$\GL_2(\Z/24\Z)$-generators: |
$\begin{bmatrix}1&0\\12&19\end{bmatrix}$, $\begin{bmatrix}9&20\\16&15\end{bmatrix}$, $\begin{bmatrix}15&20\\16&1\end{bmatrix}$, $\begin{bmatrix}23&22\\12&7\end{bmatrix}$ |
$\GL_2(\Z/24\Z)$-subgroup: |
Group 768.1089047 |
Contains $-I$: |
yes |
Quadratic refinements: |
24.192.1-24.be.2.1, 24.192.1-24.be.2.2, 24.192.1-24.be.2.3, 24.192.1-24.be.2.4, 24.192.1-24.be.2.5, 24.192.1-24.be.2.6, 24.192.1-24.be.2.7, 24.192.1-24.be.2.8, 120.192.1-24.be.2.1, 120.192.1-24.be.2.2, 120.192.1-24.be.2.3, 120.192.1-24.be.2.4, 120.192.1-24.be.2.5, 120.192.1-24.be.2.6, 120.192.1-24.be.2.7, 120.192.1-24.be.2.8, 168.192.1-24.be.2.1, 168.192.1-24.be.2.2, 168.192.1-24.be.2.3, 168.192.1-24.be.2.4, 168.192.1-24.be.2.5, 168.192.1-24.be.2.6, 168.192.1-24.be.2.7, 168.192.1-24.be.2.8, 264.192.1-24.be.2.1, 264.192.1-24.be.2.2, 264.192.1-24.be.2.3, 264.192.1-24.be.2.4, 264.192.1-24.be.2.5, 264.192.1-24.be.2.6, 264.192.1-24.be.2.7, 264.192.1-24.be.2.8, 312.192.1-24.be.2.1, 312.192.1-24.be.2.2, 312.192.1-24.be.2.3, 312.192.1-24.be.2.4, 312.192.1-24.be.2.5, 312.192.1-24.be.2.6, 312.192.1-24.be.2.7, 312.192.1-24.be.2.8 |
Cyclic 24-isogeny field degree: |
$8$ |
Cyclic 24-torsion field degree: |
$32$ |
Full 24-torsion field degree: |
$768$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 2 y^{2} - y z - 2 y w + z^{2} + 2 z w + 2 w^{2} $ |
| $=$ | $6 x^{2} - 3 y^{2} + z^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 11 x^{4} - 4 x^{3} z - 27 x^{2} y^{2} + 6 x^{2} z^{2} - 4 x z^{3} + 18 y^{4} + 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:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle x$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle -\frac{1}{3^2}\cdot\frac{435708211725yz^{23}+4706224369614yz^{22}w+15318556111404yz^{21}w^{2}-11218283846424yz^{20}w^{3}-261994825232496yz^{19}w^{4}-1080485405276832yz^{18}w^{5}-2621161826245440yz^{17}w^{6}-4159860821106048yz^{16}w^{7}-3752105822138880yz^{15}w^{8}+849970842080256yz^{14}w^{9}+10182129107798016yz^{13}w^{10}+21667420242210816yz^{12}w^{11}+30577210682204160yz^{11}w^{12}+33036307336544256yz^{10}w^{13}+28578611524829184yz^{9}w^{14}+20095813306220544yz^{8}w^{15}+11488813369786368yz^{7}w^{16}+5279724606062592yz^{6}w^{17}+1902287181053952yz^{5}w^{18}+512304271589376yz^{4}w^{19}+93619054706688yz^{3}w^{20}+8916100448256yz^{2}w^{21}-109810107319z^{24}+580930713018z^{23}w+15941619736794z^{22}w^{2}+107059724883360z^{21}w^{3}+399963634322112z^{20}w^{4}+943155108364320z^{19}w^{5}+1293296077821024z^{18}w^{6}+106546488860160z^{17}w^{7}-4522274123605632z^{16}w^{8}-13521510294377472z^{15}w^{9}-24990911486567424z^{14}w^{10}-34003820796051456z^{13}w^{11}-35169541204008960z^{12}w^{12}-26462094455881728z^{11}w^{13}-11131554370535424z^{10}w^{14}+4183462193135616z^{9}w^{15}+13705108658749440z^{8}w^{16}+15695680326598656z^{7}w^{17}+12382420249608192z^{6}w^{18}+7429712202694656z^{5}w^{19}+3454803171606528z^{4}w^{20}+1225963811635200z^{3}w^{21}+316521565913088z^{2}w^{22}+53496602689536zw^{23}+4458050224128w^{24}}{z^{4}(2608275yz^{19}+49394698yz^{18}w+430192236yz^{17}w^{2}+2278340552yz^{16}w^{3}+7991172288yz^{15}w^{4}+18153668736yz^{14}w^{5}+19106000640yz^{13}w^{6}-37842283008yz^{12}w^{7}-242431854336yz^{11}w^{8}-678605050368yz^{10}w^{9}-1324089068544yz^{9}w^{10}-1980867446784yz^{8}w^{11}-2348398780416yz^{7}w^{12}-2229443149824yz^{6}w^{13}-1690329710592yz^{5}w^{14}-1008975937536yz^{4}w^{15}-460510396416yz^{3}w^{16}-152213520384yz^{2}w^{17}-32678608896yzw^{18}-3439853568yw^{19}+2917847z^{20}+59517886z^{19}w+615426430z^{18}w^{2}+4243730688z^{17}w^{3}+21663741888z^{16}w^{4}+86320559232z^{15}w^{5}+276886961280z^{14}w^{6}+729075294720z^{13}w^{7}+1596089884032z^{12}w^{8}+2928910026240z^{11}w^{9}+4526489101824z^{10}w^{10}+5901830221824z^{9}w^{11}+6485232273408z^{8}w^{12}+5981630644224z^{7}w^{13}+4595727974400z^{6}w^{14}+2904956338176z^{5}w^{15}+1481501933568z^{4}w^{16}+590794850304z^{3}w^{17}+174572568576z^{2}w^{18}+34398535680zw^{19}+3439853568w^{20})}$ |
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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.