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Properties

Label 36.36.4259760591...0000.1
Degree $36$
Signature $[36, 0]$
Discriminant $2^{36}\cdot 5^{27}\cdot 19^{32}$
Root discriminant $91.61$
Ramified primes $2, 5, 19$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{36}$ (as 36T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -120, 2410, -4420, -182615, 717016, 3438912, -13608038, -28450501, 106401116, 116712776, -434207956, -250289952, 1018460166, 278371447, -1451486164, -130919914, 1309519986, -31268380, -772688098, 74837468, 305983704, -44357561, -82755920, 14704059, 15403666, -3089535, -1964424, 426285, 168076, -38498, -9196, 2189, 290, -71, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 4*x^35 - 71*x^34 + 290*x^33 + 2189*x^32 - 9196*x^31 - 38498*x^30 + 168076*x^29 + 426285*x^28 - 1964424*x^27 - 3089535*x^26 + 15403666*x^25 + 14704059*x^24 - 82755920*x^23 - 44357561*x^22 + 305983704*x^21 + 74837468*x^20 - 772688098*x^19 - 31268380*x^18 + 1309519986*x^17 - 130919914*x^16 - 1451486164*x^15 + 278371447*x^14 + 1018460166*x^13 - 250289952*x^12 - 434207956*x^11 + 116712776*x^10 + 106401116*x^9 - 28450501*x^8 - 13608038*x^7 + 3438912*x^6 + 717016*x^5 - 182615*x^4 - 4420*x^3 + 2410*x^2 - 120*x + 1)
 
gp: K = bnfinit(x^36 - 4*x^35 - 71*x^34 + 290*x^33 + 2189*x^32 - 9196*x^31 - 38498*x^30 + 168076*x^29 + 426285*x^28 - 1964424*x^27 - 3089535*x^26 + 15403666*x^25 + 14704059*x^24 - 82755920*x^23 - 44357561*x^22 + 305983704*x^21 + 74837468*x^20 - 772688098*x^19 - 31268380*x^18 + 1309519986*x^17 - 130919914*x^16 - 1451486164*x^15 + 278371447*x^14 + 1018460166*x^13 - 250289952*x^12 - 434207956*x^11 + 116712776*x^10 + 106401116*x^9 - 28450501*x^8 - 13608038*x^7 + 3438912*x^6 + 717016*x^5 - 182615*x^4 - 4420*x^3 + 2410*x^2 - 120*x + 1, 1)
 

Normalized defining polynomial

\( x^{36} - 4 x^{35} - 71 x^{34} + 290 x^{33} + 2189 x^{32} - 9196 x^{31} - 38498 x^{30} + 168076 x^{29} + 426285 x^{28} - 1964424 x^{27} - 3089535 x^{26} + 15403666 x^{25} + 14704059 x^{24} - 82755920 x^{23} - 44357561 x^{22} + 305983704 x^{21} + 74837468 x^{20} - 772688098 x^{19} - 31268380 x^{18} + 1309519986 x^{17} - 130919914 x^{16} - 1451486164 x^{15} + 278371447 x^{14} + 1018460166 x^{13} - 250289952 x^{12} - 434207956 x^{11} + 116712776 x^{10} + 106401116 x^{9} - 28450501 x^{8} - 13608038 x^{7} + 3438912 x^{6} + 717016 x^{5} - 182615 x^{4} - 4420 x^{3} + 2410 x^{2} - 120 x + 1 \)

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:  \(42597605919174385987291615490837462485893632000000000000000000000000000=2^{36}\cdot 5^{27}\cdot 19^{32}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $91.61$
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:  $2, 5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(380=2^{2}\cdot 5\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{380}(1,·)$, $\chi_{380}(7,·)$, $\chi_{380}(9,·)$, $\chi_{380}(267,·)$, $\chi_{380}(149,·)$, $\chi_{380}(23,·)$, $\chi_{380}(283,·)$, $\chi_{380}(161,·)$, $\chi_{380}(163,·)$, $\chi_{380}(169,·)$, $\chi_{380}(43,·)$, $\chi_{380}(301,·)$, $\chi_{380}(47,·)$, $\chi_{380}(49,·)$, $\chi_{380}(309,·)$, $\chi_{380}(329,·)$, $\chi_{380}(187,·)$, $\chi_{380}(61,·)$, $\chi_{380}(63,·)$, $\chi_{380}(321,·)$, $\chi_{380}(343,·)$, $\chi_{380}(327,·)$, $\chi_{380}(201,·)$, $\chi_{380}(207,·)$, $\chi_{380}(81,·)$, $\chi_{380}(83,·)$, $\chi_{380}(87,·)$, $\chi_{380}(347,·)$, $\chi_{380}(349,·)$, $\chi_{380}(101,·)$, $\chi_{380}(263,·)$, $\chi_{380}(229,·)$, $\chi_{380}(289,·)$, $\chi_{380}(367,·)$, $\chi_{380}(121,·)$, $\chi_{380}(123,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $\frac{1}{13681} a^{33} + \frac{5789}{13681} a^{32} - \frac{742}{13681} a^{31} + \frac{382}{13681} a^{30} + \frac{4542}{13681} a^{29} + \frac{5418}{13681} a^{28} - \frac{586}{13681} a^{27} + \frac{1417}{13681} a^{26} + \frac{3953}{13681} a^{25} - \frac{474}{13681} a^{24} - \frac{2738}{13681} a^{23} + \frac{5263}{13681} a^{22} + \frac{411}{13681} a^{21} + \frac{2191}{13681} a^{20} - \frac{692}{13681} a^{19} - \frac{3393}{13681} a^{18} - \frac{5018}{13681} a^{17} - \frac{1966}{13681} a^{16} + \frac{4968}{13681} a^{15} + \frac{5175}{13681} a^{14} - \frac{3658}{13681} a^{13} + \frac{2195}{13681} a^{12} + \frac{2123}{13681} a^{11} - \frac{498}{13681} a^{10} + \frac{3139}{13681} a^{9} - \frac{1726}{13681} a^{8} + \frac{5239}{13681} a^{7} + \frac{315}{13681} a^{6} + \frac{1160}{13681} a^{5} - \frac{5459}{13681} a^{4} - \frac{1946}{13681} a^{3} + \frac{1374}{13681} a^{2} - \frac{1505}{13681} a - \frac{2706}{13681}$, $\frac{1}{25713744943186256097910005723781} a^{34} - \frac{247080074616014414844338343}{25713744943186256097910005723781} a^{33} - \frac{11774278044537357547497560682896}{25713744943186256097910005723781} a^{32} + \frac{6223173156447996207179010336848}{25713744943186256097910005723781} a^{31} + \frac{7910313782640148266671327662000}{25713744943186256097910005723781} a^{30} + \frac{4769046438927425297675627207793}{25713744943186256097910005723781} a^{29} + \frac{12088164207156548753657612952210}{25713744943186256097910005723781} a^{28} - \frac{7395967200980431472255066148054}{25713744943186256097910005723781} a^{27} - \frac{8059343399888922222161596670811}{25713744943186256097910005723781} a^{26} + \frac{3718799816268604907716414127436}{25713744943186256097910005723781} a^{25} - \frac{10469434354987762886251151866923}{25713744943186256097910005723781} a^{24} - \frac{4892354693334994176042108361442}{25713744943186256097910005723781} a^{23} + \frac{2958988732326297685393392036307}{25713744943186256097910005723781} a^{22} + \frac{2051891521899202430873331282806}{25713744943186256097910005723781} a^{21} + \frac{1303176327765259739033783422282}{25713744943186256097910005723781} a^{20} - \frac{5270204997513703043224427394268}{25713744943186256097910005723781} a^{19} + \frac{6995965764431455786058627869809}{25713744943186256097910005723781} a^{18} - \frac{10402818553433787793381118133634}{25713744943186256097910005723781} a^{17} + \frac{4668933329807390275927067234431}{25713744943186256097910005723781} a^{16} - \frac{1980682904152126214234201514144}{25713744943186256097910005723781} a^{15} - \frac{8518487861880714927250741188653}{25713744943186256097910005723781} a^{14} + \frac{909291239869533626717340686907}{25713744943186256097910005723781} a^{13} + \frac{5732602639132211782279677313168}{25713744943186256097910005723781} a^{12} - \frac{12191211760044919253644032174557}{25713744943186256097910005723781} a^{11} + \frac{3293119690492673248152211655015}{25713744943186256097910005723781} a^{10} - \frac{2344972229807358608964523438140}{25713744943186256097910005723781} a^{9} + \frac{3591013941341453439341808084830}{25713744943186256097910005723781} a^{8} - \frac{2628317000281901701364342802877}{25713744943186256097910005723781} a^{7} - \frac{3275212441907796118021899400164}{25713744943186256097910005723781} a^{6} + \frac{10620421675483182861880014927374}{25713744943186256097910005723781} a^{5} - \frac{12310558891549680915334118963428}{25713744943186256097910005723781} a^{4} - \frac{7387181114946123230686209269000}{25713744943186256097910005723781} a^{3} + \frac{7544613927170392881144800244946}{25713744943186256097910005723781} a^{2} + \frac{359320245314112783877086018945}{25713744943186256097910005723781} a + \frac{12286036339294229397482884411156}{25713744943186256097910005723781}$, $\frac{1}{2524734846731746993693515162899957161790084429652148117441566438960756557070979} a^{35} - \frac{5442698871221067407943322788879035611326680316}{2524734846731746993693515162899957161790084429652148117441566438960756557070979} a^{34} - \frac{56617646962431320650098890613405786240376027123880216032883269221895446027}{2524734846731746993693515162899957161790084429652148117441566438960756557070979} a^{33} - \frac{1102989424142069096867626563500177293036583877316122428437913333045753213121630}{2524734846731746993693515162899957161790084429652148117441566438960756557070979} a^{32} + \frac{738497677909045999218620686486832523516904141330210343848385361242289267130110}{2524734846731746993693515162899957161790084429652148117441566438960756557070979} a^{31} + \frac{123380599393565302531652915709213594964914913851819236453111660710512172731592}{2524734846731746993693515162899957161790084429652148117441566438960756557070979} a^{30} + \frac{261746176365162902536852233688272410254109393868658249580459747965938824760685}{2524734846731746993693515162899957161790084429652148117441566438960756557070979} a^{29} - \frac{131867229434315689609879906982729887408564969705889023966883284529451966148420}{2524734846731746993693515162899957161790084429652148117441566438960756557070979} a^{28} + \frac{892325140942170029999672342564918509032424444070571681600007679717936910015806}{2524734846731746993693515162899957161790084429652148117441566438960756557070979} a^{27} + \frac{572866490880754824271908378311327423806996647368054854565019844387019721775917}{2524734846731746993693515162899957161790084429652148117441566438960756557070979} a^{26} + \frac{192393402808283946371414932528418107425389157496109994157117439054743677956205}{2524734846731746993693515162899957161790084429652148117441566438960756557070979} a^{25} + \frac{226310385346519111046105961343005675704919065403143362739295761230953338687508}{2524734846731746993693515162899957161790084429652148117441566438960756557070979} a^{24} + \frac{154605391019662691600209632647936889597668890400869371680122844167462816953913}{2524734846731746993693515162899957161790084429652148117441566438960756557070979} a^{23} + \frac{381006207953609170850720251366173042993772654978720199257632924396422124340506}{2524734846731746993693515162899957161790084429652148117441566438960756557070979} a^{22} + \frac{214812415533268133472046247861463840019169668287765509962352843715882961372877}{2524734846731746993693515162899957161790084429652148117441566438960756557070979} a^{21} - \frac{712346336972290098943139481502360282749525283193590077909179344622588491625264}{2524734846731746993693515162899957161790084429652148117441566438960756557070979} a^{20} + \frac{772999168785144866701500585807228731141592710668306292349486450278721801209728}{2524734846731746993693515162899957161790084429652148117441566438960756557070979} a^{19} - \frac{131122231580607732273295486666706237171055971298492020595805449927812364983946}{2524734846731746993693515162899957161790084429652148117441566438960756557070979} a^{18} + \frac{603822472229433349667235894218231995988625087995409062474132880123739821838466}{2524734846731746993693515162899957161790084429652148117441566438960756557070979} a^{17} - \frac{1053156857181602852766940754565048747704422246025444356171294506916742365917788}{2524734846731746993693515162899957161790084429652148117441566438960756557070979} a^{16} + \frac{1156120680687514146185982959174353455448024880008395248504490431787005342498937}{2524734846731746993693515162899957161790084429652148117441566438960756557070979} a^{15} - \frac{1138882020439815281426498649999737071791187322540164774685368346290120716567493}{2524734846731746993693515162899957161790084429652148117441566438960756557070979} a^{14} - \frac{597581597818393333611480863440218662720856362282997468055765283543103603294895}{2524734846731746993693515162899957161790084429652148117441566438960756557070979} a^{13} - \frac{1136983136412242192875507555932097150547753538502889924593955187066006804296060}{2524734846731746993693515162899957161790084429652148117441566438960756557070979} a^{12} - \frac{43578213773338942333420223703445910247275309471885148548589778841534716289253}{2524734846731746993693515162899957161790084429652148117441566438960756557070979} a^{11} + \frac{16505723137946560013390442302588086135455893466728065804154762797467339887485}{2524734846731746993693515162899957161790084429652148117441566438960756557070979} a^{10} - \frac{851356666325937805522435225857222428755821269615824705471483426127674489916485}{2524734846731746993693515162899957161790084429652148117441566438960756557070979} a^{9} + \frac{76147359370593662859393303149726027177647797297952840183789422660981642144845}{2524734846731746993693515162899957161790084429652148117441566438960756557070979} a^{8} + \frac{570278525109231839956526407981646890359242141635496797236705578697030005093092}{2524734846731746993693515162899957161790084429652148117441566438960756557070979} a^{7} + \frac{740608022069281235085406935284681937795040746456823544285394647219143838834886}{2524734846731746993693515162899957161790084429652148117441566438960756557070979} a^{6} + \frac{961413561091880342404900978168161701199491158552755604102283870334219371639553}{2524734846731746993693515162899957161790084429652148117441566438960756557070979} a^{5} + \frac{811147976204539253394792836213018134666055552625091831626263570390259502254802}{2524734846731746993693515162899957161790084429652148117441566438960756557070979} a^{4} + \frac{507001896371577019852897731316375286147232609566718302137196251454451745731229}{2524734846731746993693515162899957161790084429652148117441566438960756557070979} a^{3} + \frac{906418155958064550923482305362125324214532869463113889372050464875682477148530}{2524734846731746993693515162899957161790084429652148117441566438960756557070979} a^{2} - \frac{125426852915556802239607487913056222397338075920841082525250067804884283901252}{2524734846731746993693515162899957161790084429652148117441566438960756557070979} a - \frac{985074316210443240495752341732595573939877025568454563041499691254187733806647}{2524734846731746993693515162899957161790084429652148117441566438960756557070979}$

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:  \( 747855044096060300000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{36}$ (as 36T1):

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

Intermediate fields

\(\Q(\sqrt{5}) \), 3.3.361.1, \(\Q(\zeta_{20})^+\), 6.6.16290125.1, \(\Q(\zeta_{19})^+\), 12.12.135868504328000000000.1, 18.18.563362135874260093126953125.1

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 R $36$ R ${\href{/LocalNumberField/7.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ $36$ $36$ R $36$ $18^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{9}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{4}$ $36$ $36$ $36$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{4}$

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
2Data not computed
5Data not computed
$19$19.9.8.8$x^{9} - 19$$9$$1$$8$$C_9$$[\ ]_{9}$
19.9.8.8$x^{9} - 19$$9$$1$$8$$C_9$$[\ ]_{9}$
19.9.8.8$x^{9} - 19$$9$$1$$8$$C_9$$[\ ]_{9}$
19.9.8.8$x^{9} - 19$$9$$1$$8$$C_9$$[\ ]_{9}$