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Properties

Label 36.0.89156224773...9392.1
Degree $36$
Signature $[0, 18]$
Discriminant $2^{36}\cdot 7^{30}\cdot 13^{33}$
Root discriminant $106.27$
Ramified primes $2, 7, 13$
Class number Not computed
Class group Not computed
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![258474853, 0, 3618647942, 0, 19902563681, 0, 60003090875, 0, 113655085362, 0, 145558267218, 0, 131846926939, 0, 86981860147, 0, 42607368076, 0, 15684015256, 0, 4365494770, 0, 919646455, 0, 145978105, 0, 17271618, 0, 1493765, 0, 91364, 0, 3731, 0, 91, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 + 91*x^34 + 3731*x^32 + 91364*x^30 + 1493765*x^28 + 17271618*x^26 + 145978105*x^24 + 919646455*x^22 + 4365494770*x^20 + 15684015256*x^18 + 42607368076*x^16 + 86981860147*x^14 + 131846926939*x^12 + 145558267218*x^10 + 113655085362*x^8 + 60003090875*x^6 + 19902563681*x^4 + 3618647942*x^2 + 258474853)
 
gp: K = bnfinit(x^36 + 91*x^34 + 3731*x^32 + 91364*x^30 + 1493765*x^28 + 17271618*x^26 + 145978105*x^24 + 919646455*x^22 + 4365494770*x^20 + 15684015256*x^18 + 42607368076*x^16 + 86981860147*x^14 + 131846926939*x^12 + 145558267218*x^10 + 113655085362*x^8 + 60003090875*x^6 + 19902563681*x^4 + 3618647942*x^2 + 258474853, 1)
 

Normalized defining polynomial

\( x^{36} + 91 x^{34} + 3731 x^{32} + 91364 x^{30} + 1493765 x^{28} + 17271618 x^{26} + 145978105 x^{24} + 919646455 x^{22} + 4365494770 x^{20} + 15684015256 x^{18} + 42607368076 x^{16} + 86981860147 x^{14} + 131846926939 x^{12} + 145558267218 x^{10} + 113655085362 x^{8} + 60003090875 x^{6} + 19902563681 x^{4} + 3618647942 x^{2} + 258474853 \)

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:  \(8915622477387792008776526420485297744015759789756076873285322012614459392=2^{36}\cdot 7^{30}\cdot 13^{33}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $106.27$
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:  $2, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(364=2^{2}\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{364}(1,·)$, $\chi_{364}(261,·)$, $\chi_{364}(9,·)$, $\chi_{364}(271,·)$, $\chi_{364}(19,·)$, $\chi_{364}(277,·)$, $\chi_{364}(279,·)$, $\chi_{364}(25,·)$, $\chi_{364}(29,·)$, $\chi_{364}(31,·)$, $\chi_{364}(289,·)$, $\chi_{364}(165,·)$, $\chi_{364}(167,·)$, $\chi_{364}(171,·)$, $\chi_{364}(47,·)$, $\chi_{364}(307,·)$, $\chi_{364}(309,·)$, $\chi_{364}(59,·)$, $\chi_{364}(53,·)$, $\chi_{364}(327,·)$, $\chi_{364}(205,·)$, $\chi_{364}(337,·)$, $\chi_{364}(83,·)$, $\chi_{364}(215,·)$, $\chi_{364}(223,·)$, $\chi_{364}(225,·)$, $\chi_{364}(227,·)$, $\chi_{364}(81,·)$, $\chi_{364}(361,·)$, $\chi_{364}(111,·)$, $\chi_{364}(113,·)$, $\chi_{364}(115,·)$, $\chi_{364}(187,·)$, $\chi_{364}(233,·)$, $\chi_{364}(121,·)$, $\chi_{364}(255,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7} a^{6}$, $\frac{1}{7} a^{7}$, $\frac{1}{7} a^{8}$, $\frac{1}{7} a^{9}$, $\frac{1}{7} a^{10}$, $\frac{1}{7} a^{11}$, $\frac{1}{637} a^{12}$, $\frac{1}{637} a^{13}$, $\frac{1}{637} a^{14}$, $\frac{1}{637} a^{15}$, $\frac{1}{637} a^{16}$, $\frac{1}{637} a^{17}$, $\frac{1}{4459} a^{18}$, $\frac{1}{4459} a^{19}$, $\frac{1}{4459} a^{20}$, $\frac{1}{4459} a^{21}$, $\frac{1}{57967} a^{22} - \frac{4}{91} a^{10}$, $\frac{1}{57967} a^{23} - \frac{4}{91} a^{11}$, $\frac{1}{1217307} a^{24} + \frac{1}{173901} a^{22} + \frac{1}{13377} a^{20} + \frac{1}{13377} a^{18} + \frac{1}{1911} a^{16} + \frac{1}{1911} a^{14} + \frac{1}{1911} a^{12} - \frac{17}{273} a^{10} + \frac{1}{21} a^{8} + \frac{1}{21} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{1217307} a^{25} + \frac{1}{173901} a^{23} + \frac{1}{13377} a^{21} + \frac{1}{13377} a^{19} + \frac{1}{1911} a^{17} + \frac{1}{1911} a^{15} + \frac{1}{1911} a^{13} - \frac{17}{273} a^{11} + \frac{1}{21} a^{9} + \frac{1}{21} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{1217307} a^{26} - \frac{1}{3}$, $\frac{1}{1217307} a^{27} - \frac{1}{3} a$, $\frac{1}{1217307} a^{28} - \frac{1}{3} a^{2}$, $\frac{1}{1217307} a^{29} - \frac{1}{3} a^{3}$, $\frac{1}{8521149} a^{30} - \frac{1}{3} a^{4}$, $\frac{1}{8521149} a^{31} - \frac{1}{3} a^{5}$, $\frac{1}{110774937} a^{32} + \frac{6}{57967} a^{20} + \frac{1}{91} a^{8} - \frac{1}{21} a^{6}$, $\frac{1}{110774937} a^{33} + \frac{6}{57967} a^{21} + \frac{1}{91} a^{9} - \frac{1}{21} a^{7}$, $\frac{1}{20050263597} a^{34} - \frac{3}{6683421199} a^{32} + \frac{12}{514109323} a^{30} + \frac{76}{220332567} a^{28} - \frac{18}{73444189} a^{26} + \frac{80}{220332567} a^{24} - \frac{193}{31476081} a^{22} - \frac{370}{31476081} a^{20} + \frac{257}{2421237} a^{18} + \frac{206}{345891} a^{16} - \frac{160}{345891} a^{14} - \frac{58}{345891} a^{12} - \frac{661}{49413} a^{10} + \frac{2911}{49413} a^{8} - \frac{178}{3801} a^{6} - \frac{202}{543} a^{4} - \frac{257}{543} a^{2} - \frac{85}{543}$, $\frac{1}{20050263597} a^{35} - \frac{3}{6683421199} a^{33} + \frac{12}{514109323} a^{31} + \frac{76}{220332567} a^{29} - \frac{18}{73444189} a^{27} + \frac{80}{220332567} a^{25} - \frac{193}{31476081} a^{23} - \frac{370}{31476081} a^{21} + \frac{257}{2421237} a^{19} + \frac{206}{345891} a^{17} - \frac{160}{345891} a^{15} - \frac{58}{345891} a^{13} - \frac{661}{49413} a^{11} + \frac{2911}{49413} a^{9} - \frac{178}{3801} a^{7} - \frac{202}{543} a^{5} - \frac{257}{543} a^{3} - \frac{85}{543} a$

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

$C_3\times C_{12}$ (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

\(\Q(\sqrt{13}) \), 3.3.169.1, 3.3.8281.2, \(\Q(\zeta_{7})^+\), 3.3.8281.1, 4.0.1722448.1, \(\Q(\zeta_{13})^+\), 6.6.891474493.2, 6.6.5274997.1, 6.6.891474493.1, 9.9.567869252041.1, 12.0.863624717099249717248.1, 12.0.2073562945755298571112448.2, 12.0.12269603229321293320192.1, 12.0.2073562945755298571112448.1, 18.18.708478645847689707516501157.1

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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/5.12.0.1}{12} }^{3}$ R ${\href{/LocalNumberField/11.12.0.1}{12} }^{3}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/19.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/31.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/41.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/47.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/59.12.0.1}{12} }^{3}$

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
$2$2.12.12.25$x^{12} - 78 x^{10} - 1621 x^{8} + 460 x^{6} - 1977 x^{4} + 866 x^{2} + 749$$2$$6$$12$$C_{12}$$[2]^{6}$
2.12.12.25$x^{12} - 78 x^{10} - 1621 x^{8} + 460 x^{6} - 1977 x^{4} + 866 x^{2} + 749$$2$$6$$12$$C_{12}$$[2]^{6}$
2.12.12.25$x^{12} - 78 x^{10} - 1621 x^{8} + 460 x^{6} - 1977 x^{4} + 866 x^{2} + 749$$2$$6$$12$$C_{12}$$[2]^{6}$
7Data not computed
$13$13.12.11.1$x^{12} - 13$$12$$1$$11$$C_{12}$$[\ ]_{12}$
13.12.11.1$x^{12} - 13$$12$$1$$11$$C_{12}$$[\ ]_{12}$
13.12.11.1$x^{12} - 13$$12$$1$$11$$C_{12}$$[\ ]_{12}$