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Properties

Label 18.0.17957500407...5632.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 547^{9}$
Root discriminant $37.13$
Ramified primes $2, 547$
Class number $9$ (GRH)
Class group $[3, 3]$ (GRH)
Galois group $C_3^2 : C_2$ (as 18T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11025, 14385, -27707, 11642, 33052, -52450, 33492, -27838, 28227, -20783, 12609, -7174, 3670, -1540, 548, -172, 47, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 47*x^16 - 172*x^15 + 548*x^14 - 1540*x^13 + 3670*x^12 - 7174*x^11 + 12609*x^10 - 20783*x^9 + 28227*x^8 - 27838*x^7 + 33492*x^6 - 52450*x^5 + 33052*x^4 + 11642*x^3 - 27707*x^2 + 14385*x + 11025)
 
gp: K = bnfinit(x^18 - 9*x^17 + 47*x^16 - 172*x^15 + 548*x^14 - 1540*x^13 + 3670*x^12 - 7174*x^11 + 12609*x^10 - 20783*x^9 + 28227*x^8 - 27838*x^7 + 33492*x^6 - 52450*x^5 + 33052*x^4 + 11642*x^3 - 27707*x^2 + 14385*x + 11025, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 47 x^{16} - 172 x^{15} + 548 x^{14} - 1540 x^{13} + 3670 x^{12} - 7174 x^{11} + 12609 x^{10} - 20783 x^{9} + 28227 x^{8} - 27838 x^{7} + 33492 x^{6} - 52450 x^{5} + 33052 x^{4} + 11642 x^{3} - 27707 x^{2} + 14385 x + 11025 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-17957500407834968547439685632=-\,2^{12}\cdot 547^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.13$
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:  $2, 547$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{6} a^{8} - \frac{1}{6} a^{7} + \frac{1}{6} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{6} a^{2} - \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{5} - \frac{1}{6} a$, $\frac{1}{12} a^{10} - \frac{1}{12} a^{9} - \frac{1}{12} a^{8} - \frac{1}{6} a^{7} - \frac{1}{12} a^{6} - \frac{5}{12} a^{5} - \frac{1}{12} a^{4} + \frac{1}{3} a^{3} + \frac{1}{12} a^{2} + \frac{5}{12} a - \frac{1}{4}$, $\frac{1}{12} a^{11} - \frac{1}{12} a^{8} + \frac{1}{12} a^{7} + \frac{1}{6} a^{6} + \frac{1}{3} a^{5} + \frac{5}{12} a^{4} - \frac{1}{4} a^{3} + \frac{1}{6} a^{2} + \frac{1}{3} a + \frac{1}{4}$, $\frac{1}{36} a^{12} + \frac{1}{36} a^{9} + \frac{1}{36} a^{8} - \frac{1}{9} a^{7} - \frac{2}{9} a^{6} - \frac{7}{36} a^{5} - \frac{1}{4} a^{4} + \frac{2}{9} a^{3} - \frac{1}{18} a^{2} + \frac{1}{36} a + \frac{1}{6}$, $\frac{1}{36} a^{13} + \frac{1}{36} a^{10} + \frac{1}{36} a^{9} + \frac{1}{18} a^{8} + \frac{1}{9} a^{7} - \frac{1}{36} a^{6} - \frac{5}{12} a^{5} + \frac{7}{18} a^{4} - \frac{2}{9} a^{3} - \frac{11}{36} a^{2}$, $\frac{1}{360} a^{14} + \frac{1}{120} a^{13} - \frac{1}{90} a^{12} - \frac{1}{72} a^{11} - \frac{11}{360} a^{10} - \frac{13}{180} a^{9} + \frac{1}{24} a^{8} + \frac{3}{40} a^{7} + \frac{77}{360} a^{6} + \frac{7}{60} a^{5} - \frac{101}{360} a^{4} - \frac{97}{360} a^{3} - \frac{7}{90} a^{2} - \frac{13}{360} a + \frac{5}{24}$, $\frac{1}{360} a^{15} - \frac{1}{120} a^{13} - \frac{1}{120} a^{12} + \frac{1}{90} a^{11} - \frac{13}{360} a^{10} + \frac{1}{120} a^{9} + \frac{11}{180} a^{8} - \frac{11}{90} a^{7} - \frac{89}{360} a^{6} + \frac{23}{360} a^{5} - \frac{37}{180} a^{4} + \frac{163}{360} a^{3} + \frac{131}{360} a^{2} + \frac{37}{180} a - \frac{1}{24}$, $\frac{1}{269521509600} a^{16} - \frac{1}{33690188700} a^{15} + \frac{106238309}{134760754800} a^{14} - \frac{106238299}{19251536400} a^{13} - \frac{22458281}{16845094350} a^{12} - \frac{96876077}{26952150960} a^{11} - \frac{1908502091}{67380377400} a^{10} - \frac{1825163671}{33690188700} a^{9} - \frac{641553521}{7700614560} a^{8} + \frac{41943257}{3850307280} a^{7} - \frac{6458996681}{33690188700} a^{6} + \frac{18234552491}{67380377400} a^{5} - \frac{904964303}{8422547175} a^{4} + \frac{366791497}{5390430192} a^{3} - \frac{5822459243}{19251536400} a^{2} + \frac{956874283}{5390430192} a + \frac{41386849}{513374304}$, $\frac{1}{687010327970400} a^{17} + \frac{211}{114501721328400} a^{16} + \frac{384923055253}{343505163985200} a^{15} + \frac{5286705023}{16357388761200} a^{14} - \frac{7694716129}{4293814549815} a^{13} - \frac{36780171199}{114501721328400} a^{12} - \frac{1368517900231}{171752581992600} a^{11} + \frac{1558410777953}{57250860664200} a^{10} - \frac{191859183469}{10904925840800} a^{9} - \frac{110099695867}{1635738876120} a^{8} + \frac{17149632017129}{85876290996300} a^{7} - \frac{92282650969}{9541810110700} a^{6} - \frac{6634216083527}{17175258199260} a^{5} - \frac{17974476041833}{38167240442800} a^{4} - \frac{16000473177343}{49072166283600} a^{3} + \frac{24462101355187}{114501721328400} a^{2} - \frac{464072683049}{19628866513440} a - \frac{179874008521}{654295550448}$

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

Class group and class number

$C_{3}\times C_{3}$, which has order $9$ (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:  $8$
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:  \( 2377204.5255 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_3$ (as 18T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 6 conjugacy class representatives for $C_3^2 : C_2$
Character table for $C_3^2 : C_2$

Intermediate fields

\(\Q(\sqrt{-547}) \), 3.1.2188.1 x3, 3.1.2188.2 x3, 3.1.2188.3 x3, 3.1.547.1 x3, 6.0.2618677168.3, 6.0.2618677168.2, 6.0.2618677168.1, 6.0.163667323.1, 9.1.5729665643584.2 x9

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 9 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
547Data not computed