$\GL_2(\Z/24\Z)$-generators: |
$\begin{bmatrix}1&22\\16&17\end{bmatrix}$, $\begin{bmatrix}5&20\\16&23\end{bmatrix}$, $\begin{bmatrix}5&21\\0&5\end{bmatrix}$, $\begin{bmatrix}11&3\\0&11\end{bmatrix}$, $\begin{bmatrix}17&23\\16&5\end{bmatrix}$, $\begin{bmatrix}23&18\\0&11\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
24.144.4-24.gk.2.1, 24.144.4-24.gk.2.2, 24.144.4-24.gk.2.3, 24.144.4-24.gk.2.4, 24.144.4-24.gk.2.5, 24.144.4-24.gk.2.6, 24.144.4-24.gk.2.7, 24.144.4-24.gk.2.8, 24.144.4-24.gk.2.9, 24.144.4-24.gk.2.10, 24.144.4-24.gk.2.11, 24.144.4-24.gk.2.12, 24.144.4-24.gk.2.13, 24.144.4-24.gk.2.14, 24.144.4-24.gk.2.15, 24.144.4-24.gk.2.16, 24.144.4-24.gk.2.17, 24.144.4-24.gk.2.18, 24.144.4-24.gk.2.19, 24.144.4-24.gk.2.20, 24.144.4-24.gk.2.21, 24.144.4-24.gk.2.22, 24.144.4-24.gk.2.23, 24.144.4-24.gk.2.24, 24.144.4-24.gk.2.25, 24.144.4-24.gk.2.26, 24.144.4-24.gk.2.27, 24.144.4-24.gk.2.28, 24.144.4-24.gk.2.29, 24.144.4-24.gk.2.30, 24.144.4-24.gk.2.31, 24.144.4-24.gk.2.32, 48.144.4-24.gk.2.1, 48.144.4-24.gk.2.2, 48.144.4-24.gk.2.3, 48.144.4-24.gk.2.4, 48.144.4-24.gk.2.5, 48.144.4-24.gk.2.6, 48.144.4-24.gk.2.7, 48.144.4-24.gk.2.8, 48.144.4-24.gk.2.9, 48.144.4-24.gk.2.10, 48.144.4-24.gk.2.11, 48.144.4-24.gk.2.12, 48.144.4-24.gk.2.13, 48.144.4-24.gk.2.14, 48.144.4-24.gk.2.15, 48.144.4-24.gk.2.16, 48.144.4-24.gk.2.17, 48.144.4-24.gk.2.18, 48.144.4-24.gk.2.19, 48.144.4-24.gk.2.20, 48.144.4-24.gk.2.21, 48.144.4-24.gk.2.22, 48.144.4-24.gk.2.23, 48.144.4-24.gk.2.24, 48.144.4-24.gk.2.25, 48.144.4-24.gk.2.26, 48.144.4-24.gk.2.27, 48.144.4-24.gk.2.28, 48.144.4-24.gk.2.29, 48.144.4-24.gk.2.30, 48.144.4-24.gk.2.31, 48.144.4-24.gk.2.32, 120.144.4-24.gk.2.1, 120.144.4-24.gk.2.2, 120.144.4-24.gk.2.3, 120.144.4-24.gk.2.4, 120.144.4-24.gk.2.5, 120.144.4-24.gk.2.6, 120.144.4-24.gk.2.7, 120.144.4-24.gk.2.8, 120.144.4-24.gk.2.9, 120.144.4-24.gk.2.10, 120.144.4-24.gk.2.11, 120.144.4-24.gk.2.12, 120.144.4-24.gk.2.13, 120.144.4-24.gk.2.14, 120.144.4-24.gk.2.15, 120.144.4-24.gk.2.16, 120.144.4-24.gk.2.17, 120.144.4-24.gk.2.18, 120.144.4-24.gk.2.19, 120.144.4-24.gk.2.20, 120.144.4-24.gk.2.21, 120.144.4-24.gk.2.22, 120.144.4-24.gk.2.23, 120.144.4-24.gk.2.24, 120.144.4-24.gk.2.25, 120.144.4-24.gk.2.26, 120.144.4-24.gk.2.27, 120.144.4-24.gk.2.28, 120.144.4-24.gk.2.29, 120.144.4-24.gk.2.30, 120.144.4-24.gk.2.31, 120.144.4-24.gk.2.32, 168.144.4-24.gk.2.1, 168.144.4-24.gk.2.2, 168.144.4-24.gk.2.3, 168.144.4-24.gk.2.4, 168.144.4-24.gk.2.5, 168.144.4-24.gk.2.6, 168.144.4-24.gk.2.7, 168.144.4-24.gk.2.8, 168.144.4-24.gk.2.9, 168.144.4-24.gk.2.10, 168.144.4-24.gk.2.11, 168.144.4-24.gk.2.12, 168.144.4-24.gk.2.13, 168.144.4-24.gk.2.14, 168.144.4-24.gk.2.15, 168.144.4-24.gk.2.16, 168.144.4-24.gk.2.17, 168.144.4-24.gk.2.18, 168.144.4-24.gk.2.19, 168.144.4-24.gk.2.20, 168.144.4-24.gk.2.21, 168.144.4-24.gk.2.22, 168.144.4-24.gk.2.23, 168.144.4-24.gk.2.24, 168.144.4-24.gk.2.25, 168.144.4-24.gk.2.26, 168.144.4-24.gk.2.27, 168.144.4-24.gk.2.28, 168.144.4-24.gk.2.29, 168.144.4-24.gk.2.30, 168.144.4-24.gk.2.31, 168.144.4-24.gk.2.32, 240.144.4-24.gk.2.1, 240.144.4-24.gk.2.2, 240.144.4-24.gk.2.3, 240.144.4-24.gk.2.4, 240.144.4-24.gk.2.5, 240.144.4-24.gk.2.6, 240.144.4-24.gk.2.7, 240.144.4-24.gk.2.8, 240.144.4-24.gk.2.9, 240.144.4-24.gk.2.10, 240.144.4-24.gk.2.11, 240.144.4-24.gk.2.12, 240.144.4-24.gk.2.13, 240.144.4-24.gk.2.14, 240.144.4-24.gk.2.15, 240.144.4-24.gk.2.16, 240.144.4-24.gk.2.17, 240.144.4-24.gk.2.18, 240.144.4-24.gk.2.19, 240.144.4-24.gk.2.20, 240.144.4-24.gk.2.21, 240.144.4-24.gk.2.22, 240.144.4-24.gk.2.23, 240.144.4-24.gk.2.24, 240.144.4-24.gk.2.25, 240.144.4-24.gk.2.26, 240.144.4-24.gk.2.27, 240.144.4-24.gk.2.28, 240.144.4-24.gk.2.29, 240.144.4-24.gk.2.30, 240.144.4-24.gk.2.31, 240.144.4-24.gk.2.32, 264.144.4-24.gk.2.1, 264.144.4-24.gk.2.2, 264.144.4-24.gk.2.3, 264.144.4-24.gk.2.4, 264.144.4-24.gk.2.5, 264.144.4-24.gk.2.6, 264.144.4-24.gk.2.7, 264.144.4-24.gk.2.8, 264.144.4-24.gk.2.9, 264.144.4-24.gk.2.10, 264.144.4-24.gk.2.11, 264.144.4-24.gk.2.12, 264.144.4-24.gk.2.13, 264.144.4-24.gk.2.14, 264.144.4-24.gk.2.15, 264.144.4-24.gk.2.16, 264.144.4-24.gk.2.17, 264.144.4-24.gk.2.18, 264.144.4-24.gk.2.19, 264.144.4-24.gk.2.20, 264.144.4-24.gk.2.21, 264.144.4-24.gk.2.22, 264.144.4-24.gk.2.23, 264.144.4-24.gk.2.24, 264.144.4-24.gk.2.25, 264.144.4-24.gk.2.26, 264.144.4-24.gk.2.27, 264.144.4-24.gk.2.28, 264.144.4-24.gk.2.29, 264.144.4-24.gk.2.30, 264.144.4-24.gk.2.31, 264.144.4-24.gk.2.32, 312.144.4-24.gk.2.1, 312.144.4-24.gk.2.2, 312.144.4-24.gk.2.3, 312.144.4-24.gk.2.4, 312.144.4-24.gk.2.5, 312.144.4-24.gk.2.6, 312.144.4-24.gk.2.7, 312.144.4-24.gk.2.8, 312.144.4-24.gk.2.9, 312.144.4-24.gk.2.10, 312.144.4-24.gk.2.11, 312.144.4-24.gk.2.12, 312.144.4-24.gk.2.13, 312.144.4-24.gk.2.14, 312.144.4-24.gk.2.15, 312.144.4-24.gk.2.16, 312.144.4-24.gk.2.17, 312.144.4-24.gk.2.18, 312.144.4-24.gk.2.19, 312.144.4-24.gk.2.20, 312.144.4-24.gk.2.21, 312.144.4-24.gk.2.22, 312.144.4-24.gk.2.23, 312.144.4-24.gk.2.24, 312.144.4-24.gk.2.25, 312.144.4-24.gk.2.26, 312.144.4-24.gk.2.27, 312.144.4-24.gk.2.28, 312.144.4-24.gk.2.29, 312.144.4-24.gk.2.30, 312.144.4-24.gk.2.31, 312.144.4-24.gk.2.32 |
Cyclic 24-isogeny field degree: |
$4$ |
Cyclic 24-torsion field degree: |
$32$ |
Full 24-torsion field degree: |
$1024$ |
Canonical model in $\mathbb{P}^{ 3 }$
$ 0 $ | $=$ | $ x z + x w - y z + 2 y w $ |
| $=$ | $7 x^{3} + 4 x^{2} y + 4 x y^{2} - z^{3} + z^{2} w - z w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 15 x^{3} y^{3} - 54 x^{3} y^{2} z + 72 x^{3} y z^{2} - 48 x^{3} z^{3} + y^{3} z^{3} - y^{2} z^{4} + y z^{5} $ |
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
Maps between models of this curve
Birational map from canonical model to plane model:
$\displaystyle X$ |
$=$ |
$\displaystyle x-y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle 3z$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle 3w$ |
Maps to other modular curves
$j$-invariant map
of degree 72 from the canonical model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle -\frac{5}{2^3\cdot3^4\cdot7^3}\cdot\frac{106853165617500000000x^{2}y^{10}+823914531922482000000x^{2}y^{7}w^{3}+8298139549253594405400x^{2}y^{4}w^{6}+87563872227477701633949x^{2}yw^{9}+60924151350000000000xy^{11}+463875347639376000000xy^{8}w^{3}+4496664448634604487200xy^{5}w^{6}+49706282991727390191282xy^{2}w^{9}+61014653784000000000y^{12}+457745657031000000000y^{9}w^{3}+4555798903129311000000y^{6}w^{6}+49066156219939235269200y^{3}w^{9}-4138005984250000z^{12}+12054312907600000z^{11}w+7389030263139920000z^{10}w^{2}+223224880829713168000z^{9}w^{3}-159401804951625529600z^{8}w^{4}+966764337855332512000z^{7}w^{5}+1494157046940756101000z^{6}w^{6}+3247617381598545640240z^{5}w^{7}+6871209993418089965384z^{4}w^{8}+1433375916860923917375z^{3}w^{9}+3070849488073328769840z^{2}w^{10}+6248016438615647373384zw^{11}+14896155708984375w^{12}}{147261796875x^{2}y^{7}w^{3}+791837187750x^{2}y^{4}w^{6}+1840619667033x^{2}yw^{9}-136193906250xy^{8}w^{3}+118910592000xy^{5}w^{6}+658550237769xy^{2}w^{9}+204290859375y^{6}w^{6}+562822080750y^{3}w^{9}+128678593750z^{12}-411771500000z^{11}w+1050017325000z^{10}w^{2}-1729443579375z^{9}w^{3}+2242181461500z^{8}w^{4}-2274581955750z^{7}w^{5}+1780902877050z^{6}w^{6}-1051596151995z^{5}w^{7}+611528589528z^{4}w^{8}-159431981075z^{3}w^{9}+92907917755z^{2}w^{10}+137670839028zw^{11}}$ |
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.