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Properties

Label 36.0.10789712048...1797.1
Degree $36$
Signature $[0, 18]$
Discriminant $7^{24}\cdot 37^{33}$
Root discriminant $100.21$
Ramified primes $7, 37$
Class number $123201$ (GRH)
Class group $[3, 3, 117, 117]$ (GRH)
Galois group $C_3\times C_{12}$ (as 36T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![239345231, -600274541, 45608456, 769014408, 17515236, -648398298, -782004588, 1351057679, 1306426495, -4668590893, 5513868716, -4029586993, 2406821450, -1564444576, 971562775, -450921336, 219113520, -146575781, 71726367, -8520187, -8566260, 1162726, 2709150, -1456052, 452764, -197798, 82223, -30401, 9910, -2755, 2604, -1445, 181, 26, 16, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 9*x^35 + 16*x^34 + 26*x^33 + 181*x^32 - 1445*x^31 + 2604*x^30 - 2755*x^29 + 9910*x^28 - 30401*x^27 + 82223*x^26 - 197798*x^25 + 452764*x^24 - 1456052*x^23 + 2709150*x^22 + 1162726*x^21 - 8566260*x^20 - 8520187*x^19 + 71726367*x^18 - 146575781*x^17 + 219113520*x^16 - 450921336*x^15 + 971562775*x^14 - 1564444576*x^13 + 2406821450*x^12 - 4029586993*x^11 + 5513868716*x^10 - 4668590893*x^9 + 1306426495*x^8 + 1351057679*x^7 - 782004588*x^6 - 648398298*x^5 + 17515236*x^4 + 769014408*x^3 + 45608456*x^2 - 600274541*x + 239345231)
 
gp: K = bnfinit(x^36 - 9*x^35 + 16*x^34 + 26*x^33 + 181*x^32 - 1445*x^31 + 2604*x^30 - 2755*x^29 + 9910*x^28 - 30401*x^27 + 82223*x^26 - 197798*x^25 + 452764*x^24 - 1456052*x^23 + 2709150*x^22 + 1162726*x^21 - 8566260*x^20 - 8520187*x^19 + 71726367*x^18 - 146575781*x^17 + 219113520*x^16 - 450921336*x^15 + 971562775*x^14 - 1564444576*x^13 + 2406821450*x^12 - 4029586993*x^11 + 5513868716*x^10 - 4668590893*x^9 + 1306426495*x^8 + 1351057679*x^7 - 782004588*x^6 - 648398298*x^5 + 17515236*x^4 + 769014408*x^3 + 45608456*x^2 - 600274541*x + 239345231, 1)
 

Normalized defining polynomial

\( x^{36} - 9 x^{35} + 16 x^{34} + 26 x^{33} + 181 x^{32} - 1445 x^{31} + 2604 x^{30} - 2755 x^{29} + 9910 x^{28} - 30401 x^{27} + 82223 x^{26} - 197798 x^{25} + 452764 x^{24} - 1456052 x^{23} + 2709150 x^{22} + 1162726 x^{21} - 8566260 x^{20} - 8520187 x^{19} + 71726367 x^{18} - 146575781 x^{17} + 219113520 x^{16} - 450921336 x^{15} + 971562775 x^{14} - 1564444576 x^{13} + 2406821450 x^{12} - 4029586993 x^{11} + 5513868716 x^{10} - 4668590893 x^{9} + 1306426495 x^{8} + 1351057679 x^{7} - 782004588 x^{6} - 648398298 x^{5} + 17515236 x^{4} + 769014408 x^{3} + 45608456 x^{2} - 600274541 x + 239345231 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $36$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 18]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1078971204830304425585378172062472691337937003390705901956047673685421797=7^{24}\cdot 37^{33}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $100.21$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $7, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(259=7\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{259}(1,·)$, $\chi_{259}(134,·)$, $\chi_{259}(8,·)$, $\chi_{259}(137,·)$, $\chi_{259}(11,·)$, $\chi_{259}(142,·)$, $\chi_{259}(149,·)$, $\chi_{259}(23,·)$, $\chi_{259}(156,·)$, $\chi_{259}(29,·)$, $\chi_{259}(158,·)$, $\chi_{259}(51,·)$, $\chi_{259}(162,·)$, $\chi_{259}(36,·)$, $\chi_{259}(43,·)$, $\chi_{259}(177,·)$, $\chi_{259}(179,·)$, $\chi_{259}(184,·)$, $\chi_{259}(186,·)$, $\chi_{259}(60,·)$, $\chi_{259}(191,·)$, $\chi_{259}(64,·)$, $\chi_{259}(193,·)$, $\chi_{259}(211,·)$, $\chi_{259}(212,·)$, $\chi_{259}(85,·)$, $\chi_{259}(214,·)$, $\chi_{259}(121,·)$, $\chi_{259}(88,·)$, $\chi_{259}(100,·)$, $\chi_{259}(221,·)$, $\chi_{259}(228,·)$, $\chi_{259}(232,·)$, $\chi_{259}(233,·)$, $\chi_{259}(249,·)$, $\chi_{259}(253,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $\frac{1}{211} a^{29} - \frac{88}{211} a^{28} - \frac{19}{211} a^{27} - \frac{13}{211} a^{26} - \frac{90}{211} a^{25} - \frac{83}{211} a^{24} - \frac{98}{211} a^{23} + \frac{80}{211} a^{22} + \frac{5}{211} a^{21} + \frac{69}{211} a^{20} + \frac{36}{211} a^{19} + \frac{7}{211} a^{18} + \frac{37}{211} a^{17} + \frac{9}{211} a^{16} + \frac{86}{211} a^{15} + \frac{70}{211} a^{14} + \frac{24}{211} a^{13} - \frac{7}{211} a^{12} + \frac{59}{211} a^{11} - \frac{93}{211} a^{10} - \frac{6}{211} a^{9} + \frac{79}{211} a^{8} + \frac{27}{211} a^{7} - \frac{47}{211} a^{6} + \frac{1}{211} a^{5} - \frac{58}{211} a^{4} - \frac{2}{211} a^{3} + \frac{15}{211} a^{2} - \frac{49}{211} a + \frac{75}{211}$, $\frac{1}{633} a^{30} + \frac{1}{633} a^{29} - \frac{85}{211} a^{28} - \frac{16}{633} a^{27} - \frac{64}{211} a^{26} - \frac{25}{211} a^{25} - \frac{100}{633} a^{24} - \frac{202}{633} a^{23} + \frac{54}{211} a^{22} + \frac{92}{633} a^{21} + \frac{58}{633} a^{20} + \frac{46}{633} a^{19} + \frac{9}{211} a^{18} - \frac{74}{633} a^{17} + \frac{254}{633} a^{16} + \frac{128}{633} a^{15} - \frac{287}{633} a^{14} - \frac{64}{211} a^{13} + \frac{23}{211} a^{12} + \frac{305}{633} a^{11} - \frac{265}{633} a^{10} + \frac{178}{633} a^{9} - \frac{116}{633} a^{8} - \frac{176}{633} a^{7} + \frac{83}{211} a^{6} + \frac{242}{633} a^{5} - \frac{311}{633} a^{4} + \frac{259}{633} a^{3} + \frac{20}{633} a^{2} - \frac{277}{633} a + \frac{134}{633}$, $\frac{1}{141159} a^{31} + \frac{6}{47053} a^{30} + \frac{119}{141159} a^{29} - \frac{24373}{141159} a^{28} + \frac{6046}{141159} a^{27} - \frac{8779}{47053} a^{26} + \frac{41822}{141159} a^{25} + \frac{15653}{47053} a^{24} - \frac{16103}{141159} a^{23} + \frac{30140}{141159} a^{22} - \frac{29509}{141159} a^{21} - \frac{18875}{47053} a^{20} + \frac{61769}{141159} a^{19} - \frac{34463}{141159} a^{18} - \frac{65021}{141159} a^{17} + \frac{16479}{47053} a^{16} + \frac{57911}{141159} a^{15} - \frac{62371}{141159} a^{14} + \frac{3479}{47053} a^{13} - \frac{7351}{141159} a^{12} - \frac{20879}{47053} a^{11} - \frac{5878}{141159} a^{10} - \frac{6285}{47053} a^{9} - \frac{21277}{47053} a^{8} + \frac{43610}{141159} a^{7} + \frac{35804}{141159} a^{6} + \frac{68093}{141159} a^{5} + \frac{22861}{47053} a^{4} + \frac{36625}{141159} a^{3} + \frac{3283}{47053} a^{2} - \frac{6090}{47053} a - \frac{23}{633}$, $\frac{1}{141159} a^{32} + \frac{6}{47053} a^{30} - \frac{67}{47053} a^{29} + \frac{68113}{141159} a^{28} - \frac{69826}{141159} a^{27} - \frac{7270}{141159} a^{26} + \frac{11582}{47053} a^{25} + \frac{18760}{47053} a^{24} - \frac{23426}{141159} a^{23} - \frac{1372}{141159} a^{22} + \frac{60872}{141159} a^{21} - \frac{23369}{47053} a^{20} - \frac{55612}{141159} a^{19} + \frac{38176}{141159} a^{18} + \frac{51295}{141159} a^{17} + \frac{24142}{141159} a^{16} + \frac{38329}{141159} a^{15} - \frac{68855}{141159} a^{14} - \frac{35326}{141159} a^{13} + \frac{14530}{47053} a^{12} + \frac{46951}{141159} a^{11} + \frac{1094}{141159} a^{10} + \frac{17548}{141159} a^{9} - \frac{6256}{47053} a^{8} + \frac{19064}{47053} a^{7} - \frac{53221}{141159} a^{6} + \frac{52238}{141159} a^{5} + \frac{42680}{141159} a^{4} + \frac{61969}{141159} a^{3} + \frac{59114}{141159} a^{2} - \frac{9462}{47053} a - \frac{172}{633}$, $\frac{1}{428036294541} a^{33} + \frac{1141829}{428036294541} a^{32} + \frac{495227}{142678764847} a^{31} - \frac{174754250}{428036294541} a^{30} - \frac{52338994}{428036294541} a^{29} - \frac{182323985102}{428036294541} a^{28} + \frac{1650108167}{428036294541} a^{27} + \frac{90446406883}{428036294541} a^{26} - \frac{38442620748}{142678764847} a^{25} - \frac{55353742949}{142678764847} a^{24} + \frac{7661473680}{142678764847} a^{23} - \frac{65784483290}{142678764847} a^{22} - \frac{55690192787}{142678764847} a^{21} - \frac{24807020253}{142678764847} a^{20} + \frac{3440807829}{142678764847} a^{19} + \frac{52473756568}{142678764847} a^{18} + \frac{80380654522}{428036294541} a^{17} + \frac{183488680346}{428036294541} a^{16} - \frac{148462153345}{428036294541} a^{15} + \frac{111843701900}{428036294541} a^{14} - \frac{183682352483}{428036294541} a^{13} + \frac{36067280389}{428036294541} a^{12} + \frac{34908619411}{142678764847} a^{11} + \frac{81196071271}{428036294541} a^{10} - \frac{35948108657}{142678764847} a^{9} + \frac{59368687573}{428036294541} a^{8} - \frac{23959223081}{142678764847} a^{7} - \frac{25500426473}{142678764847} a^{6} - \frac{191431056589}{428036294541} a^{5} + \frac{34368562500}{142678764847} a^{4} - \frac{160423135948}{428036294541} a^{3} + \frac{58982723345}{142678764847} a^{2} + \frac{23296652683}{428036294541} a + \frac{188658063}{639815089}$, $\frac{1}{428036294541} a^{34} + \frac{1363078}{428036294541} a^{32} - \frac{48866}{142678764847} a^{31} - \frac{2585059}{142678764847} a^{30} - \frac{3113132}{1919445267} a^{29} - \frac{55990763171}{142678764847} a^{28} + \frac{59443087793}{142678764847} a^{27} - \frac{169903544314}{428036294541} a^{26} - \frac{46500506576}{428036294541} a^{25} - \frac{57179878718}{142678764847} a^{24} - \frac{50659658879}{428036294541} a^{23} - \frac{121056661498}{428036294541} a^{22} + \frac{91660402187}{428036294541} a^{21} - \frac{38282565664}{142678764847} a^{20} + \frac{165083299661}{428036294541} a^{19} + \frac{49439969834}{428036294541} a^{18} - \frac{18093847517}{428036294541} a^{17} + \frac{14002044511}{142678764847} a^{16} + \frac{150123297421}{428036294541} a^{15} + \frac{52395023525}{142678764847} a^{14} - \frac{38591924679}{142678764847} a^{13} + \frac{182066944553}{428036294541} a^{12} - \frac{30859156671}{142678764847} a^{11} - \frac{68046088725}{142678764847} a^{10} - \frac{4612814125}{142678764847} a^{9} + \frac{155817359695}{428036294541} a^{8} + \frac{97342638352}{428036294541} a^{7} + \frac{204306817255}{428036294541} a^{6} + \frac{128385790327}{428036294541} a^{5} + \frac{7775508564}{142678764847} a^{4} - \frac{2411517660}{142678764847} a^{3} - \frac{955903699}{2028608031} a^{2} - \frac{2405937142}{9107155203} a + \frac{158974081}{639815089}$, $\frac{1}{301615261182379810139152714274068511930920133833520459309438657598908751995667127488882109616035776371343456844614456954582034798135271135578114056654012459} a^{35} + \frac{237163296423282355126896605339699632818099820041478457611277587362431431166553914068567143573820868359906432592694106960009217829133092296832331}{301615261182379810139152714274068511930920133833520459309438657598908751995667127488882109616035776371343456844614456954582034798135271135578114056654012459} a^{34} + \frac{107178023423287897974904261001374951172015020982575232895121323713954048866965429622549288273071711218253568215498261184950178018129361464535649}{100538420394126603379717571424689503976973377944506819769812885866302917331889042496294036538678592123781152281538152318194011599378423711859371352218004153} a^{33} + \frac{66730756862344887720561311086353506693695285074374386492496863552745536290305253762305657629099475131874781883535978720213174946267026772626919472778}{100538420394126603379717571424689503976973377944506819769812885866302917331889042496294036538678592123781152281538152318194011599378423711859371352218004153} a^{32} - \frac{1000180734106163534806035956326931855179934526098219111253586300870613216887538472282527428110664588326459118550194286042660389768192602606710533514602}{301615261182379810139152714274068511930920133833520459309438657598908751995667127488882109616035776371343456844614456954582034798135271135578114056654012459} a^{31} - \frac{222311816215758394281489304300171485457531070938014261257864249821719068079563679250500436621476257919763410234235247257393097090489195247571621592201540}{301615261182379810139152714274068511930920133833520459309438657598908751995667127488882109616035776371343456844614456954582034798135271135578114056654012459} a^{30} - \frac{71242158403497465639868115874206102990263679839584282250564996780089177213748861820965999091870419547245023007355472798436323421702300446109279146577565}{301615261182379810139152714274068511930920133833520459309438657598908751995667127488882109616035776371343456844614456954582034798135271135578114056654012459} a^{29} - \frac{63189648666513364702984656750287718646124675933213749676272543154570168826471489258813410292551976714809807347669356404289592454905532033338052192866529948}{301615261182379810139152714274068511930920133833520459309438657598908751995667127488882109616035776371343456844614456954582034798135271135578114056654012459} a^{28} - \frac{4517500855993345489344145915278966377125959539400047997675964070775766773714937562783610940892897681220868962704140844315682646007967356377602148138204283}{100538420394126603379717571424689503976973377944506819769812885866302917331889042496294036538678592123781152281538152318194011599378423711859371352218004153} a^{27} + \frac{16237659841434296199066714209045829370354440002687458990043959635736500536883709274776618790385049502079709680455580999766580413542255608876557011156561910}{301615261182379810139152714274068511930920133833520459309438657598908751995667127488882109616035776371343456844614456954582034798135271135578114056654012459} a^{26} - \frac{34378977734709622175918937067256663600638760296317108927658660195024333946133001378335853335590529552568008176825267424981092575697737054946868474618356578}{301615261182379810139152714274068511930920133833520459309438657598908751995667127488882109616035776371343456844614456954582034798135271135578114056654012459} a^{25} + \frac{24958301445600906745848521473257555153512123703855283560785025343444937461546217801663827123280690801947474375551641232097012722830972522427467419031740773}{100538420394126603379717571424689503976973377944506819769812885866302917331889042496294036538678592123781152281538152318194011599378423711859371352218004153} a^{24} - \frac{145826000934077032558120657084474202709593567796711936423074258322669204473172219415311942395171247117120185269049681197085984028351959059595626633178323096}{301615261182379810139152714274068511930920133833520459309438657598908751995667127488882109616035776371343456844614456954582034798135271135578114056654012459} a^{23} - \frac{16341918464330400322081913085501466348981508593446432939327590773548690834241903613051231685146199304820395114267097516555761694652120991542164284870170062}{301615261182379810139152714274068511930920133833520459309438657598908751995667127488882109616035776371343456844614456954582034798135271135578114056654012459} a^{22} + \frac{51699488110798275129361273309155545349971686235696043005902345708189708574850579757520098968607077982212369127687429815579653489507175015842694742913655502}{301615261182379810139152714274068511930920133833520459309438657598908751995667127488882109616035776371343456844614456954582034798135271135578114056654012459} a^{21} - \frac{40683550042442545550103170260259613790180278882373829126644411504342911882814403797676553779528975968730214736851914646119776078470134967895934523249396652}{100538420394126603379717571424689503976973377944506819769812885866302917331889042496294036538678592123781152281538152318194011599378423711859371352218004153} a^{20} + \frac{32019244461415561920814679341029758163576181148294765609131849854635110737941587148841098784167244907907889139651390380479956427524158637962095127720951545}{301615261182379810139152714274068511930920133833520459309438657598908751995667127488882109616035776371343456844614456954582034798135271135578114056654012459} a^{19} + \frac{37130905065608249149036582684210458882256288777234761727563462603265659118774326142311798866651223257472005795275849972245835127070594147763532327371335704}{100538420394126603379717571424689503976973377944506819769812885866302917331889042496294036538678592123781152281538152318194011599378423711859371352218004153} a^{18} - \frac{48499545122915890028888751508579583048881427257348917203878503091182545739862293964584056191268606731108748676106951118090352778959704285239506504799880855}{301615261182379810139152714274068511930920133833520459309438657598908751995667127488882109616035776371343456844614456954582034798135271135578114056654012459} a^{17} + \frac{32796516786928917790375203034209602088228526138351586604745119258973903327151751053998656788079312011442639757761251021847471064192645223254938937196867438}{301615261182379810139152714274068511930920133833520459309438657598908751995667127488882109616035776371343456844614456954582034798135271135578114056654012459} a^{16} + \frac{95241451903321886618136498622230665966483912372819540205197872974991726922342876579457380840348622731243295090422600521271751015393348849030898142173998}{674754499289440291139044103521406066959552872110784025300757623263777968670396258364389506970997262575712431419719143075127594626700830280935378202805397} a^{15} - \frac{69528484661034089818111835777980353154911438861098520086806169293764672671546482540152778686496275281164162815766115144555388401612205733530840922850389562}{301615261182379810139152714274068511930920133833520459309438657598908751995667127488882109616035776371343456844614456954582034798135271135578114056654012459} a^{14} - \frac{34413947196007405770477984891591470908002580715328567967918150408039689092230938830228219165294270960492120673985630122774496053173839662432234686071212124}{100538420394126603379717571424689503976973377944506819769812885866302917331889042496294036538678592123781152281538152318194011599378423711859371352218004153} a^{13} - \frac{101224692255289525956092710176642636546050316200147288294602855011998571113797192274744075726671723664129030038883969252883551351515975916782379644197000358}{301615261182379810139152714274068511930920133833520459309438657598908751995667127488882109616035776371343456844614456954582034798135271135578114056654012459} a^{12} - \frac{35445530639302785072875488047737018088594809221139462751802481982275576708161608393753371953925728604521805476916351938333847448788587937786604569567345035}{100538420394126603379717571424689503976973377944506819769812885866302917331889042496294036538678592123781152281538152318194011599378423711859371352218004153} a^{11} - \frac{64468866824947981060841248343930957462385827446660648729796393052378964633883220270298500932908140378079885318624336627204157622642945032918644365745066934}{301615261182379810139152714274068511930920133833520459309438657598908751995667127488882109616035776371343456844614456954582034798135271135578114056654012459} a^{10} + \frac{26069575811151114693201773502494056147725017760095194649220550967161109949484388764599975906961663625581570151714108927374127264943779371250211713116184073}{100538420394126603379717571424689503976973377944506819769812885866302917331889042496294036538678592123781152281538152318194011599378423711859371352218004153} a^{9} - \frac{133710218242172202946427178134722988020515860802423171696002728086716854810167824609768623723282475825924355429172331661418401786191172542274883915669989904}{301615261182379810139152714274068511930920133833520459309438657598908751995667127488882109616035776371343456844614456954582034798135271135578114056654012459} a^{8} + \frac{42718761415573007149624614321940386065772406374126138329357507428557757647720822090572116838059707716349782517562886103927914233206008246980786907414175643}{301615261182379810139152714274068511930920133833520459309438657598908751995667127488882109616035776371343456844614456954582034798135271135578114056654012459} a^{7} - \frac{62544327953728986039218195189582381243379200024213495605610538415786439630429428688927695327295653543255896613343168519892735020523446689254903586949290453}{301615261182379810139152714274068511930920133833520459309438657598908751995667127488882109616035776371343456844614456954582034798135271135578114056654012459} a^{6} + \frac{82019887132103773700348080655544633497274984917998103525815355702655857175364033911110691179988446234584478821174938435305938338587646452053174110852974767}{301615261182379810139152714274068511930920133833520459309438657598908751995667127488882109616035776371343456844614456954582034798135271135578114056654012459} a^{5} - \frac{12295972099979440223917559899667655016617908503048099826550292623601832743701917829949672005198273971470360273698736570256315400361618585510087069157293586}{100538420394126603379717571424689503976973377944506819769812885866302917331889042496294036538678592123781152281538152318194011599378423711859371352218004153} a^{4} + \frac{62180085738448028225428320609258080156804095102889541634799192086094406090253915827246522778758479732843910821367352114753043833928707661525417847374905189}{301615261182379810139152714274068511930920133833520459309438657598908751995667127488882109616035776371343456844614456954582034798135271135578114056654012459} a^{3} - \frac{24401989713224589393134791880298782477070793261948352551223541491202903004471695123884528207484219063891222562530211272894795784106473713888264789420796385}{301615261182379810139152714274068511930920133833520459309438657598908751995667127488882109616035776371343456844614456954582034798135271135578114056654012459} a^{2} - \frac{40779082588047912744361001233421480692961896624815079178902518439059080503407001695565289907970081840904416772025267795242921442864402905701770569340893242}{301615261182379810139152714274068511930920133833520459309438657598908751995667127488882109616035776371343456844614456954582034798135271135578114056654012459} a + \frac{219281977000125771068369065404147601951224191410158281236873762958540452922886092410753101868824399009937212088972128854032866245487411886537543359718926}{450844934502809880626536194729549345188221425760120268026066752763690212250623508951991195240711175442964808437390817570376733629499657900714669740887911}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{3}\times C_{3}\times C_{117}\times C_{117}$, which has order $123201$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $17$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 10004315800200082 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_{12}$ (as 36T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 36
The 36 conjugacy class representatives for $C_3\times C_{12}$
Character table for $C_3\times C_{12}$ is not computed

Intermediate fields

\(\Q(\sqrt{37}) \), 3.3.67081.2, 3.3.1369.1, 3.3.67081.1, \(\Q(\zeta_{7})^+\), 4.0.50653.1, 6.6.166494840757.2, 6.6.69343957.1, 6.6.166494840757.1, 6.6.121617853.1, 9.9.301855146292441.1, 12.0.1025659683951855168322813.1, 12.0.177917621779460413.1, 12.0.1025659683951855168322813.2, 12.0.749203567532399684677.1, 18.18.4615325560822677694719842735278093.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/5.12.0.1}{12} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/13.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/17.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/19.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/23.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{9}$ ${\href{/LocalNumberField/31.12.0.1}{12} }^{3}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{9}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/59.12.0.1}{12} }^{3}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
37Data not computed