| $\GL_2(\Z/24\Z)$-generators: |
$\begin{bmatrix}7&4\\6&1\end{bmatrix}$, $\begin{bmatrix}17&11\\10&11\end{bmatrix}$, $\begin{bmatrix}19&5\\6&1\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: |
$C_2^2:C_4\times \GL(2,3)$ |
| Contains $-I$: |
yes |
| Quadratic refinements: |
48.192.1-24.cw.1.1, 48.192.1-24.cw.1.2, 48.192.1-24.cw.1.3, 48.192.1-24.cw.1.4, 48.192.1-24.cw.1.5, 48.192.1-24.cw.1.6, 48.192.1-24.cw.1.7, 48.192.1-24.cw.1.8, 240.192.1-24.cw.1.1, 240.192.1-24.cw.1.2, 240.192.1-24.cw.1.3, 240.192.1-24.cw.1.4, 240.192.1-24.cw.1.5, 240.192.1-24.cw.1.6, 240.192.1-24.cw.1.7, 240.192.1-24.cw.1.8 |
| Cyclic 24-isogeny field degree: |
$8$ |
| Cyclic 24-torsion field degree: |
$64$ |
| Full 24-torsion field degree: |
$768$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 3 x^{2} - 6 y^{2} + 2 w^{2} $ |
| $=$ | $2 x^{2} + 6 x y + 2 x z + 3 y^{2} + 2 z^{2}$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 4 x^{4} + 20 x^{2} y^{2} - 6 x^{2} z^{2} - 48 x y^{3} + 36 x y z^{2} - 23 y^{4} + 12 y^{2} z^{2} + 9 z^{4} $ |
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This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{2}{3}w$ |
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{2^4}{3}\cdot\frac{496369241078360960862403679698793274240xz^{23}+180942798737125513528295050734433587072xz^{21}w^{2}-101705021303838261896822986805819656320xz^{19}w^{4}-43460896894198787305327348864201423680xz^{17}w^{6}-1199490207080099543685405669266720064xz^{15}w^{8}-125611185041065876170869655985500480xz^{13}w^{10}-155125110302548511846784775443552384xz^{11}w^{12}+10205786532198875653681387598930976xz^{9}w^{14}-1335066240808108510046023093290504xz^{7}w^{16}+37583879468662100638682064013176xz^{5}w^{18}-1809616678171776541440770299992xz^{3}w^{20}+9806621383174931961095547420xzw^{22}+1452085221765936521731826663132748212160y^{2}z^{22}+1209238584791893790523858922504881692160y^{2}z^{20}w^{2}+278830791784609920596568666633183191808y^{2}z^{18}w^{4}+11178133103179349343450647114144302368y^{2}z^{16}w^{6}+2985896732007734568061819551131357664y^{2}z^{14}w^{8}+879759539044863550879282567856439552y^{2}z^{12}w^{10}-50669425572713733301601042160294624y^{2}z^{10}w^{12}+9576894067352039605939190240458896y^{2}z^{8}w^{14}-284497320276559059936276663818820y^{2}z^{6}w^{16}+18618024195016853140935499573440y^{2}z^{4}w^{18}-142026279002281733256727279848y^{2}z^{2}w^{20}+1405989887387931385245951330y^{2}w^{22}-436744950927534068870689110019163220480yz^{23}-598614377033738830831593979320878784768yz^{21}w^{2}-288113578054977467355103610092633618944yz^{19}w^{4}-55779605324509673460764739480678185472yz^{17}w^{6}-4435640476423138653224916396160855296yz^{15}w^{8}-807191521162180944331298876534565504yz^{13}w^{10}-136115706656991881636949228996049920yz^{11}w^{12}+896971277618569933372676612700288yz^{9}w^{14}-1149343013102029573136365950666912yz^{7}w^{16}+7766199562646018798169648682608yz^{5}w^{18}-1751340075861242735695681575264yz^{3}w^{20}-2176181567771455152527140800yzw^{22}+187543924834700068711799512095371241792z^{24}-204405543507613484142758454924501461376z^{22}w^{2}-247595919858868420356622984096356117312z^{20}w^{4}-52953913784977678124976144627960053184z^{18}w^{6}+1621533086440653924096190269126299568z^{16}w^{8}-310247997447930513086173262314212288z^{14}w^{10}-185714863234039027973360353679660448z^{12}w^{12}+24534000457112861940322597359044448z^{10}w^{14}-2546445961269491009438521806722244z^{8}w^{16}+144504518565668087344917669137640z^{6}w^{18}-4848716289720706037905161674292z^{4}w^{20}+107873991716553710888908270356z^{2}w^{22}-199901008749385868076284519w^{24}}{1532003830488768397723468147218497760xz^{23}-1899808562202692316720877314565438896xz^{21}w^{2}+936360352837269808426265539735033272xz^{19}w^{4}-247033688559570867998848492504616004xz^{17}w^{6}+37840117410486933075198160941476916xz^{15}w^{8}-3222336485305391743900309659334740xz^{13}w^{10}+105986448534228453705297882574620xz^{11}w^{12}+3958789249412465137325996173488xz^{9}w^{14}-278493753626882052016704956640xz^{7}w^{16}-6481228078102429422273323472xz^{5}w^{18}+130923533718728515853997312xz^{3}w^{20}+6269523501135545330111232xzw^{22}+4481744511623260869542674886212185840y^{2}z^{22}-3459249786111237875638992649427599752y^{2}z^{20}w^{2}+1150887832669014725347261226830135836y^{2}z^{18}w^{4}-210162807457013685786350565482251038y^{2}z^{16}w^{6}+21679161315086280819261306125280906y^{2}z^{14}w^{8}-1057888624064330884889067844507812y^{2}z^{12}w^{10}-5185370330596434961899385911000y^{2}z^{10}w^{12}+2332214471275842106305263773062y^{2}z^{8}w^{14}-5541977758098330785060071614y^{2}z^{6}w^{16}-2427466682472705270098379516y^{2}z^{4}w^{18}-54382297928110268574288168y^{2}z^{2}w^{20}-399536883134954869472256y^{2}w^{22}-1347978243603500212563855277836923520yz^{23}+315408826534300048273594924643897664yz^{21}w^{2}+186844486785946826643770905674884448yz^{19}w^{4}-102818273138510668030023658240294512yz^{17}w^{6}+21046333133803184430800097366111888yz^{15}w^{8}-2107295674878372654316377010660920yz^{13}w^{10}+79261316194251296906235922528656yz^{11}w^{12}+2561355025317923600352537305424yz^{9}w^{14}-196924723598866482418219821936yz^{7}w^{16}-4795372641374951717470764456yz^{5}w^{18}+68819884702229542154573520yz^{3}w^{20}+5411638484157782619630432yzw^{22}+578839274181173051579628123751145808z^{24}-1559694610673017316046018061994650032z^{22}w^{2}+1059107850844835710911696960716044368z^{20}w^{4}-349694342426024197641586448429516964z^{18}w^{6}+65957078473365572161770028452588783z^{16}w^{8}-7218135539750390279071757820659100z^{14}w^{10}+392022591563907191663303608565514z^{12}w^{12}-623780391098710993124938600980z^{10}w^{14}-850334081579350772386540888593z^{8}w^{16}+6529406952843831268780252068z^{6}w^{18}+1034671295237305103843023444z^{4}w^{20}+18458532478261575485358000z^{2}w^{22}+364340868170376863856688w^{24}}$ |
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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.
