$\GL_2(\Z/12\Z)$-generators: |
$\begin{bmatrix}1&5\\6&5\end{bmatrix}$, $\begin{bmatrix}7&10\\6&11\end{bmatrix}$, $\begin{bmatrix}11&3\\6&7\end{bmatrix}$ |
$\GL_2(\Z/12\Z)$-subgroup: |
$C_6:D_4$ |
Contains $-I$: |
yes |
Quadratic refinements: |
12.192.3-12.l.1.1, 12.192.3-12.l.1.2, 12.192.3-12.l.1.3, 12.192.3-12.l.1.4, 24.192.3-12.l.1.1, 24.192.3-12.l.1.2, 24.192.3-12.l.1.3, 24.192.3-12.l.1.4, 60.192.3-12.l.1.1, 60.192.3-12.l.1.2, 60.192.3-12.l.1.3, 60.192.3-12.l.1.4, 84.192.3-12.l.1.1, 84.192.3-12.l.1.2, 84.192.3-12.l.1.3, 84.192.3-12.l.1.4, 120.192.3-12.l.1.1, 120.192.3-12.l.1.2, 120.192.3-12.l.1.3, 120.192.3-12.l.1.4, 132.192.3-12.l.1.1, 132.192.3-12.l.1.2, 132.192.3-12.l.1.3, 132.192.3-12.l.1.4, 156.192.3-12.l.1.1, 156.192.3-12.l.1.2, 156.192.3-12.l.1.3, 156.192.3-12.l.1.4, 168.192.3-12.l.1.1, 168.192.3-12.l.1.2, 168.192.3-12.l.1.3, 168.192.3-12.l.1.4, 204.192.3-12.l.1.1, 204.192.3-12.l.1.2, 204.192.3-12.l.1.3, 204.192.3-12.l.1.4, 228.192.3-12.l.1.1, 228.192.3-12.l.1.2, 228.192.3-12.l.1.3, 228.192.3-12.l.1.4, 264.192.3-12.l.1.1, 264.192.3-12.l.1.2, 264.192.3-12.l.1.3, 264.192.3-12.l.1.4, 276.192.3-12.l.1.1, 276.192.3-12.l.1.2, 276.192.3-12.l.1.3, 276.192.3-12.l.1.4, 312.192.3-12.l.1.1, 312.192.3-12.l.1.2, 312.192.3-12.l.1.3, 312.192.3-12.l.1.4 |
Cyclic 12-isogeny field degree: |
$2$ |
Cyclic 12-torsion field degree: |
$8$ |
Full 12-torsion field degree: |
$48$ |
Embedded model Embedded model in $\mathbb{P}^{5}$
$ 0 $ | $=$ | $ x z - 2 x w - x u - w t + w u $ |
| $=$ | $x y - x z - x w - x t - x u - 2 y w$ |
| $=$ | $x z - x w - x u + y^{2} - 2 y w - y t + y u - z^{2} + z t + z u + t^{2} + u^{2}$ |
| $=$ | $x y - 2 x z + x w - x t - 2 y z + 2 y w + y t + y u - z t + z u - t^{2} + u^{2}$ |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 64 x^{8} - 192 x^{7} y - 448 x^{7} z + 240 x^{6} y^{2} + 1344 x^{6} y z + 1776 x^{6} z^{2} - 72 x^{5} y^{3} + \cdots + 9 z^{8} $ |
Weierstrass model Weierstrass model
$ y^{2} + x^{4} y $ | $=$ | $ -x^{8} - 30x^{4} - 108 $ |
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:
$\displaystyle X$ |
$=$ |
$\displaystyle x$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle 4u$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle 2w$ |
Birational map from embedded model to Weierstrass model:
$\displaystyle X$ |
$=$ |
$\displaystyle x^{11}-29x^{10}w-21x^{10}u+108x^{9}w^{2}+126x^{9}wu+30x^{9}u^{2}-277x^{8}w^{3}-387x^{8}w^{2}u-168x^{8}wu^{2}-18x^{8}u^{3}+494x^{7}w^{4}+714x^{7}w^{3}u+450x^{7}w^{2}u^{2}+126x^{7}wu^{3}-1281x^{6}w^{5}-1737x^{6}w^{4}u-948x^{6}w^{3}u^{2}-306x^{6}w^{2}u^{3}+2551x^{5}w^{6}+4068x^{5}w^{5}u+2058x^{5}w^{4}u^{2}+396x^{5}w^{3}u^{3}-3338x^{4}w^{7}-6960x^{4}w^{6}u-4896x^{4}w^{5}u^{2}-1044x^{4}w^{4}u^{3}+2508x^{3}w^{8}+7884x^{3}w^{7}u+8940x^{3}w^{6}u^{2}+3168x^{3}w^{5}u^{3}-816x^{2}w^{9}-4860x^{2}w^{8}u-8952x^{2}w^{7}u^{2}-4608x^{2}w^{6}u^{3}-324xw^{10}+648xw^{9}u+3888xw^{8}u^{2}+2880xw^{7}u^{3}+144w^{11}+432w^{10}u-288w^{9}u^{2}-576w^{8}u^{3}$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle -6x^{44}-24x^{43}w-12x^{42}w^{2}-24x^{41}w^{3}+114x^{40}w^{4}-312x^{39}w^{5}-2028x^{38}w^{6}+4368x^{37}w^{7}-3252x^{36}w^{8}+24312x^{35}w^{9}-9516x^{34}w^{10}-91176x^{33}w^{11}+518946x^{32}w^{12}-862152x^{31}w^{13}+2686956x^{30}w^{14}-4049520x^{29}w^{15}+3961212x^{28}w^{16}+10046568x^{27}w^{17}-38449524x^{26}w^{18}+132193128x^{25}w^{19}-328454094x^{24}w^{20}+720798456x^{23}w^{21}-1315322868x^{22}w^{22}+2316487584x^{21}w^{23}-3679920318x^{20}w^{24}+5327606352x^{19}w^{25}-6733398144x^{18}w^{26}+7777148928x^{17}w^{27}-7506640080x^{16}w^{28}+4802611392x^{15}w^{29}+1748798208x^{14}w^{30}-11749601664x^{13}w^{31}+25079883552x^{12}w^{32}-41224868352x^{11}w^{33}+58961613312x^{10}w^{34}-73874363904x^{9}w^{35}+82259753472x^{8}w^{36}-82861056000x^{7}w^{37}+76753972224x^{6}w^{38}-63977693184x^{5}w^{39}+45766176768x^{4}w^{40}-26282631168x^{3}w^{41}+11287019520x^{2}w^{42}-3224862720xw^{43}+483729408w^{44}$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle x^{11}+x^{10}w+5x^{8}w^{3}-10x^{7}w^{4}+57x^{6}w^{5}-59x^{5}w^{6}+88x^{4}w^{7}-114x^{3}w^{8}+180x^{2}w^{9}-144xw^{10}+72w^{11}$ |
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 2^2\,\frac{11022073515195888526794776915992xt^{11}+578307700969515691104742488888872xt^{10}u+8428793082565602329982965996548104xt^{9}u^{2}+63828892009394516297292532088990904xt^{8}u^{3}+305196727392672835265844076060411632xt^{7}u^{4}+1002819520814991388549742482081771152xt^{6}u^{5}+2359013354194853918424663740165962128xt^{5}u^{6}+4034888384594325402679094426840546544xt^{4}u^{7}+4990063013418465431764048027610660856xt^{3}u^{8}+4315320169564768544258587069263073352xt^{2}u^{9}+2390306550620468682649818925890175336xtu^{10}+654616250550227562874205060448282456xu^{11}+624578211990342735273198928303492ywt^{10}+45040790297008181750324483234272436ywt^{9}u+623849425367689800689545427201103924ywt^{8}u^{2}+4154975785134020055018438927509723088ywt^{7}u^{3}+16580686644976234705024074175460346600ywt^{6}u^{4}+43636993904529592847993779393014263736ywt^{5}u^{5}+78837458952639875861517594817198470312ywt^{4}u^{6}+98275655703238580827405867845280806096ywt^{3}u^{7}+81965888602175765221966357607708110548ywt^{2}u^{8}+41763425369277454260541943712690957620ywtu^{9}+9799048882287095856409249423465296484ywu^{10}+144165722378322462526323464326923yt^{11}+12449321485715680816647938349546975yt^{10}u+184859110606331704717634265464366043yt^{9}u^{2}+1320823680472930389089607528761644695yt^{8}u^{3}+5704507543503690377015967434539000350yt^{7}u^{4}+16444238360704074206402677416288728838yt^{6}u^{5}+33119012020410791170089119355928268646yt^{5}u^{6}+47287624737926322347123500127335535742yt^{4}u^{7}+47316113083251077091623028330770144871yt^{3}u^{8}+31736764732487348997042797336485233579yt^{2}u^{9}+12796058877228401798410725094764095823ytu^{10}+2318793967646278843843770457709512827yu^{11}-466721274652780122764397126916914zwt^{10}-10093927102226979277609626098656368zwt^{9}u-86325891042074114331938898548045478zwt^{8}u^{2}-401975385057228464723935429649242560zwt^{7}u^{3}-1141866684985961819633024339652357540zwt^{6}u^{4}-2008223925238031333835141393501426336zwt^{5}u^{5}-1919101223470491487350344765489700348zwt^{4}u^{6}-118501967163222539872365772659750336zwt^{3}u^{7}+2118737586132081849029549253664127958zwt^{2}u^{8}+2500620329458474348698261480379558800zwtu^{9}+1067197809743333321419313888682409122zwu^{10}-162400700117228224709378629510817zt^{11}-10334605996702189130682474268115659zt^{10}u-111780400940567075073587414345708193zt^{9}u^{2}-479815059014563787267422868229719715zt^{8}u^{3}-594361796991414600665363692730743386zt^{7}u^{4}+2824126963378920121973215940018300994zt^{6}u^{5}+15561611946169525017634426264027336878zt^{5}u^{6}+37790601839093714655592557688667833578zt^{4}u^{7}+55553175502814244149497117686791717835zt^{3}u^{8}+51341988431184545065486510821792593081zt^{2}u^{9}+27865015107811442931744770509769669059ztu^{10}+6659555764248985696607400074042389737zu^{11}-168965425805889245563783172509730w^{2}t^{10}+4993393070684357892811180037934668w^{2}t^{9}u+104456812495644990757593304101567150w^{2}t^{8}u^{2}+797052844845233439384523532413658736w^{2}t^{7}u^{3}+3435710884029417515751079647676021260w^{2}t^{6}u^{4}+9584735626295965874484197959157692296w^{2}t^{5}u^{5}+18272769347876877622345526414924600124w^{2}t^{4}u^{6}+24115410156062937446116749634578123504w^{2}t^{3}u^{7}+21502295968127072468887707927116923446w^{2}t^{2}u^{8}+11920325821673148657708837059158511756w^{2}tu^{9}+3160270217966736834920421081288085558w^{2}u^{10}-468278402636162392259473676300895wt^{11}-5732200559470000567489012129467109wt^{10}u-5831309185686707002657218239922527wt^{9}u^{2}+261205896613418709394199422741535763wt^{8}u^{3}+2111660674774797810723186217676471706wt^{7}u^{4}+8604054939572900734871249297795655582wt^{6}u^{5}+22339552113020456103708962016917869458wt^{5}u^{6}+39570292304492599961605094039954125494wt^{4}u^{7}+48436411073290718866676746529047013109wt^{3}u^{8}+39808313139901242905799571428347679255wt^{2}u^{9}+20020187885217223010895021511643506429wtu^{10}+4656057975300261404296143154483051271wu^{11}-100266652731110901140834774777084t^{12}-3238099241756518108430100837655728t^{11}u-24260260222191748562069978992095412t^{10}u^{2}-14289796866021266581572015859035792t^{9}u^{3}+718330182054651779869948207675145016t^{8}u^{4}+4801420691103604817355084390733344672t^{7}u^{5}+16718462692163081371571535925680127992t^{6}u^{6}+37575886232019243221162016302393590368t^{5}u^{7}+57818217164755977807807492621371304468t^{4}u^{8}+61237870666833597628801573274251785488t^{3}u^{9}+43042091411239918650302080935269963196t^{2}u^{10}+18073451456343157840447545974011678768tu^{11}+3398277101444050185556213457667729072u^{12}}{6790545851092466687357858944xt^{11}-42271444362693698731779244096xt^{10}u-556313592980723923089259075920xt^{9}u^{2}-1791853512893535182720465189040xt^{8}u^{3}-858708114930095552563111282296xt^{7}u^{4}+6513964945873111301234651049912xt^{6}u^{5}+10212169947499465439918980629672xt^{5}u^{6}-8162031548901368265897371201448xt^{4}u^{7}-23226909183960530762395982290152xt^{3}u^{8}+5216906540194654581469107720296xt^{2}u^{9}+26018097033183613300612003051144xtu^{10}-13593045159858199488957319132296xu^{11}+677915226751271422203328324476ywt^{10}-5291441523153070907409841648500ywt^{9}u-47207856076027324183600129050372ywt^{8}u^{2}-68596394460228625360358055750048ywt^{7}u^{3}+139322224954146081939330019440168ywt^{6}u^{4}+370987732560401859001557438578016ywt^{5}u^{5}-116085625131278561406160795178184ywt^{4}u^{6}-759975493222680034080577194306288ywt^{3}u^{7}+729878123244698356005967640940ywt^{2}u^{8}+909985213062939993598986468617364ywtu^{9}-431689620253388322199400697580452ywu^{10}+203693077306658060354001678759yt^{11}-1278241989555786287542767201741yt^{10}u-14731565484639906921668638354773yt^{9}u^{2}-31911464382013842433946674729569yt^{8}u^{3}+21019561769091589583593844173986yt^{7}u^{4}+140970556260108802974294146987586yt^{6}u^{5}+65844742313538822893854271919678yt^{5}u^{6}-242057771141792389502499345687858yt^{4}u^{7}-201635196883769941731609058291941yt^{3}u^{8}+249266696384725671718657177879455yt^{2}u^{9}+127441975313138797071703484213451ytu^{10}-116846565026806437512734631908713yu^{11}+146433434311569191960648070470zwt^{10}+1920578045195548034084492860168zwt^{9}u+5616519615902482059014154949482zwt^{8}u^{2}+245443968997416548601317509872zwt^{7}u^{3}-21840407901098539881204951235020zwt^{6}u^{4}-23512602007626124162438209452472zwt^{5}u^{5}+28858034236376296580268784072020zwt^{4}u^{6}+54646125152882821833243584296128zwt^{3}u^{7}-12383064009732928193170689326418zwt^{2}u^{8}-71367377664666546674018949170720zwtu^{9}+37670317129458004663659817426490zwu^{10}-137478840855566059517445097437zt^{11}+2009019091841936283022332721705zt^{10}u+7553102417788141674526207141511zt^{9}u^{2}-19757464276148685423310756401555zt^{8}u^{3}-83736909204939083097281805544278zt^{7}u^{4}+19624969459148430466515107675814zt^{6}u^{5}+295566957126303492479456062103574zt^{5}u^{6}+76421559083720506471063730863914zt^{4}u^{7}-543062290417559566428603365949753zt^{3}u^{8}-180159759066766882422024113132595zt^{2}u^{9}+720673758913213056571015190694023ztu^{10}-299607676662803867322004570552043zu^{11}+289888286545985790395289291434w^{2}t^{10}-171733572045783436250971223348w^{2}t^{9}u-8374140300616078615235086410030w^{2}t^{8}u^{2}-16233277130709607315096988400864w^{2}t^{7}u^{3}+21567961780736030423087810985228w^{2}t^{6}u^{4}+75701448481106214979665121347216w^{2}t^{5}u^{5}-10838843532534179356155252919500w^{2}t^{4}u^{6}-150259374825411956064465344514480w^{2}t^{3}u^{7}-12143576026354714227032546533998w^{2}t^{2}u^{8}+183203918337402353310082458420868w^{2}tu^{9}-83895324592382787190780346411806w^{2}u^{10}+268230313731838183660121731017wt^{11}+1823507005133953061999663099835wt^{10}u-593332195751994425169770006571wt^{9}u^{2}-21463368379017966632068295211513wt^{8}u^{3}-28828164661701796907465292947682wt^{7}u^{4}+57836566145934890243099546986098wt^{6}u^{5}+128752654190260652074165252379778wt^{5}u^{6}-62813795012413043395558327278786wt^{4}u^{7}-244786907158776972601684369183611wt^{3}u^{8}+52871433561351441705874120497399wt^{2}u^{9}+233172373622287573785949885084101wtu^{10}-118545303619567618496374247888625wu^{11}+7586524794394157100043792929t^{12}+869029002612862434839298932920t^{11}u+2070101255096190975047303583186t^{10}u^{2}-12331173771197470155555560185480t^{9}u^{3}-46301364671052130032409509703389t^{8}u^{4}-2869580992778991441139196731008t^{7}u^{5}+163567169107744257330151083926076t^{6}u^{6}+127717650259570489944776744997936t^{5}u^{7}-260242760678715872928336487620009t^{4}u^{8}-287630898968554859995472088010488t^{3}u^{9}+277321801173411037016139813382082t^{2}u^{10}+171938868281818925808979088257560tu^{11}-138728638899806919921263960099435u^{12}}$ 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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.