$\GL_2(\Z/24\Z)$-generators: |
$\begin{bmatrix}3&10\\22&21\end{bmatrix}$, $\begin{bmatrix}7&16\\18&1\end{bmatrix}$, $\begin{bmatrix}11&4\\16&19\end{bmatrix}$, $\begin{bmatrix}11&6\\12&19\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
24.96.1-24.j.1.1, 24.96.1-24.j.1.2, 24.96.1-24.j.1.3, 24.96.1-24.j.1.4, 120.96.1-24.j.1.1, 120.96.1-24.j.1.2, 120.96.1-24.j.1.3, 120.96.1-24.j.1.4, 168.96.1-24.j.1.1, 168.96.1-24.j.1.2, 168.96.1-24.j.1.3, 168.96.1-24.j.1.4, 264.96.1-24.j.1.1, 264.96.1-24.j.1.2, 264.96.1-24.j.1.3, 264.96.1-24.j.1.4, 312.96.1-24.j.1.1, 312.96.1-24.j.1.2, 312.96.1-24.j.1.3, 312.96.1-24.j.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 $ | $=$ | $ 3 x^{2} - 3 y^{2} - w^{2} $ |
| $=$ | $2 x^{2} - x y + 2 x z + 2 y^{2} - 2 y z + 2 z^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} - 4 x^{3} y + 16 x^{2} y^{2} + 3 x^{2} z^{2} - 24 x y^{3} - 6 x y z^{2} + 60 y^{4} + 45 y^{2} z^{2} + 9 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:
$\displaystyle X$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{2}y$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{3}w$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle 2^2\cdot3^2\,\frac{72235836288xz^{11}+197034467328xz^{9}w^{2}+231428361600xz^{7}w^{4}+126866808000xz^{5}w^{6}+22172025000xz^{3}w^{8}+792125000xzw^{10}+186224282496y^{2}z^{10}+427311126000y^{2}z^{8}w^{2}+376566732000y^{2}z^{6}w^{4}+161842500000y^{2}z^{4}w^{6}+21090375000y^{2}z^{2}w^{8}+462890625y^{2}w^{10}-72235836288yz^{11}-134075388672yz^{9}w^{2}-103055025600yz^{7}w^{4}-39964392000yz^{5}w^{6}-2145525000yz^{3}w^{8}+204625000yzw^{10}+73524521664z^{12}+189348964416z^{10}w^{2}+230320355040z^{8}w^{4}+160502112000z^{6}w^{6}+49345387500z^{4}w^{8}+5225687500z^{2}w^{10}+92734375w^{12}}{1504913256xz^{11}-33988464xz^{9}w^{2}-659632500xz^{7}w^{4}+130266000xz^{5}w^{6}+11625000xz^{3}w^{8}+187500xzw^{10}+3879672552y^{2}z^{10}-1780463025y^{2}z^{8}w^{2}-1258989750y^{2}z^{6}w^{4}+532828125y^{2}z^{4}w^{6}-10968750y^{2}z^{2}w^{8}-3515625y^{2}w^{10}-1504913256yz^{11}+1345635936yz^{9}w^{2}+124743600yz^{7}w^{4}-232479000yz^{5}w^{6}+26400000yz^{3}w^{8}-750000yzw^{10}+1531760868z^{12}-13019508z^{10}w^{2}-590604795z^{8}w^{4}-66210750z^{6}w^{6}+72684375z^{4}w^{8}+1031250z^{2}w^{10}-859375w^{12}}$ |
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.