Minimal polynomials of monstrous McKay–Thompson CM values

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Polynomials

$g$

$r$

$P_{g,r}(x)$

x - 544

comment: The selected root is $T_{2A}(i\sqrt{1})$; its order and level constraints are $H_{-4}$ and $\Phi^c_{2}$.

x^2 - 7104*x - 903296

comment: The selected root is $T_{2A}(i\sqrt{2})$; its order and level constraints are $H_{-8}$ and $\Phi^c_{2}$.

x^2 - 53504*x + 13399504

comment: The selected root is $T_{2A}(i\sqrt{3})$; its order and level constraints are $H_{-12}$ and $\Phi^c_{2}$.

x^2 - 286208*x - 155800136

comment: The selected root is $T_{2A}(i\sqrt{4})$; its order and level constraints are $H_{-16}$ and $\Phi^c_{2}$.

x^4 - 1264768*x^3 + 1229814784*x^2 + 207817474048*x + 14703821520896

comment: The selected root is $T_{2A}(i\sqrt{5})$; its order and level constraints are $H_{-20}$ and $\Phi^c_{2}$.

x^4 - 4829056*x^3 - 10183657728*x^2 + 1080905547776*x + 224648877457408

comment: The selected root is $T_{2A}(i\sqrt{6})$; its order and level constraints are $H_{-24}$ and $\Phi^c_{2}$.

x^2 - 16584704*x + 67533102529

comment: The selected root is $T_{2A}(i\sqrt{7})$; its order and level constraints are $H_{-28}$ and $\Phi^c_{2}$.

x^4 - 52241408*x^3 - 397880386576*x^2 - 152969416056832*x - 51376886272158784

comment: The selected root is $T_{2A}(i\sqrt{8})$; its order and level constraints are $H_{-32}$ and $\Phi^c_{2}$.

x^4 - 153565312*x^3 + 1900463769600*x^2 + 28471281188864*x + 10235564632834048

comment: The selected root is $T_{2A}(i\sqrt{9})$; its order and level constraints are $H_{-36}$ and $\Phi^c_{2}$.

10:

x^4 - 425650048*x^3 - 8778444001536*x^2 + 95340937330688*x + 267218949448155136

comment: The selected root is $T_{2A}(i\sqrt{10})$; its order and level constraints are $H_{-40}$ and $\Phi^c_{2}$.

x - 536

comment: The selected root is $T_{2B}(i\sqrt{1})$; its order and level constraints are $H_{-4}$ and $\Phi^c_{2}$.

x^2 - 7216*x - 89536

comment: The selected root is $T_{2B}(i\sqrt{2})$; its order and level constraints are $H_{-8}$ and $\Phi^c_{2}$.

x^2 - 53296*x + 2327104

comment: The selected root is $T_{2B}(i\sqrt{3})$; its order and level constraints are $H_{-12}$ and $\Phi^c_{2}$.

x^2 - 286768*x + 4784704

comment: The selected root is $T_{2B}(i\sqrt{4})$; its order and level constraints are $H_{-16}$ and $\Phi^c_{2}$.

x^4 - 1263712*x^3 - 104575616*x^2 - 10513471488*x + 398747570176

comment: The selected root is $T_{2B}(i\sqrt{5})$; its order and level constraints are $H_{-20}$ and $\Phi^c_{2}$.

x^4 - 4831328*x^3 + 792972672*x^2 + 8403625984*x - 522931859456

comment: The selected root is $T_{2B}(i\sqrt{6})$; its order and level constraints are $H_{-24}$ and $\Phi^c_{2}$.

x^2 - 16580656*x + 414712384

comment: The selected root is $T_{2B}(i\sqrt{7})$; its order and level constraints are $H_{-28}$ and $\Phi^c_{2}$.

x^4 - 52248672*x^3 - 18346275456*x^2 - 6828753737728*x + 170781783166976

comment: The selected root is $T_{2B}(i\sqrt{8})$; its order and level constraints are $H_{-32}$ and $\Phi^c_{2}$.

x^4 - 153552992*x^3 + 8691797376*x^2 - 358562060288*x + 5790449864704

comment: The selected root is $T_{2B}(i\sqrt{9})$; its order and level constraints are $H_{-36}$ and $\Phi^c_{2}$.

10:

x^4 - 425670752*x^3 + 34641792384*x^2 - 579891550208*x - 83083587584

comment: The selected root is $T_{2B}(i\sqrt{10})$; its order and level constraints are $H_{-40}$ and $\Phi^c_{2}$.

x^2 - 492*x - 24156

comment: The selected root is $T_{3A}(i\sqrt{1})$; its order and level constraints are $H_{-4}$ and $\Phi^c_{3}$.

x^2 - 7324*x + 690628

comment: The selected root is $T_{3A}(i\sqrt{2})$; its order and level constraints are $H_{-8}$ and $\Phi^c_{3}$.

x^3 - 53190*x^2 - 3316788*x - 72403848

comment: The selected root is $T_{3A}(i\sqrt{3})$; its order and level constraints are $H_{-12}$ and $\Phi^c_{3}$.

x^4 - 286752*x^3 + 193752*x^2 + 694752768*x + 90243793872

comment: The selected root is $T_{3A}(i\sqrt{4})$; its order and level constraints are $H_{-16}$ and $\Phi^c_{3}$.

x^4 - 1262744*x^3 - 1328026024*x^2 + 111537135776*x + 6940896633616

comment: The selected root is $T_{3A}(i\sqrt{5})$; its order and level constraints are $H_{-20}$ and $\Phi^c_{3}$.

x^6 - 4833108*x^5 + 9391930236*x^4 + 2489619005856*x^3 + 449819274128112*x^2 + 26577209630446272*x + 481968777016337472

comment: The selected root is $T_{3A}(i\sqrt{6})$; its order and level constraints are $H_{-24}$ and $\Phi^c_{3}$.

x^4 - 16580631*x^3 + 193752*x^2 + 38952780660*x + 275349242998593

comment: The selected root is $T_{3A}(i\sqrt{7})$; its order and level constraints are $H_{-28}$ and $\Phi^c_{3}$.

x^4 - 52249264*x^3 + 12584968136*x^2 + 2450481438016*x + 76858764318736

comment: The selected root is $T_{3A}(i\sqrt{8})$; its order and level constraints are $H_{-32}$ and $\Phi^c_{3}$.

x^6 - 153540036*x^5 - 1980733474404*x^4 - 1007618010819552*x^3 - 514360176572693520*x^2 + 15202462891650457536*x - 378882975916804506048

comment: The selected root is $T_{3A}(i\sqrt{9})$; its order and level constraints are $H_{-36}$ and $\Phi^c_{3}$.

10:

x^8 - 425691312*x^7 + 8786430969840*x^6 - 81478976963904*x^5 + 130810615161383520*x^4 - 3740807756378064542976*x^3 + 35179030223697153814272*x^2 + 12672396737947816461480960*x + 275437127516115003225940224

comment: The selected root is $T_{3A}(i\sqrt{10})$; its order and level constraints are $H_{-40}$ and $\Phi^c_{3}$.

x^2 - 510*x - 13707

comment: The selected root is $T_{3B}(i\sqrt{1})$; its order and level constraints are $H_{-4}$ and $\Phi^c_{3}$.

x^2 - 7314*x + 619065

comment: The selected root is $T_{3B}(i\sqrt{2})$; its order and level constraints are $H_{-8}$ and $\Phi^c_{3}$.

x^3 - 53253*x^2 + 37611*x - 7133535

comment: The selected root is $T_{3B}(i\sqrt{3})$; its order and level constraints are $H_{-12}$ and $\Phi^c_{3}$.

x^4 - 286788*x^3 + 10520334*x^2 - 109470636*x + 681688953

comment: The selected root is $T_{3B}(i\sqrt{4})$; its order and level constraints are $H_{-16}$ and $\Phi^c_{3}$.

x^4 - 1262676*x^3 - 1413881514*x^2 + 8134032204*x + 390601971441

comment: The selected root is $T_{3B}(i\sqrt{5})$; its order and level constraints are $H_{-20}$ and $\Phi^c_{3}$.

x^6 - 4833342*x^5 + 10522894815*x^4 + 211787549532*x^3 - 1517402686257*x^2 - 61219174946814*x + 576058272972849

comment: The selected root is $T_{3B}(i\sqrt{6})$; its order and level constraints are $H_{-24}$ and $\Phi^c_{3}$.

x^4 - 16580667*x^3 + 597099978*x^2 - 7148426364*x + 28837511865

comment: The selected root is $T_{3B}(i\sqrt{7})$; its order and level constraints are $H_{-28}$ and $\Phi^c_{3}$.

x^4 - 52248936*x^3 - 4552663914*x^2 + 4761546984*x + 971160716961

comment: The selected root is $T_{3B}(i\sqrt{8})$; its order and level constraints are $H_{-32}$ and $\Phi^c_{3}$.

x^6 - 153540594*x^5 - 1895058117981*x^4 + 95235273277524*x^3 - 4850305047892521*x^2 + 84125160495684990*x - 441849480897259611

comment: The selected root is $T_{3B}(i\sqrt{9})$; its order and level constraints are $H_{-36}$ and $\Phi^c_{3}$.

10:

x^8 - 425691384*x^7 + 8817080752620*x^6 - 717331301493480*x^5 + 15876496057995990*x^4 - 198438093184519944*x^3 + 3082241831999697036*x^2 - 36754088238464935320*x + 163224906103868562225

comment: The selected root is $T_{3B}(i\sqrt{10})$; its order and level constraints are $H_{-40}$ and $\Phi^c_{3}$.

x^2 - 508*x - 14864

comment: The selected root is $T_{5A}(i\sqrt{1})$; its order and level constraints are $H_{-4}$ and $\Phi^c_{5}$.

x^6 - 7256*x^5 + 196080*x^4 + 26350720*x^3 + 786484480*x^2 + 10477891584*x + 53802582016

comment: The selected root is $T_{5A}(i\sqrt{2})$; its order and level constraints are $H_{-8}$ and $\Phi^c_{5}$.

x^6 - 53256*x^5 + 196080*x^4 + 57170720*x^3 + 961284480*x^2 + 7117315584*x + 35981814016

comment: The selected root is $T_{5A}(i\sqrt{3})$; its order and level constraints are $H_{-12}$ and $\Phi^c_{5}$.

x^4 - 286748*x^3 - 950916*x^2 + 210956368*x + 2696198416

comment: The selected root is $T_{5A}(i\sqrt{4})$; its order and level constraints are $H_{-16}$ and $\Phi^c_{5}$.

x^10 - 1262500*x^9 - 1636091840*x^8 - 266883393600*x^7 - 28874769290240*x^6 - 1241926354076672*x^5 - 23284838221107200*x^4 - 171402078955765760*x^3 - 481639385268224000*x^2 - 12451365281291304960*x - 135018793727745327104

comment: The selected root is $T_{5A}(i\sqrt{5})$; its order and level constraints are $H_{-20}$ and $\Phi^c_{5}$.

x^8 - 4833544*x^7 + 11499235504*x^6 - 1941314777728*x^5 + 16562640519424*x^4 + 892779401928704*x^3 - 5780036768989184*x^2 + 98022668053250048*x + 3391513805917782016

comment: The selected root is $T_{5A}(i\sqrt{6})$; its order and level constraints are $H_{-24}$ and $\Phi^c_{5}$.

x^6 - 16580631*x^5 + 196080*x^4 + 11130511970*x^3 + 63765309480*x^2 - 1200306592416*x + 267547528160641

comment: The selected root is $T_{5A}(i\sqrt{7})$; its order and level constraints are $H_{-28}$ and $\Phi^c_{5}$.

x^12 - 52248512*x^11 - 26706054304*x^10 - 10209838751520*x^9 - 1086753576273280*x^8 - 32580329756859392*x^7 + 203616851638788224*x^6 + 28379177217454184448*x^5 + 988004934875775825920*x^4 + 58560995177973354055680*x^3 + 2018190388681669716156416*x^2 + 29677618654640311162994688*x + 154897491530439295455072256

comment: The selected root is $T_{5A}(i\sqrt{8})$; its order and level constraints are $H_{-32}$ and $\Phi^c_{5}$.

x^8 - 153540472*x^7 - 1913790066704*x^6 - 136549110115456*x^5 - 598538802818816*x^4 + 265523504471293952*x^3 + 9432709397055668224*x^2 + 121322897698073870336*x + 542851946877137453056

comment: The selected root is $T_{5A}(i\sqrt{9})$; its order and level constraints are $H_{-36}$ and $\Phi^c_{5}$.

10:

x^10 - 425691200*x^9 + 8738753556960*x^8 + 896891069683200*x^7 + 52127143519906560*x^6 + 2691652648588259328*x^5 + 145164353933835264000*x^4 + 4350577195671267901440*x^3 + 70487420895857398579200*x^2 + 647165490995574200074240*x + 2780896827656066245328896

comment: The selected root is $T_{5A}(i\sqrt{10})$; its order and level constraints are $H_{-40}$ and $\Phi^c_{5}$.

x^2 - 512*x - 12589

comment: The selected root is $T_{5B}(i\sqrt{1})$; its order and level constraints are $H_{-4}$ and $\Phi^c_{5}$.

x^6 - 7286*x^5 + 414915*x^4 + 12973180*x^3 + 307813815*x^2 + 1918029594*x + 4644775781

comment: The selected root is $T_{5B}(i\sqrt{2})$; its order and level constraints are $H_{-8}$ and $\Phi^c_{5}$.

x^6 - 53286*x^5 + 1794915*x^4 - 3586820*x^3 + 407173815*x^2 + 1619949594*x + 5002471781

comment: The selected root is $T_{5B}(i\sqrt{3})$; its order and level constraints are $H_{-12}$ and $\Phi^c_{5}$.

x^4 - 286774*x^3 + 6505466*x^2 + 3686386*x + 49767421

comment: The selected root is $T_{5B}(i\sqrt{4})$; its order and level constraints are $H_{-16}$ and $\Phi^c_{5}$.

x^10 - 1262560*x^9 - 1560339005*x^8 - 172139350920*x^7 - 16982523396590*x^6 - 39216466284552*x^5 - 26711042533490*x^4 - 14010078736767320*x^3 - 86205904359389155*x^2 + 100551104040548040*x - 741574797211224449

comment: The selected root is $T_{5B}(i\sqrt{5})$; its order and level constraints are $H_{-20}$ and $\Phi^c_{5}$.

x^8 - 4833548*x^7 + 11518570508*x^6 - 1990709700596*x^5 + 30972863069470*x^4 - 53407048307956*x^3 - 1026399961540132*x^2 + 5595522726641012*x + 49320700216101241

comment: The selected root is $T_{5B}(i\sqrt{6})$; its order and level constraints are $H_{-24}$ and $\Phi^c_{5}$.

x^6 - 16580661*x^5 + 497616165*x^4 - 5953441820*x^3 + 36106303815*x^2 - 105477440406*x + 133519339781

comment: The selected root is $T_{5B}(i\sqrt{7})$; its order and level constraints are $H_{-28}$ and $\Phi^c_{5}$.

x^12 - 52248572*x^11 - 23571140374*x^10 - 8768662265020*x^9 - 547702517834105*x^8 + 6150156226236408*x^7 + 77454473621529644*x^6 - 900286804734826312*x^5 - 2109248367316800705*x^4 + 45074733199553429380*x^3 - 24264337502194246134*x^2 - 734387694131004098172*x + 1883614578580027263961

comment: The selected root is $T_{5B}(i\sqrt{8})$; its order and level constraints are $H_{-32}$ and $\Phi^c_{5}$.

x^8 - 153540548*x^7 - 1902120985492*x^6 + 8098221182404*x^5 + 68927334239470*x^4 + 754243054047044*x^3 - 5216129088268132*x^2 - 5859264987777988*x + 56111127341103241

comment: The selected root is $T_{5B}(i\sqrt{9})$; its order and level constraints are $H_{-36}$ and $\Phi^c_{5}$.

10:

x^10 - 425691310*x^9 + 8785579594745*x^8 - 66781234010920*x^7 + 1933277296843410*x^6 - 32670698545632052*x^5 + 162206023921364010*x^4 + 68468869925142680*x^3 - 558383707789479155*x^2 - 10634934503102285710*x + 34144445974374154301

comment: The selected root is $T_{5B}(i\sqrt{10})$; its order and level constraints are $H_{-40}$ and $\Phi^c_{5}$.

x^4 - 492*x^3 - 22794*x^2 - 292940*x - 1157103

comment: The selected root is $T_{7A}(i\sqrt{1})$; its order and level constraints are $H_{-4}$ and $\Phi^c_{7}$.

x^8 - 7256*x^7 + 196476*x^6 + 24082520*x^5 + 814461606*x^4 + 14294648088*x^3 + 137279399740*x^2 + 679096087400*x + 1352298318625

comment: The selected root is $T_{7A}(i\sqrt{2})$; its order and level constraints are $H_{-8}$ and $\Phi^c_{7}$.

x^6 - 53270*x^5 + 942207*x^4 + 29923852*x^3 + 415047535*x^2 + 5561951850*x + 29654312625

comment: The selected root is $T_{7A}(i\sqrt{3})$; its order and level constraints are $H_{-12}$ and $\Phi^c_{7}$.

x^8 - 286752*x^7 + 196476*x^6 + 123862592*x^5 + 1213581894*x^4 + 5449438176*x^3 + 80040854908*x^2 + 731349262080*x + 2029795504641

comment: The selected root is $T_{7A}(i\sqrt{4})$; its order and level constraints are $H_{-16}$ and $\Phi^c_{7}$.

x^12 - 1262556*x^11 - 1565389782*x^10 - 177755745660*x^9 - 16957229861049*x^8 - 28345714131960*x^7 + 6868585780651692*x^6 + 70711328984353128*x^5 - 141763187110897249*x^4 - 3613836579936448940*x^3 + 3055364465292226250*x^2 + 152897188750435460500*x + 480332815506881205625

comment: The selected root is $T_{7A}(i\sqrt{5})$; its order and level constraints are $H_{-20}$ and $\Phi^c_{7}$.

x^12 - 4833460*x^11 + 11093221522*x^10 - 992929476292*x^9 - 106239407211713*x^8 - 3279750451389160*x^7 - 30751328059215812*x^6 + 536414452252512344*x^5 + 18162925029134765839*x^4 + 225179936316707744412*x^3 + 1518633085704718231890*x^2 + 5649253859188063657836*x + 9306644288810301834129

comment: The selected root is $T_{7A}(i\sqrt{6})$; its order and level constraints are $H_{-24}$ and $\Phi^c_{7}$.

x^7 - 16580613*x^6 - 298254558*x^5 + 572195351*x^4 + 34780757424*x^3 + 115850683095*x^2 - 1171502917550*x - 17309481105375

comment: The selected root is $T_{7A}(i\sqrt{7})$; its order and level constraints are $H_{-28}$ and $\Phi^c_{7}$.

x^16 - 52248512*x^15 - 26706053512*x^14 - 10246882940672*x^13 - 1103748483044452*x^12 - 38265124383157056*x^11 - 347652691016622264*x^10 + 9029153366481975936*x^9 + 259748531742268911686*x^8 + 2081957680429999045056*x^7 - 8407604905550973064376*x^6 - 236160952436100980520960*x^5 - 41051931278351717286500*x^4 + 33437041132415152145480000*x^3 + 387721925210694078662075000*x^2 + 1940214659516139654024400000*x + 3796849200701563094896890625

comment: The selected root is $T_{7A}(i\sqrt{8})$; its order and level constraints are $H_{-32}$ and $\Phi^c_{7}$.

x^16 - 153540528*x^15 - 1905191795400*x^14 - 30112171826576*x^13 - 1939345991587620*x^12 - 107247470909036976*x^11 - 1371573434926878456*x^10 + 23844831341666629104*x^9 + 798992527088898400518*x^8 + 6789214446773889920368*x^7 - 23554715345533390591608*x^6 - 500100781513598538198576*x^5 + 11830366566610058231005276*x^4 + 333274888085587856916222960*x^3 + 3357014841664634771350548408*x^2 + 16185710755894562958690928080*x + 31330403149004888827627234881

comment: The selected root is $T_{7A}(i\sqrt{9})$; its order and level constraints are $H_{-36}$ and $\Phi^c_{7}$.

10:

x^12 - 425691332*x^11 + 8794944801378*x^10 - 259150877233332*x^9 - 15074161988715537*x^8 + 66921889001086584*x^7 + 6552700616284056156*x^6 - 762891703068155784*x^5 - 1208237200725542776209*x^4 + 2539957114639205353036*x^3 + 249802578270020208889954*x^2 + 2012349263895383993932860*x + 4968260689627048834692225

comment: The selected root is $T_{7A}(i\sqrt{10})$; its order and level constraints are $H_{-40}$ and $\Phi^c_{7}$.

x^4 - 506*x^3 - 15633*x^2 - 86202*x - 196479

comment: The selected root is $T_{7B}(i\sqrt{1})$; its order and level constraints are $H_{-4}$ and $\Phi^c_{7}$.

x^8 - 7284*x^7 + 400386*x^6 + 13540800*x^5 + 345283435*x^4 + 2837831360*x^3 + 13708770306*x^2 + 33693934732*x + 38706757889

comment: The selected root is $T_{7B}(i\sqrt{2})$; its order and level constraints are $H_{-8}$ and $\Phi^c_{7}$.

x^6 - 53287*x^5 + 1848226*x^4 - 6340851*x^3 + 428533242*x^2 + 1273229505*x + 1879067709

comment: The selected root is $T_{7B}(i\sqrt{3})$; its order and level constraints are $H_{-12}$ and $\Phi^c_{7}$.

x^8 - 286780*x^7 + 8226274*x^6 - 80369856*x^5 + 971354475*x^4 + 333547200*x^3 + 19719052290*x^2 + 25680225420*x + 43286020353

comment: The selected root is $T_{7B}(i\sqrt{4})$; its order and level constraints are $H_{-16}$ and $\Phi^c_{7}$.

x^12 - 1262582*x^11 - 1532562487*x^10 - 138053044202*x^9 - 13479681400998*x^8 + 307232868886242*x^7 - 1730918526103543*x^6 + 9271811614609834*x^5 + 115197586352148618*x^4 + 604012326454057854*x^3 + 3044155982315838073*x^2 + 10629057252595027026*x + 25694146001044019689

comment: The selected root is $T_{7B}(i\sqrt{5})$; its order and level constraints are $H_{-20}$ and $\Phi^c_{7}$.

x^12 - 4833506*x^11 + 11315562285*x^10 - 1510728901790*x^9 - 43692505489886*x^8 - 224969137005594*x^7 + 10190315658585573*x^6 + 28317382517247246*x^5 + 191193735386928690*x^4 + 399153207231179370*x^3 + 3477639464533466685*x^2 - 1891210692827219130*x + 5209418506116787929

comment: The selected root is $T_{7B}(i\sqrt{6})$; its order and level constraints are $H_{-24}$ and $\Phi^c_{7}$.

x^7 - 16580648*x^6 + 282067758*x^5 - 2452367862*x^4 + 10493596953*x^3 - 30230042157*x^2 + 37572162579*x - 28204084899

comment: The selected root is $T_{7B}(i\sqrt{7})$; its order and level constraints are $H_{-28}$ and $\Phi^c_{7}$.

x^16 - 52248568*x^15 - 23780134572*x^14 - 8867283457648*x^13 - 584584979610534*x^12 + 3347452022945240*x^11 + 65808981357875952*x^10 - 393498707387278424*x^9 - 1777487944365640541*x^8 + 16876955826134560952*x^7 + 47597675704277929328*x^6 - 155815871821121112440*x^5 + 183164986180319718106*x^4 + 3391592964626548998064*x^3 + 6324163531706854303892*x^2 - 26144374091513602678504*x + 34514605560880841720321

comment: The selected root is $T_{7B}(i\sqrt{8})$; its order and level constraints are $H_{-32}$ and $\Phi^c_{7}$.

x^16 - 153540584*x^15 - 1896593523564*x^14 + 76237862218672*x^13 - 4377661809608486*x^12 + 52379884781706888*x^11 - 275461877962667088*x^10 + 2565883660823724744*x^9 - 28092588141459139101*x^8 + 147742472759452364232*x^7 - 255387824424328130640*x^6 + 302610070003883547528*x^5 - 2165991723219247472934*x^4 + 10539815552414042845104*x^3 + 11539756474489576564884*x^2 - 80185471639381806653160*x + 98049571947843050373633

comment: The selected root is $T_{7B}(i\sqrt{9})$; its order and level constraints are $H_{-36}$ and $\Phi^c_{7}$.

10:

x^12 - 425691370*x^11 + 8811121073509*x^10 - 593982843217494*x^9 + 7251937500282642*x^8 - 29603302589246850*x^7 + 70184361146982957*x^6 - 638352423812710170*x^5 + 2940616462301756514*x^4 + 1667578774750166898*x^3 - 4902712475217213771*x^2 - 33655947136102165122*x + 67552529765690485881

comment: The selected root is $T_{7B}(i\sqrt{10})$; its order and level constraints are $H_{-40}$ and $\Phi^c_{7}$.

13A

x^6 - 496*x^5 - 20690*x^4 - 249412*x^3 - 1327607*x^2 - 3279892*x - 3072432

comment: The selected root is $T_{13A}(i\sqrt{1})$; its order and level constraints are $H_{-4}$ and $\Phi^c_{13}$.

13A

x^14 - 7256*x^13 + 196716*x^12 + 22625304*x^11 + 838600022*x^10 + 17280881592*x^9 + 223968590092*x^8 + 1951170845416*x^7 + 11867390699697*x^6 + 51268134122400*x^5 + 157198957676800*x^4 + 335193970800000*x^3 + 473289506560000*x^2 + 398193664000000*x + 151191616000000

comment: The selected root is $T_{13A}(i\sqrt{2})$; its order and level constraints are $H_{-8}$ and $\Phi^c_{13}$.

13A

x^12 - 53266*x^11 + 729351*x^10 + 23839444*x^9 + 598715807*x^8 + 10306645422*x^7 + 104204824697*x^6 + 658346618896*x^5 + 2732229213312*x^4 + 7519391374880*x^3 + 13257025619200*x^2 + 13550133696000*x + 6092998560000

comment: The selected root is $T_{13A}(i\sqrt{3})$; its order and level constraints are $H_{-12}$ and $\Phi^c_{13}$.

13A

x^12 - 286756*x^11 + 1343736*x^10 + 61998760*x^9 + 686966582*x^8 + 11908745232*x^7 + 163077811220*x^6 + 1293089207464*x^5 + 6367334869281*x^4 + 20822175292628*x^3 + 45753575286988*x^2 + 62275440704496*x + 39535774911504

comment: The selected root is $T_{13A}(i\sqrt{4})$; its order and level constraints are $H_{-16}$ and $\Phi^c_{13}$.

13A

x^28 - 1262512*x^27 - 1620941032*x^26 - 248116371984*x^25 - 26588950010948*x^24 - 1007004018512752*x^23 - 16847411755759000*x^22 - 68241626825979152*x^21 + 2634920100803998566*x^20 + 66641368050160483632*x^19 + 964949779336857346344*x^18 + 11559224617527861872208*x^17 + 128085000706054610356476*x^16 + 1280951420501588585103792*x^15 + 10903928555039247211494104*x^14 + 76468398906781883626058704*x^13 + 438331178443881663411955809*x^12 + 2060862285010494514158043200*x^11 + 8001882507352649335998323200*x^10 + 25826682898653851066164000000*x^9 + 69580389744099923314669056000*x^8 + 156470023757141039444721664000*x^7 + 291857622973475205709257728000*x^6 + 444914615208169345527808000000*x^5 + 540084927033667762552832000000*x^4 + 500862509870285409566720000000*x^3 + 331925374219948509102080000000*x^2 + 139349629568344391680000000000*x + 27731006769226202742784000000

comment: The selected root is $T_{13A}(i\sqrt{5})$; its order and level constraints are $H_{-20}$ and $\Phi^c_{13}$.

13A

x^28 - 4833456*x^27 + 11073888024*x^26 - 950021124624*x^25 - 107300050352196*x^24 - 3891854367129456*x^23 - 66023147766790552*x^22 - 378947690667491088*x^21 + 5704114145593655142*x^20 + 143336341161754024752*x^19 + 1534398775386061763880*x^18 + 11553443042436364165200*x^17 + 94338850752881129407740*x^16 + 1004436711661006335753648*x^15 + 10130047722454699873672920*x^14 + 79841158229452186351713744*x^13 + 480206352029913203035004001*x^12 + 2242288489176867167359831872*x^11 + 8311319858191460883124891136*x^10 + 25029691972600075986152368896*x^9 + 62844741510670070494319382528*x^8 + 135096723584524577102207385600*x^7 + 252827710688880826341629190144*x^6 + 408400973994055076523223547904*x^5 + 547338568905273230355607846912*x^4 + 571212900230207371351535124480*x^3 + 426629058400091671024600350720*x^2 + 200414607835697319543570432000*x + 44277168107811077314433777664

comment: The selected root is $T_{13A}(i\sqrt{6})$; its order and level constraints are $H_{-24}$ and $\Phi^c_{13}$.

13A

x^14 - 16580631*x^13 + 196716*x^12 + 2608071804*x^11 + 6871308522*x^10 - 123625952658*x^9 - 399124016408*x^8 + 4433630393166*x^7 + 31116039892197*x^6 + 62874849822525*x^5 - 2153313719700*x^4 - 57089363028750*x^3 + 376180136020000*x^2 + 934905840000000*x + 833440812890625

comment: The selected root is $T_{13A}(i\sqrt{7})$; its order and level constraints are $H_{-28}$ and $\Phi^c_{13}$.

13A

x^28 - 52248512*x^27 - 26706053032*x^26 - 10269924531984*x^25 - 1114604697010948*x^24 - 42046729985960752*x^23 - 727509600798487000*x^22 - 4543457298178223152*x^21 + 47038748409208342566*x^20 + 1165870344428448511632*x^19 + 9076930668992334778344*x^18 + 11243525216993275076208*x^17 - 353738540005681156915524*x^16 - 2660220002709630797432208*x^15 - 126726333031174034849896*x^14 + 124533997667616684598014704*x^13 + 1035671049329264455939003809*x^12 + 4651184817569535708161353200*x^11 + 12428044200665943950267731200*x^10 + 14504258693871229119281900000*x^9 - 26060079744056546935437600000*x^8 - 144092778016380863990944000000*x^7 - 236900682239120904698798000000*x^6 + 47584637617660065754000000000*x^5 + 1024343020157722861752000000000*x^4 + 2316583112620222700020000000000*x^3 + 2828439471682114099520000000000*x^2 + 1973068127397703424000000000000*x + 628936409083222321000000000000

comment: The selected root is $T_{13A}(i\sqrt{8})$; its order and level constraints are $H_{-32}$ and $\Phi^c_{13}$.

13A

x^24 - 153540532*x^23 - 1904577632774*x^22 - 22564336395380*x^21 - 2649618658189897*x^20 - 112139057210183528*x^19 - 1814851782022342708*x^18 - 8652328943767403336*x^17 + 137738918989384715135*x^16 + 2613701970148246010716*x^15 + 16891063165062601499354*x^14 - 4860063615088466276804*x^13 - 889103324118002950146007*x^12 - 6532878574253883013742528*x^11 - 14004064098987035648215552*x^10 + 119112437687228150117348608*x^9 + 1309080211412436514978770944*x^8 + 6878371296181590319577202688*x^7 + 24272606529809726759838834688*x^6 + 62321205319884153588273643520*x^5 + 119178816756405075025906892800*x^4 + 168643075120582085199296200704*x^3 + 169638362143318430884596547584*x^2 + 109511259389731647227066056704*x + 34256231055743974520899239936

comment: The selected root is $T_{13A}(i\sqrt{9})$; its order and level constraints are $H_{-36}$ and $\Phi^c_{13}$.

13A

10:

x^24 - 425691324*x^23 + 8791539271586*x^22 - 189142514166540*x^21 - 10236321782150481*x^20 - 214906092390437688*x^19 - 2229306268680613668*x^18 + 8314413261283360008*x^17 + 449980111189509367791*x^16 + 3570082544611869064884*x^15 - 6087057878698988990814*x^14 - 251962940746786085526780*x^13 - 1291706796989913825987519*x^12 + 1519581138724892466734400*x^11 + 43603267266813615806630400*x^10 + 187253907268596051778272000*x^9 + 199103885982151764374016000*x^8 - 1197552054873485408292864000*x^7 - 5647981289986801651704832000*x^6 - 9495755854085872285286400000*x^5 + 5375269327413611207065600000*x^4 + 67319074535247873490944000000*x^3 + 166319574943657616474112000000*x^2 + 201017286639993391939584000000*x + 100564020517846712057856000000

comment: The selected root is $T_{13A}(i\sqrt{10})$; its order and level constraints are $H_{-40}$ and $\Phi^c_{13}$.

13B

x^6 - 506*x^5 - 15618*x^4 - 92252*x^3 - 345222*x^2 - 684954*x - 1071981

comment: The selected root is $T_{13B}(i\sqrt{1})$; its order and level constraints are $H_{-4}$ and $\Phi^c_{13}$.

13B

x^14 - 7282*x^13 + 385853*x^12 + 14108588*x^11 + 383306664*x^10 + 3909112116*x^9 + 28385230406*x^8 + 141603613580*x^7 + 552690905390*x^6 + 1657594013868*x^5 + 3948212112520*x^4 + 7238823978100*x^3 + 10133700032625*x^2 + 9640529296006*x + 5745767702029

comment: The selected root is $T_{13B}(i\sqrt{2})$; its order and level constraints are $H_{-8}$ and $\Phi^c_{13}$.

13B

x^12 - 53285*x^11 + 1741705*x^10 - 5308672*x^9 + 499255565*x^8 + 1901031437*x^7 + 23078833400*x^6 + 61612115069*x^5 + 308721867983*x^4 + 531436352712*x^3 + 1444577130435*x^2 + 1297311784659*x + 1915381511343

comment: The selected root is $T_{13B}(i\sqrt{3})$; its order and level constraints are $H_{-12}$ and $\Phi^c_{13}$.

13B

x^12 - 286778*x^11 + 7652740*x^10 - 70513162*x^9 + 953918852*x^8 + 1346296334*x^7 + 31310724002*x^6 + 98791577870*x^5 + 332285126780*x^4 + 676383706926*x^3 + 1059963867180*x^2 + 1069516299222*x + 638673037029

comment: The selected root is $T_{13B}(i\sqrt{4})$; its order and level constraints are $H_{-16}$ and $\Phi^c_{13}$.

13B

x^28 - 1262564*x^27 - 1555288770*x^26 - 165879057916*x^25 - 16253210148695*x^24 + 36865413916864*x^23 + 1142695520765916*x^22 + 12574910340356936*x^21 + 255087709678669608*x^20 + 4014755102102374760*x^19 + 40993604068232061532*x^18 + 349280178182027684800*x^17 + 2380589172882833902054*x^16 + 13752380851604375409776*x^15 + 67650541684386980219540*x^14 + 287151978852640075297536*x^13 + 1064645514181619012280950*x^12 + 3444608318971520429448512*x^11 + 9827653311065411356534428*x^10 + 24524380360649546603975320*x^9 + 53914521578351098398716008*x^8 + 102891036520929818854675320*x^7 + 171007070155407324102963036*x^6 + 241423428453316665688235584*x^5 + 288315489749837971885981985*x^4 + 277796382878926161023905300*x^3 + 209999954599264371836368286*x^2 + 110501227752055360331176348*x + 33072588288247685007772841

comment: The selected root is $T_{13B}(i\sqrt{5})$; its order and level constraints are $H_{-20}$ and $\Phi^c_{13}$.

13B

x^28 - 4833508*x^27 + 11325229374*x^26 - 1533717705468*x^25 - 39869266817367*x^24 - 236533505003328*x^23 + 7341460316357596*x^22 - 26645949132839608*x^21 + 449814980632289064*x^20 + 2873906996895570792*x^19 + 46710060934505701980*x^18 + 326990793980669307840*x^17 + 2459567744458146969062*x^16 + 13489559182094520905584*x^15 + 68411752456000041619092*x^14 + 285126778107139601133056*x^13 + 1070141884650467005467766*x^12 + 3429996060517660667352384*x^11 + 9864055297537408485054108*x^10 + 24435655652052368810467992*x^9 + 54122018410508072891256936*x^8 + 102459760278480357277422456*x^7 + 171798354523548136478388060*x^6 + 240097013166430055694966336*x^5 + 290263451245258867305021729*x^4 + 275571935110572066328325652*x^3 + 211740818733336787070137758*x^2 + 109690722758345323292383644*x + 33240669339371141291543721

comment: The selected root is $T_{13B}(i\sqrt{6})$; its order and level constraints are $H_{-24}$ and $\Phi^c_{13}$.

13B

x^14 - 16580657*x^13 + 431293603*x^12 - 5156784412*x^11 + 38303188664*x^10 - 185690297884*x^9 + 710943106406*x^8 - 1678550722420*x^7 + 4192999577390*x^6 - 3802868994132*x^5 + 10015393232520*x^4 + 2385079082100*x^3 + 12781197248625*x^2 + 8758030224006*x + 5881536790029

comment: The selected root is $T_{13B}(i\sqrt{7})$; its order and level constraints are $H_{-28}$ and $\Phi^c_{13}$.

13B

x^28 - 52248564*x^27 - 23989128770*x^26 - 8966113643916*x^25 - 621688404616695*x^24 + 425186618668864*x^23 + 40291735794485916*x^22 - 113079330333319064*x^21 - 4986028023186392*x^20 + 5099380870418282760*x^19 + 40088319014283773532*x^18 + 361131785256123044800*x^17 + 2385907194931883014054*x^16 + 13264795823597212791776*x^15 + 70356653954145331907540*x^14 + 275816828540851953127536*x^13 + 1110472913494212392684950*x^12 + 3288214037642422322296512*x^11 + 10275571780918464801494428*x^10 + 23342072995226896320743320*x^9 + 56792659371755759740348008*x^8 + 96804429466945825475859320*x^7 + 182255228206501084818739036*x^6 + 222513313146523245005483584*x^5 + 316115578917635686153117985*x^4 + 246040179689597746725793300*x^3 + 234854917620504409859536286*x^2 + 98929003493630585208232348*x + 35472438586753615497628841

comment: The selected root is $T_{13B}(i\sqrt{8})$; its order and level constraints are $H_{-32}$ and $\Phi^c_{13}$.

13B

x^24 - 153540578*x^23 - 1897514766993*x^22 + 64847246155646*x^21 - 4051158241922933*x^20 + 30466281969451416*x^19 - 215397603876078176*x^18 + 2821195206085837812*x^17 - 19950936467442765795*x^16 + 93664856069958750094*x^15 - 343901912351801575727*x^14 + 1904755585991053814466*x^13 - 936387263143689033794*x^12 + 20510009104353116362974*x^11 + 30296632119929357458905*x^10 + 129383898819514768780938*x^9 + 372766891394446517617533*x^8 + 477839872650970681828596*x^7 + 1808472272000012705680704*x^6 + 965875685420995611784848*x^5 + 4164379561539702806137947*x^4 + 1144586005057623786303978*x^3 + 3509823280498141105764711*x^2 + 843951053010353306753970*x + 551233019562736981240161

comment: The selected root is $T_{13B}(i\sqrt{9})$; its order and level constraints are $H_{-36}$ and $\Phi^c_{13}$.

13B

10:

x^24 - 425691366*x^23 + 8809418308339*x^22 - 558869046174758*x^21 + 7554896837089955*x^20 - 185962702416531264*x^19 + 2666476777265294376*x^18 - 20846031775461962956*x^17 + 107060320858335858965*x^16 - 416080849757454977630*x^15 + 1410248896099832692533*x^14 - 3580006522720599505970*x^13 + 9923735597064266677830*x^12 - 11085841777356819821350*x^11 + 61330510982382916419917*x^10 + 27165721309477571717262*x^9 + 401421718840744420303005*x^8 + 439219131371738870199732*x^7 + 2098287400829708976975288*x^6 + 1724324125116134624881176*x^5 + 5239459305991790227199403*x^4 + 1174811779188083707533558*x^3 + 5032779415104217444305291*x^2 + 320374723092970542301566*x + 654945064233404943878361

comment: The selected root is $T_{13B}(i\sqrt{10})$; its order and level constraints are $H_{-40}$ and $\Phi^c_{13}$.

Definition

For the zero-constant McKay–Thompson series $T_g(\tau)=q^{-1}+\sum_{n\geq1}c_g(n)q^n$, where $q=e^{2\pi i\tau}$, let $P_{g,r}(x)$ be the monic irreducible polynomial over $\mathbb Q$ of $T_g(i\sqrt r)$, with the positive square root [1].

Parameters

$g$

— Monster conjugacy class (a Monster conjugacy class in the ATLAS notation)

$r$

— rectangular CM index ($r>0$)

Formulas

(1)

For $p\in\{2,3,5,7,13\}$, put $b_p=24/(p-1)$, $C_p=p^{12/(p-1)}$ and $R_p(\tau)=(\eta(\tau)/\eta(p\tau))^{b_p}$. Then $T_{pB}(\tau)=R_p(\tau)+b_p$ and $T_{pA}(\tau)=R_p(\tau)+b_p+C_p/R_p(\tau)$ [1], where $\eta(\tau)=q^{1/24}\prod_{n\geq1}(1-q^n)$.

(2)

With $m_p=C_p/R_p$, the canonical modular polynomial satisfies $\Phi^c_p(m_p,j)=A_p(m_p)-m_pj=0$ for these five levels [3]. At $\tau=i\sqrt r$, the Hilbert class polynomial satisfies $H_{-4r}(j(\tau))=0$. Thus $m_p^h H_{-4r}(A_p(m_p)/m_p)=0$, where $h=\deg H_{-4r}$.

(3)

If $B=T_{pB}(i\sqrt r)$ and $A=T_{pA}(i\sqrt r)$, then $(B-b_p)(A-B)=C_p$. Consequently $P_{pA,r}(x)$ divides $\operatorname{Res}_t(P_{pB,r}(t),(t-b_p)^2-(x-b_p)(t-b_p)+C_p)$ in $\mathbb Q[x]$.

(4)

$P_{2A,1}(x)=x-544$ and $P_{2B,1}(x)=x-536$, corresponding to $T_{2A}(i)$ and $T_{2B}(i)$. The level-two equation at $j(i)=1728$ is $(m_2-8)^2(m_2+64)=0$, and positivity selects $m_2=8$.

Comments

(5)

The class labels follow the ATLAS notation. Every series has constant coefficient zero, including $T_{13A}$ [1]. The selected primes $p=2,3,5,7,13$ are precisely those for which $X_0(p)$ has genus zero; their classes $pA$ and $pB$ give the Fricke and non-Fricke Hauptmoduln, respectively.

(6)

The lattice $L_r=\mathbb Z+\mathbb Z i\sqrt r$ has endomorphism order $\mathbb Z[\sqrt{-r}]$ of discriminant $-4r$. Square factors of $r$ are retained. At prime level $p$, the cyclic level structure is represented by $\mathbb Z+\mathbb Z p i\sqrt r\subset L_r$, as for the numerical CM values.

(7)

Each entry is the minimal polynomial of one specified value. A resultant can have extraneous irreducible factors or repeated factors and is not itself the entry. These polynomials are not products over a complete CM orbit or over all cyclic level structures. Distinct pairs $(g,r)$ retain their contexts even when their minimal polynomials agree.

(8)

The argument $i\sqrt r$ is an algebraic integer, so these Hauptmodul values are algebraic integers [2], Proposition 3.7 and Theorem 3.9. Their minimal polynomials therefore belong to $\mathbb Z[x]$.

(9)

The Weber class polynomials concern $\mathfrak f(i\sqrt{-D})/\sqrt2$ with $D\equiv1\pmod8$ and $3\nmid D$, a different invariant and indexing. The Hecke-type Faber polynomials are universal transformations of $T_g$, rather than minimal polynomials of its CM values.

Programs

(P1)

Sage

R.<x> = PolynomialRing(QQ)
f = x^3 + 48*x^2 + (768-1728)*x + 4096
assert f == (x-8)^2*(x+64)
m = QQ(8)  # The unique positive root.
B = 4096/m + 24
A = B + m
assert x-A == x-544 and x-B == x-536

References

[1]

Toshiki Matsusaka, The Fourier coefficients of the McKay–Thompson series and the traces of CM values, Research in Number Theory 3 (2017), article 23, §1. (doi)

[2]

Kazuki Tomiyama, Generalized modular equations and the CM values of Hauptmoduln, version 4 (2026), Proposition 3.7, Theorem 3.9 and Corollary 3.10. (arXiv)

Links

[3]

PARI/GP: canonical modular equations and eta normalization

[4]

SageMath: certified Hilbert class polynomial computation

Similar tables

McKay–Thompson values at rectangular CM points — The entry with the same $(g,r)$ gives the real value whose minimal polynomial is listed here, in the same zero-constant normalization.

Hilbert class polynomials — The entry $H_{-4r}$ specifies the underlying CM order and supplies the exact constraint on $j(i\sqrt r)$.

Canonical modular polynomials — The entry at level $p$ connects $j$ to the eta quotient used for both $pA$ and $pB$.

Weber class polynomials — These are minimal polynomials of a different normalized eta invariant with different discriminant restrictions.

Hecke-type Faber polynomials of monstrous McKay–Thompson series — These transform the same normalized series for arbitrary arguments, rather than giving the minimal polynomial of one CM specialization.

Data properties

Entries are of type: integral polynomial

How well the digits are known: exact

Table is complete: no (it holds both Fricke and non-Fricke classes $g=2A,2B,3A,3B,5A,5B,7A,7B,13A,13B$ at every $1\leq r\leq10$. These include every prime level with genus-zero $X_0(p)$ and a consecutive prefix of principal rectangular CM orders of discriminant $-4r$, across fields and conductors with square factors retained. All five levels are included before increasing discriminant depth. The noncanonical cutoff keeps the initial elimination degree at most $28$ because $\deg H_{-4r}\leq2$ throughout this prefix; the longest resulting entry has $888$ characters. Other Monster classes, larger rectangular indices, nonprincipal CM classes and other level structures are outside this coverage)

How they were obtained:

Exact elimination combines frozen Hilbert class polynomials and canonical modular equations. Factorization is over $\mathbb Q$, and every selected factor is required to be monic with integer coefficients and irreducible. Arb evaluations of the eta quotient at 640 bits enclose the specified real value and exclude every other irreducible factor. The embedding enclosures retain at least 630 bits of relative accuracy. No polynomial is inferred from a rounded numerical approximation.

The independent checker substitutes directly into the canonical equation and uses PARI resultants. It verifies exact divisibility and irreducibility of every polynomial, removes every copy of the selected factor, and proves that the remaining quotient is nonzero throughout an independently computed enclosure of the CM value. These enclosures use 160-factor real Euler products at 768 bits. For $0<q<1$, the omitted product lies between $1-q^{161}/(1-q)$ and $1$; propagating this bound gives at least 753 bits of relative accuracy. All 100 enclosures agree with the corresponding numerical companion entries and fit inside its 98 decimal intervals; the two exact integer entries are separately checked.

All ten input Hilbert polynomials agree with PARI polclass and FLINT's certified computation [4]; all five canonical equations agree with PARI data [3]. The exact A/B relation is checked for all 50 level-index pairs. Known square-lattice values and deliberately wrong factors, wide balls and nonfinite balls test the checking method first. The attached checker repeats the arithmetic, enumeration and embedding tests on values read from the database.