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Properties

Label 18.0.72558852188...6319.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{27}\cdot 7^{9}\cdot 11^{9}$
Root discriminant $45.60$
Ramified primes $3, 7, 11$
Class number $324$ (GRH)
Class group $[3, 6, 18]$ (GRH)
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5184, 23328, 60912, 94392, 113724, 96822, 68469, 28818, 14067, -444, 3081, 126, 1141, -438, -3, 40, 3, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 3*x^16 + 40*x^15 - 3*x^14 - 438*x^13 + 1141*x^12 + 126*x^11 + 3081*x^10 - 444*x^9 + 14067*x^8 + 28818*x^7 + 68469*x^6 + 96822*x^5 + 113724*x^4 + 94392*x^3 + 60912*x^2 + 23328*x + 5184)
 
gp: K = bnfinit(x^18 - 6*x^17 + 3*x^16 + 40*x^15 - 3*x^14 - 438*x^13 + 1141*x^12 + 126*x^11 + 3081*x^10 - 444*x^9 + 14067*x^8 + 28818*x^7 + 68469*x^6 + 96822*x^5 + 113724*x^4 + 94392*x^3 + 60912*x^2 + 23328*x + 5184, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 3 x^{16} + 40 x^{15} - 3 x^{14} - 438 x^{13} + 1141 x^{12} + 126 x^{11} + 3081 x^{10} - 444 x^{9} + 14067 x^{8} + 28818 x^{7} + 68469 x^{6} + 96822 x^{5} + 113724 x^{4} + 94392 x^{3} + 60912 x^{2} + 23328 x + 5184 \)

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:  \(-725588521883853198293196716319=-\,3^{27}\cdot 7^{9}\cdot 11^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.60$
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:  $3, 7, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{6} a^{12} + \frac{1}{6} a^{9} - \frac{1}{3} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{13} + \frac{1}{6} a^{10} + \frac{1}{6} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{36} a^{14} - \frac{1}{12} a^{12} + \frac{1}{9} a^{11} + \frac{1}{12} a^{10} + \frac{1}{6} a^{9} + \frac{7}{36} a^{8} + \frac{1}{6} a^{7} + \frac{5}{12} a^{6} + \frac{1}{6} a^{5} - \frac{1}{4} a^{4} - \frac{1}{12} a^{2}$, $\frac{1}{936} a^{15} + \frac{1}{78} a^{14} + \frac{7}{104} a^{13} + \frac{29}{468} a^{12} - \frac{61}{312} a^{11} + \frac{3}{13} a^{10} + \frac{205}{936} a^{9} - \frac{3}{13} a^{8} - \frac{7}{104} a^{7} - \frac{17}{156} a^{6} + \frac{11}{104} a^{5} + \frac{4}{13} a^{4} - \frac{61}{312} a^{3} + \frac{5}{13} a^{2} + \frac{1}{26} a + \frac{3}{13}$, $\frac{1}{120704688} a^{16} + \frac{2983}{6705816} a^{15} + \frac{169873}{40234896} a^{14} + \frac{2029195}{30176172} a^{13} - \frac{3182633}{40234896} a^{12} + \frac{601673}{20117448} a^{11} - \frac{21276803}{120704688} a^{10} - \frac{2700353}{20117448} a^{9} + \frac{2373211}{40234896} a^{8} - \frac{553793}{5029362} a^{7} + \frac{1247641}{4470544} a^{6} - \frac{103417}{6705816} a^{5} + \frac{15996815}{40234896} a^{4} + \frac{815249}{6705816} a^{3} - \frac{489901}{1117636} a^{2} - \frac{13739}{279409} a - \frac{81969}{279409}$, $\frac{1}{8998483755824434864710735264} a^{17} + \frac{1686612044265885577}{2249620938956108716177683816} a^{16} + \frac{248004019842617190495305}{2999494585274811621570245088} a^{15} + \frac{42656507843055648312284615}{4499241877912217432355367632} a^{14} - \frac{353747693803372190816120711}{8998483755824434864710735264} a^{13} + \frac{8755127886735387752132767}{249957882106234301797520424} a^{12} + \frac{885602993777792760467510413}{8998483755824434864710735264} a^{11} + \frac{58365417291371323968216323}{1124810469478054358088841908} a^{10} + \frac{77680522205000777212645459}{2999494585274811621570245088} a^{9} - \frac{4622545398509484407848043}{115365176356723523906547888} a^{8} - \frac{696359436751309860943347883}{2999494585274811621570245088} a^{7} - \frac{14732532810470248242849800}{31244735263279287724690053} a^{6} + \frac{455787547456187483223933911}{2999494585274811621570245088} a^{5} - \frac{80492881054199794542858289}{187468411579675726348140318} a^{4} - \frac{27690059634899928859339831}{83319294035411433932506808} a^{3} + \frac{41855273625961222815192107}{124978941053117150898760212} a^{2} + \frac{1577000076370896385058795}{20829823508852858483126702} a + \frac{1281666284346300786144709}{10414911754426429241563351}$

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

Class group and class number

$C_{3}\times C_{6}\times C_{18}$, which has order $324$ (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:  \( 429572.316092 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 18T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-231}) \), 3.1.231.1 x3, \(\Q(\zeta_{9})^+\), 6.0.12326391.1, 6.0.8985939039.8, 6.0.8985939039.4 x2, 9.3.727861062159.3 x3

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

Sibling fields

Degree 6 sibling: data not computed
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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R R ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$3$3.6.9.3$x^{6} + 3 x^{4} + 24$$6$$1$$9$$C_6$$[2]_{2}$
3.6.9.3$x^{6} + 3 x^{4} + 24$$6$$1$$9$$C_6$$[2]_{2}$
3.6.9.3$x^{6} + 3 x^{4} + 24$$6$$1$$9$$C_6$$[2]_{2}$
$7$7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$11$11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$