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Properties

Label 36.36.8226549975...8125.1
Degree $36$
Signature $[36, 0]$
Discriminant $5^{27}\cdot 7^{30}\cdot 19^{24}$
Root discriminant $120.50$
Ramified primes $5, 7, 19$
Class number $1$ (GRH)
Class group Trivial (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![4591091, -12769152, -136694687, 363308426, 1545247949, -3909998733, -8861546511, 21853214932, 27858490170, -71694298121, -46806011656, 145220963746, 32402227403, -183495454624, 15401207070, 142697506526, -44892472008, -66363236954, 34707820890, 17229198563, -14012133789, -1874661149, 3313200544, -170531710, -475474126, 82185945, 41000690, -11622649, -1940661, 883449, 29909, -38540, 1441, 909, -74, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 9*x^35 - 74*x^34 + 909*x^33 + 1441*x^32 - 38540*x^31 + 29909*x^30 + 883449*x^29 - 1940661*x^28 - 11622649*x^27 + 41000690*x^26 + 82185945*x^25 - 475474126*x^24 - 170531710*x^23 + 3313200544*x^22 - 1874661149*x^21 - 14012133789*x^20 + 17229198563*x^19 + 34707820890*x^18 - 66363236954*x^17 - 44892472008*x^16 + 142697506526*x^15 + 15401207070*x^14 - 183495454624*x^13 + 32402227403*x^12 + 145220963746*x^11 - 46806011656*x^10 - 71694298121*x^9 + 27858490170*x^8 + 21853214932*x^7 - 8861546511*x^6 - 3909998733*x^5 + 1545247949*x^4 + 363308426*x^3 - 136694687*x^2 - 12769152*x + 4591091)
 
gp: K = bnfinit(x^36 - 9*x^35 - 74*x^34 + 909*x^33 + 1441*x^32 - 38540*x^31 + 29909*x^30 + 883449*x^29 - 1940661*x^28 - 11622649*x^27 + 41000690*x^26 + 82185945*x^25 - 475474126*x^24 - 170531710*x^23 + 3313200544*x^22 - 1874661149*x^21 - 14012133789*x^20 + 17229198563*x^19 + 34707820890*x^18 - 66363236954*x^17 - 44892472008*x^16 + 142697506526*x^15 + 15401207070*x^14 - 183495454624*x^13 + 32402227403*x^12 + 145220963746*x^11 - 46806011656*x^10 - 71694298121*x^9 + 27858490170*x^8 + 21853214932*x^7 - 8861546511*x^6 - 3909998733*x^5 + 1545247949*x^4 + 363308426*x^3 - 136694687*x^2 - 12769152*x + 4591091, 1)
 

Normalized defining polynomial

\( x^{36} - 9 x^{35} - 74 x^{34} + 909 x^{33} + 1441 x^{32} - 38540 x^{31} + 29909 x^{30} + 883449 x^{29} - 1940661 x^{28} - 11622649 x^{27} + 41000690 x^{26} + 82185945 x^{25} - 475474126 x^{24} - 170531710 x^{23} + 3313200544 x^{22} - 1874661149 x^{21} - 14012133789 x^{20} + 17229198563 x^{19} + 34707820890 x^{18} - 66363236954 x^{17} - 44892472008 x^{16} + 142697506526 x^{15} + 15401207070 x^{14} - 183495454624 x^{13} + 32402227403 x^{12} + 145220963746 x^{11} - 46806011656 x^{10} - 71694298121 x^{9} + 27858490170 x^{8} + 21853214932 x^{7} - 8861546511 x^{6} - 3909998733 x^{5} + 1545247949 x^{4} + 363308426 x^{3} - 136694687 x^{2} - 12769152 x + 4591091 \)

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

Invariants

Degree:  $36$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[36, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(822654997589852745462607600081733488063942633416481504805386066436767578125=5^{27}\cdot 7^{30}\cdot 19^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $120.50$
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:  $5, 7, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(665=5\cdot 7\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{665}(68,·)$, $\chi_{665}(1,·)$, $\chi_{665}(258,·)$, $\chi_{665}(134,·)$, $\chi_{665}(647,·)$, $\chi_{665}(11,·)$, $\chi_{665}(144,·)$, $\chi_{665}(657,·)$, $\chi_{665}(153,·)$, $\chi_{665}(292,·)$, $\chi_{665}(39,·)$, $\chi_{665}(296,·)$, $\chi_{665}(429,·)$, $\chi_{665}(558,·)$, $\chi_{665}(178,·)$, $\chi_{665}(571,·)$, $\chi_{665}(191,·)$, $\chi_{665}(64,·)$, $\chi_{665}(577,·)$, $\chi_{665}(324,·)$, $\chi_{665}(83,·)$, $\chi_{665}(596,·)$, $\chi_{665}(87,·)$, $\chi_{665}(353,·)$, $\chi_{665}(482,·)$, $\chi_{665}(106,·)$, $\chi_{665}(239,·)$, $\chi_{665}(552,·)$, $\chi_{665}(467,·)$, $\chi_{665}(628,·)$, $\chi_{665}(501,·)$, $\chi_{665}(248,·)$, $\chi_{665}(121,·)$, $\chi_{665}(634,·)$, $\chi_{665}(362,·)$, $\chi_{665}(254,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $\frac{1}{4026062671353929720720724005110191889019627759353182087002690569050567759205178329456821397158464781876946977609984501916281146087139432559856082031109051341} a^{35} - \frac{1494823723554092975480181061387300095550229689846470419158790180239378127821811152277985157909483411924803661088770813092367342206504860486299558879023301896}{4026062671353929720720724005110191889019627759353182087002690569050567759205178329456821397158464781876946977609984501916281146087139432559856082031109051341} a^{34} + \frac{1772657534353756536942379551439502005548683608374146495593354028087153488824338530619837848972991553560345974304018294999181166141668649935021258554924991659}{4026062671353929720720724005110191889019627759353182087002690569050567759205178329456821397158464781876946977609984501916281146087139432559856082031109051341} a^{33} + \frac{1939947890298402759269400315865576032234700095117034885610014937814680920872321057113192441116645089912947704951701055240430895336190827341264104300820862152}{4026062671353929720720724005110191889019627759353182087002690569050567759205178329456821397158464781876946977609984501916281146087139432559856082031109051341} a^{32} + \frac{895387070084359734283081491127900342452973674791278685166571325128316181832523564451156076012883579145924753480650795920335035456223920967776730279883954445}{4026062671353929720720724005110191889019627759353182087002690569050567759205178329456821397158464781876946977609984501916281146087139432559856082031109051341} a^{31} - \frac{1991070571043792691669880618866191109176068109229413245460703753415538230806563401634419851698225370396031723153224913147734965127524384992312840613090731019}{4026062671353929720720724005110191889019627759353182087002690569050567759205178329456821397158464781876946977609984501916281146087139432559856082031109051341} a^{30} + \frac{1822474821470678428263810602337379106781572377034713581538242229914300563308660391213090859074423701022408340210892006335975485009437744823872142429397962273}{4026062671353929720720724005110191889019627759353182087002690569050567759205178329456821397158464781876946977609984501916281146087139432559856082031109051341} a^{29} + \frac{127211491277621307448793205842784502972120429274185979457968446150770383746771317997343107594067130040217695710057133345975823911816206740173968857768611052}{4026062671353929720720724005110191889019627759353182087002690569050567759205178329456821397158464781876946977609984501916281146087139432559856082031109051341} a^{28} + \frac{1559558880843409473671633275435463771526170693758054330160777855575153128369529944783041926606877793362406331469276139299500534957625700428554827370747081051}{4026062671353929720720724005110191889019627759353182087002690569050567759205178329456821397158464781876946977609984501916281146087139432559856082031109051341} a^{27} - \frac{1062721987159628361761154884594364178544401715340666647382047129005460858578789678948399184872982005168928413020862956370450140425638005119655534534201619661}{4026062671353929720720724005110191889019627759353182087002690569050567759205178329456821397158464781876946977609984501916281146087139432559856082031109051341} a^{26} - \frac{1046989054437241169040668751772174669916105365923719375841800187981113765039881680831954202240588418129300522364800336885118401129481197522645060598750640447}{4026062671353929720720724005110191889019627759353182087002690569050567759205178329456821397158464781876946977609984501916281146087139432559856082031109051341} a^{25} + \frac{1250933359338300342851480906335574461487935962913674985889968218362591752809742182018936078384062983233360518832113736549412779887806022728490964109961745456}{4026062671353929720720724005110191889019627759353182087002690569050567759205178329456821397158464781876946977609984501916281146087139432559856082031109051341} a^{24} + \frac{1565391932262999545558900227853360807394510009572152299133148449888585499903175201586100882548993841736732578688813163714419817953182101370153016379697312904}{4026062671353929720720724005110191889019627759353182087002690569050567759205178329456821397158464781876946977609984501916281146087139432559856082031109051341} a^{23} - \frac{295899897074210735984489366706801705306480522914736075840232762408764048343944465242672988174003532869406176509432012174930099902248634894672708075541695504}{4026062671353929720720724005110191889019627759353182087002690569050567759205178329456821397158464781876946977609984501916281146087139432559856082031109051341} a^{22} + \frac{1226491800209930142771105720748434002028005851182457077893786921264703517230166671977356467121186424656406740746778815646293090075101430435898887276189599427}{4026062671353929720720724005110191889019627759353182087002690569050567759205178329456821397158464781876946977609984501916281146087139432559856082031109051341} a^{21} - \frac{331931993330674967078206270357067861501159514039339018759411003526794686668216548255181015276897689674182611481348995046462510902453538147489917356894836168}{4026062671353929720720724005110191889019627759353182087002690569050567759205178329456821397158464781876946977609984501916281146087139432559856082031109051341} a^{20} + \frac{1557218901812946055603467917427515542363239565788698530395377084788481877738193917362653165222685361300390529389343301366630937044151695330274792698717399256}{4026062671353929720720724005110191889019627759353182087002690569050567759205178329456821397158464781876946977609984501916281146087139432559856082031109051341} a^{19} + \frac{678356845295100811272426239876278009618163960118040570415491570978604475164431305634909699450767337221607861407134419139735068387635625078061418465525432304}{4026062671353929720720724005110191889019627759353182087002690569050567759205178329456821397158464781876946977609984501916281146087139432559856082031109051341} a^{18} - \frac{143893195671771519634635482906438538844262721562467864209265230346159327995875161119410488855926158951305969332772765572430228747274355035615259743737864612}{4026062671353929720720724005110191889019627759353182087002690569050567759205178329456821397158464781876946977609984501916281146087139432559856082031109051341} a^{17} - \frac{729611102005204382761791269532009987006929251016816955132888814575494762914779500301393300115222871575389001226291112219573819972229436826815517378632286067}{4026062671353929720720724005110191889019627759353182087002690569050567759205178329456821397158464781876946977609984501916281146087139432559856082031109051341} a^{16} - \frac{321466354811775761075777759185811452902742969411603828870769265123129363100927209777391046064632753851025767520298911390492324540664580184237646190469166645}{4026062671353929720720724005110191889019627759353182087002690569050567759205178329456821397158464781876946977609984501916281146087139432559856082031109051341} a^{15} + \frac{902840775420203511600889845411337071625135248671090627421084269077851486957058021942832849330886488047577786448654422717187371161531537573486819657328311417}{4026062671353929720720724005110191889019627759353182087002690569050567759205178329456821397158464781876946977609984501916281146087139432559856082031109051341} a^{14} + \frac{1245637031485768242570839997601844571478153887956914793703816775614065680527153772120862090688550277344549255017339721184743342861026259633168618792422848042}{4026062671353929720720724005110191889019627759353182087002690569050567759205178329456821397158464781876946977609984501916281146087139432559856082031109051341} a^{13} - \frac{1102376565391968171602480579963881684751962087886215792835805284166749662490216519478274643306431626763297472441696149130259532224616436746055535290622063465}{4026062671353929720720724005110191889019627759353182087002690569050567759205178329456821397158464781876946977609984501916281146087139432559856082031109051341} a^{12} - \frac{1440418223709371597153061852652645103336712057697286404215891141533428229325260791735210221546022812411271099400606427844159281821447868087400954735430457452}{4026062671353929720720724005110191889019627759353182087002690569050567759205178329456821397158464781876946977609984501916281146087139432559856082031109051341} a^{11} + \frac{1492238483485231109343105239272065231391550002857615716466838529878897265042954740771477937287123353082524506276739204283009140970496503919171988400658510235}{4026062671353929720720724005110191889019627759353182087002690569050567759205178329456821397158464781876946977609984501916281146087139432559856082031109051341} a^{10} - \frac{1850578125711683033070250895074147609855144118523707709157254599684382790113603083465126083371756263112479270259559757249966259343152861523189082994277500618}{4026062671353929720720724005110191889019627759353182087002690569050567759205178329456821397158464781876946977609984501916281146087139432559856082031109051341} a^{9} + \frac{1240497890866260614537315340019947589365226739231016812201881167721740291378813427796323931099940748602180796199920459437088740962191100299819990686384317298}{4026062671353929720720724005110191889019627759353182087002690569050567759205178329456821397158464781876946977609984501916281146087139432559856082031109051341} a^{8} + \frac{760336125806141971846161962749135094832466568528738551190298237790897868857354844437691153898636128500813287120936826072138529497505321929349376488632479771}{4026062671353929720720724005110191889019627759353182087002690569050567759205178329456821397158464781876946977609984501916281146087139432559856082031109051341} a^{7} - \frac{1077903101233440213368649064175968718771009676770230717439194589609248753551259546576687451620880192468703372978247246313958601373039009794023537359763982200}{4026062671353929720720724005110191889019627759353182087002690569050567759205178329456821397158464781876946977609984501916281146087139432559856082031109051341} a^{6} - \frac{329014585266609374689434602968556381081401605330672900484398812358271478542807384465325473318670958845503685158696925157463460768225672133742482712678404612}{4026062671353929720720724005110191889019627759353182087002690569050567759205178329456821397158464781876946977609984501916281146087139432559856082031109051341} a^{5} - \frac{1261089186012255001205514881478188573598753163661957160420826380415629898308827927281281114830905501656163979781134905105176027626479141364149252348745667965}{4026062671353929720720724005110191889019627759353182087002690569050567759205178329456821397158464781876946977609984501916281146087139432559856082031109051341} a^{4} - \frac{378149735088067572715601262863030124572701995529799290991500065176815356818848907867567105886825818348076747209801836276496851900061868949508359647532061725}{4026062671353929720720724005110191889019627759353182087002690569050567759205178329456821397158464781876946977609984501916281146087139432559856082031109051341} a^{3} + \frac{1722901320895628014951632611238579748744355255772400033649378245539696638732235190255758958041470749958957905171622870032758437307914615892194108764707835591}{4026062671353929720720724005110191889019627759353182087002690569050567759205178329456821397158464781876946977609984501916281146087139432559856082031109051341} a^{2} - \frac{1423952810116396284243237260067184100497625730074237591032277102263428997080815998691681186310612682789394312467738490803961587027043209233824839362578990049}{4026062671353929720720724005110191889019627759353182087002690569050567759205178329456821397158464781876946977609984501916281146087139432559856082031109051341} a - \frac{1718510362335763793626865771519278213435868575455280574308152000676103206713973707412756305770742746339616335534227885115611375022068469889075595595085502569}{4026062671353929720720724005110191889019627759353182087002690569050567759205178329456821397158464781876946977609984501916281146087139432559856082031109051341}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $35$
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:  \( 92625495286396430000000000 \) (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

\(\Q(\sqrt{5}) \), 3.3.17689.1, 3.3.17689.2, \(\Q(\zeta_{7})^+\), 3.3.361.1, 4.4.6125.1, 6.6.39112590125.2, 6.6.39112590125.1, 6.6.300125.1, 6.6.16290125.1, 9.9.5534900853769.1, 12.12.9369992576003266033203125.2, 12.12.9369992576003266033203125.1, \(\Q(\zeta_{35})^+\), 12.12.3902537516036345703125.1, 18.18.59834233322368760002940158203125.1

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 ${\href{/LocalNumberField/2.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/3.12.0.1}{12} }^{3}$ R R ${\href{/LocalNumberField/11.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/13.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/17.12.0.1}{12} }^{3}$ R ${\href{/LocalNumberField/23.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/43.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/47.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/53.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{12}$

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
$5$5.12.9.1$x^{12} - 10 x^{8} - 375 x^{4} - 2000$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
5.12.9.1$x^{12} - 10 x^{8} - 375 x^{4} - 2000$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
5.12.9.1$x^{12} - 10 x^{8} - 375 x^{4} - 2000$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
$7$7.12.10.5$x^{12} + 56 x^{6} + 1323$$6$$2$$10$$C_{12}$$[\ ]_{6}^{2}$
7.12.10.5$x^{12} + 56 x^{6} + 1323$$6$$2$$10$$C_{12}$$[\ ]_{6}^{2}$
7.12.10.5$x^{12} + 56 x^{6} + 1323$$6$$2$$10$$C_{12}$$[\ ]_{6}^{2}$
$19$19.9.6.2$x^{9} + 228 x^{6} + 16967 x^{3} + 438976$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
19.9.6.2$x^{9} + 228 x^{6} + 16967 x^{3} + 438976$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
19.9.6.2$x^{9} + 228 x^{6} + 16967 x^{3} + 438976$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
19.9.6.2$x^{9} + 228 x^{6} + 16967 x^{3} + 438976$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$