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Properties

Label 18.0.29888055988...0208.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{21}\cdot 17^{8}$
Root discriminant $20.15$
Ramified primes $2, 3, 17$
Class number $3$
Class group $[3]$
Galois group $C_2\times C_3:S_3$ (as 18T12)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -12, 48, -56, 36, 30, -9, -84, 216, -64, -168, 180, 15, -48, -12, 4, 12, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 12*x^16 + 4*x^15 - 12*x^14 - 48*x^13 + 15*x^12 + 180*x^11 - 168*x^10 - 64*x^9 + 216*x^8 - 84*x^7 - 9*x^6 + 30*x^5 + 36*x^4 - 56*x^3 + 48*x^2 - 12*x + 1)
 
gp: K = bnfinit(x^18 - 6*x^17 + 12*x^16 + 4*x^15 - 12*x^14 - 48*x^13 + 15*x^12 + 180*x^11 - 168*x^10 - 64*x^9 + 216*x^8 - 84*x^7 - 9*x^6 + 30*x^5 + 36*x^4 - 56*x^3 + 48*x^2 - 12*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 12 x^{16} + 4 x^{15} - 12 x^{14} - 48 x^{13} + 15 x^{12} + 180 x^{11} - 168 x^{10} - 64 x^{9} + 216 x^{8} - 84 x^{7} - 9 x^{6} + 30 x^{5} + 36 x^{4} - 56 x^{3} + 48 x^{2} - 12 x + 1 \)

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:  \(-298880559887628015710208=-\,2^{12}\cdot 3^{21}\cdot 17^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.15$
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:  $2, 3, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{12} - \frac{1}{2}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{12} - \frac{1}{2} a^{6} - \frac{1}{4} a^{3} - \frac{1}{4}$, $\frac{1}{28} a^{16} - \frac{3}{28} a^{15} - \frac{1}{28} a^{13} - \frac{1}{28} a^{12} - \frac{3}{14} a^{11} + \frac{1}{14} a^{9} + \frac{3}{14} a^{8} - \frac{5}{14} a^{7} + \frac{1}{14} a^{5} + \frac{11}{28} a^{4} + \frac{1}{28} a^{3} + \frac{3}{14} a^{2} - \frac{13}{28} a + \frac{5}{28}$, $\frac{1}{1154747330244004} a^{17} + \frac{129748888471}{31209387303892} a^{16} + \frac{143931477796033}{1154747330244004} a^{15} - \frac{38109220685537}{1154747330244004} a^{14} + \frac{3434031793947}{31209387303892} a^{13} + \frac{4424751698783}{23566272045796} a^{12} - \frac{32456973785869}{577373665122002} a^{11} + \frac{24664091436461}{577373665122002} a^{10} + \frac{124445055392557}{577373665122002} a^{9} + \frac{111315920691261}{288686832561001} a^{8} - \frac{143253708074224}{288686832561001} a^{7} - \frac{116471645253806}{288686832561001} a^{6} - \frac{33206567326507}{1154747330244004} a^{5} - \frac{427129682620657}{1154747330244004} a^{4} - \frac{52166890865685}{164963904320572} a^{3} - \frac{19574787844597}{164963904320572} a^{2} - \frac{419910158379055}{1154747330244004} a + \frac{20297995675165}{1154747330244004}$

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:  \( -\frac{8903324665}{15545034331} a^{17} + \frac{2827845183}{840272126} a^{16} - \frac{398746661003}{62180137324} a^{15} - \frac{205018269167}{62180137324} a^{14} + \frac{2826092114}{420136063} a^{13} + \frac{898841970611}{31090068662} a^{12} - \frac{318683077969}{62180137324} a^{11} - \frac{6595534691465}{62180137324} a^{10} + \frac{5047691414789}{62180137324} a^{9} + \frac{3342223319131}{62180137324} a^{8} - \frac{7222684569349}{62180137324} a^{7} + \frac{1774566407551}{62180137324} a^{6} + \frac{824983814541}{62180137324} a^{5} - \frac{879834709775}{62180137324} a^{4} - \frac{731339059111}{31090068662} a^{3} + \frac{939340710315}{31090068662} a^{2} - \frac{1357041871107}{62180137324} a + \frac{213285326825}{62180137324} \) (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:  \( 63321.4880156 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_3:S_3$ (as 18T12):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 12 conjugacy class representatives for $C_2\times C_3:S_3$
Character table for $C_2\times C_3:S_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.204.1, 3.1.459.1, 3.1.1836.1, 3.1.1836.2, 6.0.632043.1, 6.0.124848.1, 6.0.10112688.3, 6.0.10112688.2, 9.1.315637217856.1

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 18 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 R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$$[\ ]_{3}^{2}$
$3$3.6.7.5$x^{6} + 6 x^{2} + 3$$6$$1$$7$$D_{6}$$[3/2]_{2}^{2}$
3.12.14.11$x^{12} + 6 x^{11} + 21 x^{10} + 36 x^{9} + 30 x^{8} + 36 x^{7} + 3 x^{6} + 36 x^{5} + 27 x^{4} - 9 x^{2} + 36$$6$$2$$14$$D_6$$[3/2]_{2}^{2}$
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$