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Properties

Label 18.0.18283008447...6512.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{26}\cdot 3^{9}\cdot 7^{12}$
Root discriminant $17.25$
Ramified primes $2, 3, 7$
Class number $3$
Class group $[3]$
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![1, 1, 9, 36, 86, 166, 506, -158, -181, 299, -3, -74, 40, -4, 0, 8, -3, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 3*x^16 + 8*x^15 - 4*x^13 + 40*x^12 - 74*x^11 - 3*x^10 + 299*x^9 - 181*x^8 - 158*x^7 + 506*x^6 + 166*x^5 + 86*x^4 + 36*x^3 + 9*x^2 + x + 1)
 
gp: K = bnfinit(x^18 - x^17 - 3*x^16 + 8*x^15 - 4*x^13 + 40*x^12 - 74*x^11 - 3*x^10 + 299*x^9 - 181*x^8 - 158*x^7 + 506*x^6 + 166*x^5 + 86*x^4 + 36*x^3 + 9*x^2 + x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} - 3 x^{16} + 8 x^{15} - 4 x^{13} + 40 x^{12} - 74 x^{11} - 3 x^{10} + 299 x^{9} - 181 x^{8} - 158 x^{7} + 506 x^{6} + 166 x^{5} + 86 x^{4} + 36 x^{3} + 9 x^{2} + x + 1 \)

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:  \(-18283008447003871936512=-\,2^{26}\cdot 3^{9}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.25$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{4} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{10} - \frac{1}{3} a^{7} - \frac{1}{2} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{2} a^{2} - \frac{1}{3} a - \frac{1}{6}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{11} - \frac{1}{6} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{2} a^{3} - \frac{1}{3} a^{2} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{18} a^{14} - \frac{1}{18} a^{10} - \frac{1}{9} a^{9} - \frac{1}{18} a^{8} - \frac{2}{9} a^{7} + \frac{1}{18} a^{6} - \frac{1}{9} a^{5} - \frac{1}{2} a^{4} + \frac{4}{9} a^{2} + \frac{1}{9} a + \frac{7}{18}$, $\frac{1}{18} a^{15} - \frac{1}{18} a^{11} - \frac{1}{9} a^{10} - \frac{1}{18} a^{9} + \frac{1}{9} a^{8} + \frac{7}{18} a^{7} - \frac{1}{9} a^{6} - \frac{1}{2} a^{5} + \frac{1}{3} a^{4} + \frac{4}{9} a^{3} + \frac{1}{9} a^{2} + \frac{1}{18} a + \frac{1}{3}$, $\frac{1}{918} a^{16} + \frac{7}{918} a^{15} - \frac{11}{918} a^{14} - \frac{23}{306} a^{13} - \frac{1}{918} a^{12} + \frac{20}{153} a^{11} + \frac{73}{459} a^{10} - \frac{7}{918} a^{9} - \frac{35}{459} a^{8} + \frac{1}{27} a^{7} - \frac{86}{459} a^{6} + \frac{421}{918} a^{5} - \frac{265}{918} a^{4} - \frac{287}{918} a^{3} - \frac{397}{918} a^{2} - \frac{7}{17} a + \frac{361}{918}$, $\frac{1}{892443743838} a^{17} + \frac{116991155}{297481247946} a^{16} + \frac{917158316}{148740623973} a^{15} + \frac{94295551}{3734074242} a^{14} - \frac{206168921}{3734074242} a^{13} - \frac{37246972451}{892443743838} a^{12} + \frac{34480183991}{892443743838} a^{11} + \frac{764644733}{6466983651} a^{10} + \frac{36867991495}{297481247946} a^{9} - \frac{28510051646}{446221871919} a^{8} + \frac{355141804543}{892443743838} a^{7} - \frac{19754763872}{446221871919} a^{6} - \frac{27112564195}{446221871919} a^{5} + \frac{157817220161}{892443743838} a^{4} + \frac{10089029663}{38801901906} a^{3} + \frac{162334114921}{892443743838} a^{2} - \frac{162094802527}{446221871919} a - \frac{25895071651}{892443743838}$

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

Class group and class number

$C_{3}$, which has order $3$

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:  \( -\frac{3624966722}{26248345407} a^{17} + \frac{699944263}{2916482823} a^{16} + \frac{2844671768}{8749448469} a^{15} - \frac{156287396}{109825713} a^{14} + \frac{84953176}{109825713} a^{13} + \frac{17157904432}{26248345407} a^{12} - \frac{155269516996}{26248345407} a^{11} + \frac{5409210236}{380410803} a^{10} - \frac{19355383378}{2916482823} a^{9} - \frac{1115445049474}{26248345407} a^{8} + \frac{1450174831468}{26248345407} a^{7} + \frac{192424131008}{26248345407} a^{6} - \frac{2307495793820}{26248345407} a^{5} + \frac{664545306056}{26248345407} a^{4} + \frac{13389354656}{1141232409} a^{3} + \frac{185544828760}{26248345407} a^{2} + \frac{32238400738}{26248345407} a + \frac{29430407450}{26248345407} \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 19737.264553893918 \)
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

\(\Q(\sqrt{-3}) \), 3.1.1176.1, 3.1.588.1 x3, 6.0.4148928.1, 6.0.1037232.1, 9.1.78066229248.1 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 ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.12.22.60$x^{12} - 84 x^{10} + 444 x^{8} + 32 x^{6} - 272 x^{4} - 320 x^{2} + 64$$6$$2$$22$$D_6$$[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}$