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Properties

Label 36.0.19036714782...6048.1
Degree $36$
Signature $[0, 18]$
Discriminant $2^{99}\cdot 19^{34}$
Root discriminant $108.53$
Ramified primes $2, 19$
Class number Not computed
Class group Not computed
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![184832, 0, 5544960, 0, 57390336, 0, 304972800, 0, 974896384, 0, 2040175616, 0, 2943264768, 0, 3025838464, 0, 2263903200, 0, 1247889600, 0, 509408848, 0, 153873248, 0, 34166712, 0, 5506048, 0, 630800, 0, 49704, 0, 2546, 0, 76, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 + 76*x^34 + 2546*x^32 + 49704*x^30 + 630800*x^28 + 5506048*x^26 + 34166712*x^24 + 153873248*x^22 + 509408848*x^20 + 1247889600*x^18 + 2263903200*x^16 + 3025838464*x^14 + 2943264768*x^12 + 2040175616*x^10 + 974896384*x^8 + 304972800*x^6 + 57390336*x^4 + 5544960*x^2 + 184832)
 
gp: K = bnfinit(x^36 + 76*x^34 + 2546*x^32 + 49704*x^30 + 630800*x^28 + 5506048*x^26 + 34166712*x^24 + 153873248*x^22 + 509408848*x^20 + 1247889600*x^18 + 2263903200*x^16 + 3025838464*x^14 + 2943264768*x^12 + 2040175616*x^10 + 974896384*x^8 + 304972800*x^6 + 57390336*x^4 + 5544960*x^2 + 184832, 1)
 

Normalized defining polynomial

\( x^{36} + 76 x^{34} + 2546 x^{32} + 49704 x^{30} + 630800 x^{28} + 5506048 x^{26} + 34166712 x^{24} + 153873248 x^{22} + 509408848 x^{20} + 1247889600 x^{18} + 2263903200 x^{16} + 3025838464 x^{14} + 2943264768 x^{12} + 2040175616 x^{10} + 974896384 x^{8} + 304972800 x^{6} + 57390336 x^{4} + 5544960 x^{2} + 184832 \)

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:  \(19036714782161565107424425435655777110146017378670996611401194085493506048=2^{99}\cdot 19^{34}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $108.53$
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:  $2, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(304=2^{4}\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{304}(1,·)$, $\chi_{304}(261,·)$, $\chi_{304}(9,·)$, $\chi_{304}(13,·)$, $\chi_{304}(17,·)$, $\chi_{304}(269,·)$, $\chi_{304}(21,·)$, $\chi_{304}(73,·)$, $\chi_{304}(25,·)$, $\chi_{304}(153,·)$, $\chi_{304}(29,·)$, $\chi_{304}(161,·)$, $\chi_{304}(37,·)$, $\chi_{304}(49,·)$, $\chi_{304}(169,·)$, $\chi_{304}(173,·)$, $\chi_{304}(177,·)$, $\chi_{304}(53,·)$, $\chi_{304}(137,·)$, $\chi_{304}(189,·)$, $\chi_{304}(181,·)$, $\chi_{304}(69,·)$, $\chi_{304}(289,·)$, $\chi_{304}(201,·)$, $\chi_{304}(205,·)$, $\chi_{304}(141,·)$, $\chi_{304}(81,·)$, $\chi_{304}(221,·)$, $\chi_{304}(165,·)$, $\chi_{304}(225,·)$, $\chi_{304}(293,·)$, $\chi_{304}(273,·)$, $\chi_{304}(233,·)$, $\chi_{304}(109,·)$, $\chi_{304}(117,·)$, $\chi_{304}(121,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{4} a^{8}$, $\frac{1}{4} a^{9}$, $\frac{1}{4} a^{10}$, $\frac{1}{4} a^{11}$, $\frac{1}{8} a^{12}$, $\frac{1}{8} a^{13}$, $\frac{1}{8} a^{14}$, $\frac{1}{8} a^{15}$, $\frac{1}{16} a^{16}$, $\frac{1}{16} a^{17}$, $\frac{1}{304} a^{18}$, $\frac{1}{304} a^{19}$, $\frac{1}{608} a^{20}$, $\frac{1}{608} a^{21}$, $\frac{1}{608} a^{22}$, $\frac{1}{608} a^{23}$, $\frac{1}{1216} a^{24}$, $\frac{1}{1216} a^{25}$, $\frac{1}{1216} a^{26}$, $\frac{1}{1216} a^{27}$, $\frac{1}{2432} a^{28}$, $\frac{1}{2432} a^{29}$, $\frac{1}{274816} a^{30} - \frac{9}{274816} a^{28} + \frac{1}{68704} a^{26} + \frac{53}{137408} a^{24} + \frac{3}{34352} a^{22} - \frac{37}{68704} a^{20} + \frac{1}{1808} a^{18} - \frac{9}{1808} a^{16} - \frac{13}{226} a^{14} - \frac{33}{904} a^{12} - \frac{13}{113} a^{10} - \frac{1}{226} a^{8} - \frac{53}{226} a^{6} - \frac{53}{226} a^{4} + \frac{8}{113} a^{2} - \frac{31}{113}$, $\frac{1}{274816} a^{31} - \frac{9}{274816} a^{29} + \frac{1}{68704} a^{27} + \frac{53}{137408} a^{25} + \frac{3}{34352} a^{23} - \frac{37}{68704} a^{21} + \frac{1}{1808} a^{19} - \frac{9}{1808} a^{17} - \frac{13}{226} a^{15} - \frac{33}{904} a^{13} - \frac{13}{113} a^{11} - \frac{1}{226} a^{9} - \frac{53}{226} a^{7} - \frac{53}{226} a^{5} + \frac{8}{113} a^{3} - \frac{31}{113} a$, $\frac{1}{549632} a^{32} + \frac{9}{137408} a^{28} - \frac{21}{137408} a^{26} - \frac{1}{3616} a^{24} - \frac{3}{4294} a^{22} + \frac{11}{34352} a^{20} - \frac{9}{452} a^{16} + \frac{4}{113} a^{14} - \frac{31}{904} a^{12} - \frac{9}{452} a^{10} + \frac{51}{452} a^{8} - \frac{39}{226} a^{6} + \frac{26}{113} a^{4} - \frac{36}{113} a^{2} + \frac{30}{113}$, $\frac{1}{549632} a^{33} + \frac{9}{137408} a^{29} - \frac{21}{137408} a^{27} - \frac{1}{3616} a^{25} - \frac{3}{4294} a^{23} + \frac{11}{34352} a^{21} - \frac{9}{452} a^{17} + \frac{4}{113} a^{15} - \frac{31}{904} a^{13} - \frac{9}{452} a^{11} + \frac{51}{452} a^{9} - \frac{39}{226} a^{7} + \frac{26}{113} a^{5} - \frac{36}{113} a^{3} + \frac{30}{113} a$, $\frac{1}{339323332987860253952} a^{34} + \frac{14217559119613}{84830833246965063488} a^{32} - \frac{17787852072535}{169661666493930126976} a^{30} - \frac{3614511314850045}{84830833246965063488} a^{28} + \frac{12645366535069587}{42415416623482531744} a^{26} - \frac{12584017317739045}{42415416623482531744} a^{24} + \frac{3962521614519619}{5301927077935316468} a^{22} - \frac{2546079326769937}{42415416623482531744} a^{20} + \frac{3162316844394401}{5301927077935316468} a^{18} - \frac{5075619312631185}{558097587151085944} a^{16} + \frac{7435642064240857}{558097587151085944} a^{14} - \frac{27472035453660613}{558097587151085944} a^{12} + \frac{17617232076941885}{279048793575542972} a^{10} - \frac{32336573929162137}{279048793575542972} a^{8} + \frac{15238149087898669}{139524396787771486} a^{6} - \frac{3404210171901985}{69762198393885743} a^{4} - \frac{30482966091161973}{69762198393885743} a^{2} - \frac{13401713776841009}{69762198393885743}$, $\frac{1}{339323332987860253952} a^{35} + \frac{14217559119613}{84830833246965063488} a^{33} - \frac{17787852072535}{169661666493930126976} a^{31} - \frac{3614511314850045}{84830833246965063488} a^{29} + \frac{12645366535069587}{42415416623482531744} a^{27} - \frac{12584017317739045}{42415416623482531744} a^{25} + \frac{3962521614519619}{5301927077935316468} a^{23} - \frac{2546079326769937}{42415416623482531744} a^{21} + \frac{3162316844394401}{5301927077935316468} a^{19} - \frac{5075619312631185}{558097587151085944} a^{17} + \frac{7435642064240857}{558097587151085944} a^{15} - \frac{27472035453660613}{558097587151085944} a^{13} + \frac{17617232076941885}{279048793575542972} a^{11} - \frac{32336573929162137}{279048793575542972} a^{9} + \frac{15238149087898669}{139524396787771486} a^{7} - \frac{3404210171901985}{69762198393885743} a^{5} - \frac{30482966091161973}{69762198393885743} a^{3} - \frac{13401713776841009}{69762198393885743} a$

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

Class group and class number

Not computed

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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
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

\(\Q(\sqrt{2}) \), 3.3.361.1, 4.0.739328.2, 6.6.66724352.1, \(\Q(\zeta_{19})^+\), 12.0.52665458133728799752192.46, 18.18.38713951190154487490850848768.1

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 $36$ $36$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/11.12.0.1}{12} }^{3}$ $36$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{4}$ R $18^{2}$ $36$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{9}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{4}$ $36$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{4}$ $36$ $36$

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
19Data not computed