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Properties

Label 36.36.4438795073...5625.1
Degree $36$
Signature $[36, 0]$
Discriminant $3^{18}\cdot 5^{18}\cdot 19^{34}$
Root discriminant $62.48$
Ramified primes $3, 5, 19$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_{18}$ (as 36T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 20, -340, -1460, 14445, 26779, -206783, -252736, 1470387, 1424143, -6107727, -5027655, 16278597, 11597407, -29492311, -18102232, 37621312, 19586929, -34467709, -14890129, 22875412, 7985283, -10994500, -3009217, 3797668, 788451, -929619, -141168, 157962, 16794, -18056, -1262, 1316, 54, -55, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - x^35 - 55*x^34 + 54*x^33 + 1316*x^32 - 1262*x^31 - 18056*x^30 + 16794*x^29 + 157962*x^28 - 141168*x^27 - 929619*x^26 + 788451*x^25 + 3797668*x^24 - 3009217*x^23 - 10994500*x^22 + 7985283*x^21 + 22875412*x^20 - 14890129*x^19 - 34467709*x^18 + 19586929*x^17 + 37621312*x^16 - 18102232*x^15 - 29492311*x^14 + 11597407*x^13 + 16278597*x^12 - 5027655*x^11 - 6107727*x^10 + 1424143*x^9 + 1470387*x^8 - 252736*x^7 - 206783*x^6 + 26779*x^5 + 14445*x^4 - 1460*x^3 - 340*x^2 + 20*x + 1)
 
gp: K = bnfinit(x^36 - x^35 - 55*x^34 + 54*x^33 + 1316*x^32 - 1262*x^31 - 18056*x^30 + 16794*x^29 + 157962*x^28 - 141168*x^27 - 929619*x^26 + 788451*x^25 + 3797668*x^24 - 3009217*x^23 - 10994500*x^22 + 7985283*x^21 + 22875412*x^20 - 14890129*x^19 - 34467709*x^18 + 19586929*x^17 + 37621312*x^16 - 18102232*x^15 - 29492311*x^14 + 11597407*x^13 + 16278597*x^12 - 5027655*x^11 - 6107727*x^10 + 1424143*x^9 + 1470387*x^8 - 252736*x^7 - 206783*x^6 + 26779*x^5 + 14445*x^4 - 1460*x^3 - 340*x^2 + 20*x + 1, 1)
 

Normalized defining polynomial

\( x^{36} - x^{35} - 55 x^{34} + 54 x^{33} + 1316 x^{32} - 1262 x^{31} - 18056 x^{30} + 16794 x^{29} + 157962 x^{28} - 141168 x^{27} - 929619 x^{26} + 788451 x^{25} + 3797668 x^{24} - 3009217 x^{23} - 10994500 x^{22} + 7985283 x^{21} + 22875412 x^{20} - 14890129 x^{19} - 34467709 x^{18} + 19586929 x^{17} + 37621312 x^{16} - 18102232 x^{15} - 29492311 x^{14} + 11597407 x^{13} + 16278597 x^{12} - 5027655 x^{11} - 6107727 x^{10} + 1424143 x^{9} + 1470387 x^{8} - 252736 x^{7} - 206783 x^{6} + 26779 x^{5} + 14445 x^{4} - 1460 x^{3} - 340 x^{2} + 20 x + 1 \)

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:  \(44387950739803436916061690678602018428037077267652774810791015625=3^{18}\cdot 5^{18}\cdot 19^{34}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $62.48$
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:  $3, 5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(285=3\cdot 5\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{285}(256,·)$, $\chi_{285}(1,·)$, $\chi_{285}(4,·)$, $\chi_{285}(86,·)$, $\chi_{285}(139,·)$, $\chi_{285}(269,·)$, $\chi_{285}(14,·)$, $\chi_{285}(271,·)$, $\chi_{285}(16,·)$, $\chi_{285}(146,·)$, $\chi_{285}(281,·)$, $\chi_{285}(284,·)$, $\chi_{285}(29,·)$, $\chi_{285}(164,·)$, $\chi_{285}(41,·)$, $\chi_{285}(71,·)$, $\chi_{285}(49,·)$, $\chi_{285}(179,·)$, $\chi_{285}(56,·)$, $\chi_{285}(116,·)$, $\chi_{285}(59,·)$, $\chi_{285}(61,·)$, $\chi_{285}(64,·)$, $\chi_{285}(196,·)$, $\chi_{285}(199,·)$, $\chi_{285}(214,·)$, $\chi_{285}(89,·)$, $\chi_{285}(221,·)$, $\chi_{285}(224,·)$, $\chi_{285}(226,·)$, $\chi_{285}(229,·)$, $\chi_{285}(106,·)$, $\chi_{285}(236,·)$, $\chi_{285}(244,·)$, $\chi_{285}(169,·)$, $\chi_{285}(121,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $\frac{1}{229} a^{33} + \frac{91}{229} a^{32} + \frac{73}{229} a^{31} - \frac{39}{229} a^{30} + \frac{73}{229} a^{29} + \frac{60}{229} a^{28} - \frac{30}{229} a^{27} - \frac{62}{229} a^{26} - \frac{31}{229} a^{25} + \frac{23}{229} a^{24} + \frac{58}{229} a^{23} + \frac{14}{229} a^{22} + \frac{104}{229} a^{21} - \frac{104}{229} a^{20} + \frac{7}{229} a^{19} - \frac{58}{229} a^{18} - \frac{86}{229} a^{17} - \frac{19}{229} a^{16} - \frac{22}{229} a^{15} - \frac{70}{229} a^{14} + \frac{5}{229} a^{13} + \frac{23}{229} a^{12} + \frac{48}{229} a^{11} + \frac{72}{229} a^{10} - \frac{85}{229} a^{9} - \frac{43}{229} a^{8} - \frac{90}{229} a^{7} + \frac{38}{229} a^{6} - \frac{65}{229} a^{5} + \frac{60}{229} a^{4} + \frac{56}{229} a^{3} + \frac{28}{229} a^{2} + \frac{71}{229} a + \frac{15}{229}$, $\frac{1}{229} a^{34} + \frac{36}{229} a^{32} - \frac{41}{229} a^{31} - \frac{42}{229} a^{30} + \frac{58}{229} a^{29} + \frac{6}{229} a^{28} - \frac{80}{229} a^{27} - \frac{114}{229} a^{26} + \frac{96}{229} a^{25} + \frac{26}{229} a^{24} + \frac{3}{229} a^{23} - \frac{25}{229} a^{22} + \frac{50}{229} a^{21} + \frac{82}{229} a^{20} - \frac{8}{229} a^{19} - \frac{75}{229} a^{18} + \frac{21}{229} a^{17} + \frac{104}{229} a^{16} + \frac{100}{229} a^{15} - \frac{37}{229} a^{14} + \frac{26}{229} a^{13} + \frac{16}{229} a^{12} + \frac{55}{229} a^{11} + \frac{4}{229} a^{10} - \frac{94}{229} a^{9} - \frac{70}{229} a^{8} - \frac{16}{229} a^{7} - \frac{88}{229} a^{6} + \frac{21}{229} a^{5} + \frac{92}{229} a^{4} - \frac{30}{229} a^{3} + \frac{42}{229} a^{2} - \frac{34}{229} a + \frac{9}{229}$, $\frac{1}{441740965663486218662287302998401975424365701930646103174713} a^{35} + \frac{357595585314283523591323451245890101023030415391987307200}{441740965663486218662287302998401975424365701930646103174713} a^{34} - \frac{94111012951706176219291454608164720680340498265650698086}{441740965663486218662287302998401975424365701930646103174713} a^{33} + \frac{31534299450091099861835309367928985581200338884495074418720}{441740965663486218662287302998401975424365701930646103174713} a^{32} + \frac{186776652920464282901399911333288274120692099374664522112865}{441740965663486218662287302998401975424365701930646103174713} a^{31} - \frac{95392592962537822627342846066355354573607242462356573706385}{441740965663486218662287302998401975424365701930646103174713} a^{30} + \frac{188582167313035840459179668623583819521282149962144406824740}{441740965663486218662287302998401975424365701930646103174713} a^{29} + \frac{104541034748463702696217527874242463082095951462416995047411}{441740965663486218662287302998401975424365701930646103174713} a^{28} + \frac{62396213494574351753814279967753208830275525396011703147277}{441740965663486218662287302998401975424365701930646103174713} a^{27} - \frac{15099485814470154362504447992816097221137249313492907648137}{441740965663486218662287302998401975424365701930646103174713} a^{26} - \frac{62007554570502671187737297350478749107978379168465485543747}{441740965663486218662287302998401975424365701930646103174713} a^{25} - \frac{151926908726739423829058201293441729572623937156095067997904}{441740965663486218662287302998401975424365701930646103174713} a^{24} - \frac{25650382700087484460317016142575425136260552000768825057742}{441740965663486218662287302998401975424365701930646103174713} a^{23} + \frac{8670115152629109058766587941647751193920992139014276868570}{441740965663486218662287302998401975424365701930646103174713} a^{22} - \frac{135797387437641697036351524098037991507121134757122231147571}{441740965663486218662287302998401975424365701930646103174713} a^{21} - \frac{121058526373069653228694515731132470427750128054691702331663}{441740965663486218662287302998401975424365701930646103174713} a^{20} - \frac{77182864840896914036580098186678011866489710004900067287514}{441740965663486218662287302998401975424365701930646103174713} a^{19} - \frac{175505519112945877699778066809722448905951901805436473722586}{441740965663486218662287302998401975424365701930646103174713} a^{18} - \frac{57703247960868025806864646499904164406958087605526428676575}{441740965663486218662287302998401975424365701930646103174713} a^{17} - \frac{194269433050581122820665664265420256783961211877730611435127}{441740965663486218662287302998401975424365701930646103174713} a^{16} + \frac{219502153449508094597671318115930083471489975957147966379311}{441740965663486218662287302998401975424365701930646103174713} a^{15} - \frac{151266686686050599742491277289492826987984305793974173624131}{441740965663486218662287302998401975424365701930646103174713} a^{14} + \frac{3208277628672821374942148477608668444259182239058753038614}{441740965663486218662287302998401975424365701930646103174713} a^{13} + \frac{119665018566510022312498550374317295428560640092223015628309}{441740965663486218662287302998401975424365701930646103174713} a^{12} + \frac{39893887416477180610787461230946552703465391477855234134140}{441740965663486218662287302998401975424365701930646103174713} a^{11} - \frac{181662172802038593113230274627003329019560885423254169591926}{441740965663486218662287302998401975424365701930646103174713} a^{10} - \frac{90326223578878651225277250991394786691857820851639935267202}{441740965663486218662287302998401975424365701930646103174713} a^{9} - \frac{179272789193083403132327587300236935743904643859389635355889}{441740965663486218662287302998401975424365701930646103174713} a^{8} - \frac{178124971446421370943302552805252113412343012193860651340700}{441740965663486218662287302998401975424365701930646103174713} a^{7} - \frac{67719797599206296583153368398395141732728831222271946146538}{441740965663486218662287302998401975424365701930646103174713} a^{6} + \frac{55165792419466959479704101582569941665968697690926882021027}{441740965663486218662287302998401975424365701930646103174713} a^{5} - \frac{51465552479406017519341929321194670742359614378815539105513}{441740965663486218662287302998401975424365701930646103174713} a^{4} - \frac{156842035679304892105137195140783928498383407091866892474514}{441740965663486218662287302998401975424365701930646103174713} a^{3} - \frac{73895714639313847079535252078815804904091412785923539498052}{441740965663486218662287302998401975424365701930646103174713} a^{2} + \frac{76171081353096991602199208274581985037070275077740092910833}{441740965663486218662287302998401975424365701930646103174713} a - \frac{188150945813755343231925846108372910072924874142424466967563}{441740965663486218662287302998401975424365701930646103174713}$

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:  \( 736998020653083800000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{18}$ (as 36T2):

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 $C_2\times C_{18}$
Character table for $C_2\times C_{18}$ is not computed

Intermediate fields

\(\Q(\sqrt{57}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{285}) \), 3.3.361.1, \(\Q(\sqrt{5}, \sqrt{57})\), 6.6.66854673.1, 6.6.16290125.1, 6.6.8356834125.1, \(\Q(\zeta_{19})^+\), 12.12.69836676592764515625.1, \(\Q(\zeta_{57})^+\), 18.18.563362135874260093126953125.1, 18.18.210684481487848166847338548828125.1

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 $18^{2}$ R R ${\href{/LocalNumberField/7.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ $18^{2}$ $18^{2}$ R $18^{2}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{18}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{4}$ $18^{2}$ $18^{2}$ $18^{2}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{4}$

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
3Data not computed
5Data not computed
19Data not computed