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Properties

Label 18.0.54742910338...1875.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{21}\cdot 5^{12}\cdot 11^{8}$
Root discriminant $30.58$
Ramified primes $3, 5, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3^2:S_3$ (as 18T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![27, 243, 891, 1458, 2214, 1809, -378, -1890, 2970, -2241, 1341, -873, 351, -63, -9, 0, 12, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 12*x^16 - 9*x^14 - 63*x^13 + 351*x^12 - 873*x^11 + 1341*x^10 - 2241*x^9 + 2970*x^8 - 1890*x^7 - 378*x^6 + 1809*x^5 + 2214*x^4 + 1458*x^3 + 891*x^2 + 243*x + 27)
 
gp: K = bnfinit(x^18 - 6*x^17 + 12*x^16 - 9*x^14 - 63*x^13 + 351*x^12 - 873*x^11 + 1341*x^10 - 2241*x^9 + 2970*x^8 - 1890*x^7 - 378*x^6 + 1809*x^5 + 2214*x^4 + 1458*x^3 + 891*x^2 + 243*x + 27, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 12 x^{16} - 9 x^{14} - 63 x^{13} + 351 x^{12} - 873 x^{11} + 1341 x^{10} - 2241 x^{9} + 2970 x^{8} - 1890 x^{7} - 378 x^{6} + 1809 x^{5} + 2214 x^{4} + 1458 x^{3} + 891 x^{2} + 243 x + 27 \)

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:  \(-547429103383751426513671875=-\,3^{21}\cdot 5^{12}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.58$
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:  $3, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6}$, $\frac{1}{3} a^{7}$, $\frac{1}{9} a^{8} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{9} a^{9} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{9} a^{10} + \frac{1}{3} a^{4}$, $\frac{1}{9} a^{11} + \frac{1}{3} a^{5}$, $\frac{1}{54} a^{12} - \frac{1}{18} a^{11} - \frac{1}{18} a^{9} - \frac{1}{18} a^{8} - \frac{1}{18} a^{6} - \frac{1}{3} a^{5} + \frac{1}{6} a^{4} - \frac{1}{6} a^{3} + \frac{1}{3} a^{2} - \frac{1}{2} a + \frac{1}{6}$, $\frac{1}{54} a^{13} - \frac{1}{18} a^{11} - \frac{1}{18} a^{10} - \frac{1}{18} a^{8} - \frac{1}{18} a^{7} - \frac{1}{6} a^{6} + \frac{1}{6} a^{5} - \frac{1}{3} a^{4} - \frac{1}{2} a^{3} - \frac{1}{6} a^{2} - \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{54} a^{14} - \frac{1}{6} a^{7} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{2}$, $\frac{1}{162} a^{15} + \frac{1}{27} a^{11} + \frac{1}{27} a^{9} - \frac{1}{18} a^{8} + \frac{1}{9} a^{5} + \frac{1}{3} a^{4} - \frac{1}{9} a^{3} + \frac{1}{3} a^{2} - \frac{1}{6} a$, $\frac{1}{32238} a^{16} + \frac{67}{32238} a^{15} + \frac{1}{3582} a^{14} + \frac{19}{5373} a^{13} + \frac{29}{10746} a^{12} - \frac{1}{54} a^{11} + \frac{238}{5373} a^{10} + \frac{214}{5373} a^{9} + \frac{163}{3582} a^{7} - \frac{109}{3582} a^{6} + \frac{130}{1791} a^{5} - \frac{239}{3582} a^{4} - \frac{1559}{3582} a^{3} + \frac{203}{1194} a^{2} - \frac{277}{597} a + \frac{52}{199}$, $\frac{1}{16586646470954298} a^{17} + \frac{81215175545}{5528882156984766} a^{16} - \frac{13587928031513}{5528882156984766} a^{15} + \frac{2832397031194}{307160119832487} a^{14} + \frac{12752792498135}{1842960718994922} a^{13} - \frac{44296567200637}{5528882156984766} a^{12} + \frac{79533206343355}{2764441078492383} a^{11} + \frac{17066380377113}{921480359497461} a^{10} + \frac{9114508324477}{307160119832487} a^{9} - \frac{65415727637203}{1842960718994922} a^{8} - \frac{297488433467933}{1842960718994922} a^{7} + \frac{108644354026607}{921480359497461} a^{6} + \frac{73348608946765}{1842960718994922} a^{5} - \frac{130645777179247}{614320239664974} a^{4} + \frac{297101812294501}{614320239664974} a^{3} + \frac{127297949884642}{307160119832487} a^{2} + \frac{119243530428089}{307160119832487} a - \frac{66521622169961}{307160119832487}$

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:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{278297132452}{13891663711017} a^{17} + \frac{576704578472}{4630554570339} a^{16} - \frac{3713088970880}{13891663711017} a^{15} + \frac{262572285671}{4630554570339} a^{14} + \frac{2404203160006}{13891663711017} a^{13} + \frac{16886970072002}{13891663711017} a^{12} - \frac{33761547778060}{4630554570339} a^{11} + \frac{88285618296230}{4630554570339} a^{10} - \frac{143221311986834}{4630554570339} a^{9} + \frac{237468741706384}{4630554570339} a^{8} - \frac{323379659947972}{4630554570339} a^{7} + \frac{239113010539348}{4630554570339} a^{6} - \frac{2245089040334}{1543518190113} a^{5} - \frac{59717239346510}{1543518190113} a^{4} - \frac{17325719200816}{514506063371} a^{3} - \frac{34888442478302}{1543518190113} a^{2} - \frac{21261392464478}{1543518190113} a - \frac{2626860916942}{1543518190113} \) (order $6$)
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:  \( 7783705.8504994 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3^2:S_3$ (as 18T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 10 conjugacy class representatives for $C_3^2:S_3$
Character table for $C_3^2:S_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.675.1 x3, 6.0.1366875.1, 9.3.4502793796875.1

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

Sibling fields

Degree 9 siblings: data not computed
Degree 18 siblings: 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R R ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
$3$3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
5Data not computed
$11$11.6.4.2$x^{6} - 11 x^{3} + 847$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
11.6.4.2$x^{6} - 11 x^{3} + 847$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
11.6.0.1$x^{6} + x^{2} - 2 x + 8$$1$$6$$0$$C_6$$[\ ]^{6}$