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Properties

Label 18.0.18188661188...7904.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 97^{12}$
Root discriminant $42.22$
Ramified primes $2, 97$
Class number $12$ (GRH)
Class group $[2, 6]$ (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![297992, 461656, 357604, -16740, -151144, -102816, 22565, 50102, 26834, 1120, -4099, -2090, 18, 178, 86, -6, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^15 + 86*x^14 + 178*x^13 + 18*x^12 - 2090*x^11 - 4099*x^10 + 1120*x^9 + 26834*x^8 + 50102*x^7 + 22565*x^6 - 102816*x^5 - 151144*x^4 - 16740*x^3 + 357604*x^2 + 461656*x + 297992)
 
gp: K = bnfinit(x^18 - 6*x^15 + 86*x^14 + 178*x^13 + 18*x^12 - 2090*x^11 - 4099*x^10 + 1120*x^9 + 26834*x^8 + 50102*x^7 + 22565*x^6 - 102816*x^5 - 151144*x^4 - 16740*x^3 + 357604*x^2 + 461656*x + 297992, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{15} + 86 x^{14} + 178 x^{13} + 18 x^{12} - 2090 x^{11} - 4099 x^{10} + 1120 x^{9} + 26834 x^{8} + 50102 x^{7} + 22565 x^{6} - 102816 x^{5} - 151144 x^{4} - 16740 x^{3} + 357604 x^{2} + 461656 x + 297992 \)

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:  \(-181886611880788099149343227904=-\,2^{18}\cdot 97^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.22$
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:  $2, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{582} a^{15} - \frac{18}{97} a^{14} + \frac{8}{291} a^{13} + \frac{101}{582} a^{12} + \frac{65}{582} a^{10} - \frac{7}{291} a^{9} + \frac{11}{97} a^{8} - \frac{89}{582} a^{7} + \frac{17}{97} a^{6} - \frac{47}{97} a^{5} + \frac{8}{291} a^{4} - \frac{85}{582} a^{3} - \frac{53}{582} a^{2} - \frac{101}{291}$, $\frac{1}{330217870956} a^{16} + \frac{41484317}{82554467739} a^{15} - \frac{24816591667}{165108935478} a^{14} - \frac{12482462479}{165108935478} a^{13} + \frac{14204055533}{165108935478} a^{12} - \frac{12879262114}{82554467739} a^{11} + \frac{19236886127}{82554467739} a^{10} - \frac{511373953}{82554467739} a^{9} + \frac{11925186649}{330217870956} a^{8} + \frac{1283998207}{165108935478} a^{7} + \frac{16693768063}{55036311826} a^{6} + \frac{16685008487}{165108935478} a^{5} - \frac{32733401489}{330217870956} a^{4} - \frac{78442321343}{165108935478} a^{3} + \frac{45606412609}{165108935478} a^{2} + \frac{25391960822}{82554467739} a + \frac{12270092581}{82554467739}$, $\frac{1}{15283877065187024383320876315252} a^{17} - \frac{33935730793488363}{26397024292205568883110321788} a^{16} + \frac{2544087272129293626098758}{19797768219154176662332741341} a^{15} - \frac{985854968231537050702028525159}{7641938532593512191660438157626} a^{14} + \frac{555200014347195021305769680189}{2547312844197837397220146052542} a^{13} + \frac{912248336329656515309237328584}{3820969266296756095830219078813} a^{12} + \frac{35670177498622384355765682311}{7641938532593512191660438157626} a^{11} - \frac{288931727190098850643300767134}{1273656422098918698610073026271} a^{10} - \frac{2375700778252404829402379733989}{15283877065187024383320876315252} a^{9} - \frac{929418683921919706949221786359}{5094625688395674794440292105084} a^{8} - \frac{4470991921409010124951141609}{134069097063044073537902423818} a^{7} - \frac{41136734332873592668563659417}{3820969266296756095830219078813} a^{6} + \frac{6555647138451884412312636483791}{15283877065187024383320876315252} a^{5} + \frac{3725659569036246945454612295071}{15283877065187024383320876315252} a^{4} - \frac{12218891001509801138111545059}{2547312844197837397220146052542} a^{3} - \frac{1451940308983272815853503714497}{3820969266296756095830219078813} a^{2} + \frac{270242603329262179163033863700}{1273656422098918698610073026271} a + \frac{104599574769077391935619629}{6599256073051392220777580447}$

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

Class group and class number

$C_{2}\times C_{6}$, which has order $12$ (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:  \( -\frac{262772347500904124}{13468708441167459610197} a^{17} + \frac{660483232590877}{69786054099313262229} a^{16} + \frac{637590621589554}{23262018033104420743} a^{15} + \frac{275814076250056216}{4489569480389153203399} a^{14} - \frac{23506331866165775597}{13468708441167459610197} a^{13} - \frac{23907037205010450731}{8979138960778306406798} a^{12} + \frac{17226821748187793237}{4489569480389153203399} a^{11} + \frac{1101247547595769372573}{26937416882334919220394} a^{10} + \frac{445660355879809803873}{8979138960778306406798} a^{9} - \frac{2853057155641720287973}{26937416882334919220394} a^{8} - \frac{353098498140368006959}{708879391640392611063} a^{7} - \frac{6662143721729295230312}{13468708441167459610197} a^{6} + \frac{3960947796725685247857}{8979138960778306406798} a^{5} + \frac{52287719296288435178209}{26937416882334919220394} a^{4} + \frac{17819099423409981041179}{26937416882334919220394} a^{3} - \frac{32202519276020250310577}{13468708441167459610197} a^{2} - \frac{66629354318518059330445}{13468708441167459610197} a - \frac{172033821085040243135}{69786054099313262229} \) (order $4$)
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:  \( 1883402.18484 \) (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

\(\Q(\sqrt{-1}) \), 3.1.37636.1 x3, 3.3.9409.1, 6.0.5665873984.1, 6.0.5665873984.2, 6.0.602176.1 x2, 9.3.53310208315456.2 x3

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

Sibling fields

Degree 6 sibling: 6.0.602176.1
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 R ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
$97$97.3.2.1$x^{3} - 97$$3$$1$$2$$C_3$$[\ ]_{3}$
97.3.2.1$x^{3} - 97$$3$$1$$2$$C_3$$[\ ]_{3}$
97.3.2.1$x^{3} - 97$$3$$1$$2$$C_3$$[\ ]_{3}$
97.3.2.1$x^{3} - 97$$3$$1$$2$$C_3$$[\ ]_{3}$
97.3.2.1$x^{3} - 97$$3$$1$$2$$C_3$$[\ ]_{3}$
97.3.2.1$x^{3} - 97$$3$$1$$2$$C_3$$[\ ]_{3}$