{9}w^{2}-183735107136yz^{7}w^{4}+617914368288yz^{5}w^{6}-325410276744yz^{3}w^{8}+34775363700yzw^{10}-9922751424z^{12}+148246687296z^{10}w^{2}-246392680464z^{8}w^{4}-173682720288z^{6}w^{6}+327577832316z^{4}w^{8}-80670286620z^{2}w^{10}+2796264625w^{12}}{3451680yz^{11}+3681888yz^{9}w^{2}+1155616yz^{7}w^{4}-19208yz^{5}w^{6}-638666yz^{3}w^{8}-336140yzw^{10}-1134288z^{12}-263232z^{10}w^{2}+2570792z^{8}w^{4}+2472344z^{6}w^{6}+948395z^{4}w^{8}+292922z^{2}w^{10}+67228w^{12}}$ |
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.