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Properties

Label 18.0.66376707777...8928.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{9}\cdot 7^{12}\cdot 29^{6}$
Root discriminant $30.91$
Ramified primes $2, 3, 7, 29$
Class number $18$ (GRH)
Class group $[3, 6]$ (GRH)
Galois group $S_3^2$ (as 18T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![27, 0, 252, 0, 1092, 0, 2575, 0, 3556, 0, 2772, 0, 1157, 0, 252, 0, 28, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 28*x^16 + 252*x^14 + 1157*x^12 + 2772*x^10 + 3556*x^8 + 2575*x^6 + 1092*x^4 + 252*x^2 + 27)
 
gp: K = bnfinit(x^18 + 28*x^16 + 252*x^14 + 1157*x^12 + 2772*x^10 + 3556*x^8 + 2575*x^6 + 1092*x^4 + 252*x^2 + 27, 1)
 

Normalized defining polynomial

\( x^{18} + 28 x^{16} + 252 x^{14} + 1157 x^{12} + 2772 x^{10} + 3556 x^{8} + 2575 x^{6} + 1092 x^{4} + 252 x^{2} + 27 \)

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:  \(-663767077778191870430588928=-\,2^{12}\cdot 3^{9}\cdot 7^{12}\cdot 29^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.91$
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:  $2, 3, 7, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} + \frac{1}{4} a$, $\frac{1}{20} a^{12} - \frac{1}{2} a^{5} + \frac{1}{10} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{9}{20}$, $\frac{1}{20} a^{13} - \frac{1}{2} a^{6} + \frac{1}{10} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{9}{20} a$, $\frac{1}{60} a^{14} + \frac{1}{60} a^{12} - \frac{1}{6} a^{8} + \frac{1}{5} a^{6} - \frac{1}{2} a^{5} - \frac{2}{15} a^{4} - \frac{1}{2} a^{3} - \frac{29}{60} a^{2} + \frac{7}{20}$, $\frac{1}{120} a^{15} - \frac{1}{120} a^{14} - \frac{1}{60} a^{13} + \frac{1}{60} a^{12} - \frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{5}{24} a^{9} - \frac{1}{24} a^{8} - \frac{1}{40} a^{7} + \frac{11}{40} a^{6} - \frac{29}{120} a^{5} + \frac{59}{120} a^{4} - \frac{11}{30} a^{3} + \frac{7}{60} a^{2} - \frac{9}{40} a + \frac{19}{40}$, $\frac{1}{3080880} a^{16} + \frac{23593}{3080880} a^{14} + \frac{8167}{1026960} a^{12} + \frac{1031}{308088} a^{10} - \frac{71933}{513480} a^{8} - \frac{309217}{1540440} a^{6} - \frac{1}{2} a^{5} - \frac{524147}{3080880} a^{4} - \frac{1}{2} a^{3} - \frac{399769}{1026960} a^{2} - \frac{66991}{342320}$, $\frac{1}{6161760} a^{17} - \frac{1}{6161760} a^{16} + \frac{23593}{6161760} a^{15} - \frac{23593}{6161760} a^{14} - \frac{43181}{2053920} a^{13} + \frac{43181}{2053920} a^{12} - \frac{75991}{616176} a^{11} + \frac{75991}{616176} a^{10} - \frac{200303}{1026960} a^{9} + \frac{200303}{1026960} a^{8} - \frac{694327}{3080880} a^{7} - \frac{846113}{3080880} a^{6} + \frac{295685}{1232352} a^{5} - \frac{295685}{1232352} a^{4} + \frac{370451}{2053920} a^{3} + \frac{656509}{2053920} a^{2} + \frac{1473}{684640} a - \frac{1473}{684640}$

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

Class group and class number

$C_{3}\times C_{6}$, which has order $18$ (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:  \( \frac{70441}{616176} a^{17} + \frac{1936369}{616176} a^{15} + \frac{5585587}{205392} a^{13} + \frac{36396031}{308088} a^{11} + \frac{26124319}{102696} a^{9} + \frac{82218311}{308088} a^{7} + \frac{83935117}{616176} a^{5} + \frac{6611879}{205392} a^{3} + \frac{122213}{68464} a + \frac{1}{2} \) (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:  \( 892484.3013079121 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3^2$ (as 18T11):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 9 conjugacy class representatives for $S_3^2$
Character table for $S_3^2$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.3.5684.1, 3.1.588.1 x3, 6.0.872312112.3, 6.0.1037232.1, 9.3.4958222044608.1 x3

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

Sibling fields

Galois closure: data not computed
Degree 6 sibling: data not computed
Degree 9 sibling: data not computed
Degree 12 sibling: 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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
$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$$[\ ]_{3}^{2}$
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$