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Properties

Label 18.0.11464615459...2608.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 17^{9}\cdot 79^{14}$
Root discriminant $246.71$
Ramified primes $2, 17, 79$
Class number $14676480$ (GRH)
Class group $[2, 2, 2, 4, 84, 5460]$ (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3057477097, 337394588, 3062291750, 450056094, 1909020302, 187135216, 310395661, 49846322, 19205829, 6714132, 727903, 389884, 101877, -2892, 3926, -30, 73, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 73*x^16 - 30*x^15 + 3926*x^14 - 2892*x^13 + 101877*x^12 + 389884*x^11 + 727903*x^10 + 6714132*x^9 + 19205829*x^8 + 49846322*x^7 + 310395661*x^6 + 187135216*x^5 + 1909020302*x^4 + 450056094*x^3 + 3062291750*x^2 + 337394588*x + 3057477097)
 
gp: K = bnfinit(x^18 - 4*x^17 + 73*x^16 - 30*x^15 + 3926*x^14 - 2892*x^13 + 101877*x^12 + 389884*x^11 + 727903*x^10 + 6714132*x^9 + 19205829*x^8 + 49846322*x^7 + 310395661*x^6 + 187135216*x^5 + 1909020302*x^4 + 450056094*x^3 + 3062291750*x^2 + 337394588*x + 3057477097, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} + 73 x^{16} - 30 x^{15} + 3926 x^{14} - 2892 x^{13} + 101877 x^{12} + 389884 x^{11} + 727903 x^{10} + 6714132 x^{9} + 19205829 x^{8} + 49846322 x^{7} + 310395661 x^{6} + 187135216 x^{5} + 1909020302 x^{4} + 450056094 x^{3} + 3062291750 x^{2} + 337394588 x + 3057477097 \)

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:  \(-11464615459611696074365702834006806143172608=-\,2^{18}\cdot 17^{9}\cdot 79^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $246.71$
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:  $2, 17, 79$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{4} a^{12} - \frac{1}{2} a^{11} + \frac{1}{4} a^{10} - \frac{1}{2} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{12} a^{13} + \frac{1}{12} a^{12} + \frac{1}{4} a^{11} + \frac{1}{12} a^{10} + \frac{5}{12} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{5}{12} a^{4} - \frac{1}{12} a^{3} - \frac{1}{4} a^{2} + \frac{5}{12} a - \frac{5}{12}$, $\frac{1}{12} a^{14} - \frac{1}{12} a^{12} + \frac{1}{3} a^{11} + \frac{1}{12} a^{10} + \frac{1}{3} a^{9} + \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{6} a^{5} + \frac{1}{12} a^{4} - \frac{1}{6} a^{3} - \frac{1}{12} a^{2} + \frac{1}{6} a + \frac{1}{6}$, $\frac{1}{12} a^{15} - \frac{1}{12} a^{12} + \frac{1}{3} a^{11} - \frac{1}{12} a^{10} - \frac{1}{3} a^{9} + \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{5}{12} a^{6} - \frac{1}{6} a^{5} - \frac{1}{12} a^{4} - \frac{1}{6} a^{3} + \frac{5}{12} a^{2} - \frac{5}{12} a + \frac{1}{12}$, $\frac{1}{84} a^{16} + \frac{1}{42} a^{15} - \frac{1}{84} a^{14} + \frac{1}{42} a^{13} + \frac{1}{14} a^{12} + \frac{2}{7} a^{11} - \frac{4}{21} a^{10} - \frac{1}{14} a^{9} + \frac{3}{14} a^{8} + \frac{17}{42} a^{7} + \frac{1}{14} a^{6} - \frac{2}{7} a^{5} + \frac{1}{3} a^{4} + \frac{3}{7} a^{3} - \frac{5}{28} a^{2} + \frac{1}{21} a + \frac{11}{28}$, $\frac{1}{4223933870573809842514110384207048909820661537199800202525622984467758191742678796} a^{17} + \frac{938681207464468112745389627992922640445743031589122624059285790240508117066111}{2111966935286904921257055192103524454910330768599900101262811492233879095871339398} a^{16} - \frac{548781920833740976736705753528431390553219158993066515208645272575657533028897}{14367122008754455246646633959887921461975039242176191165053139402951558475315234} a^{15} - \frac{3701107244238528389257798769688315410583430776605467800231794593169728430120754}{117331496404828051180947510672418025272796153811105561181267305124104394215074411} a^{14} - \frac{6179504176368055727172906113718718016637732091560104756773554710569661507807347}{603419124367687120359158626315292701402951648171400028932231854923965455963239828} a^{13} + \frac{71704287146854521819527700686043045788365366417543345089922602619875706263121919}{2111966935286904921257055192103524454910330768599900101262811492233879095871339398} a^{12} + \frac{1773583001623087689084854875998828353443026790297131304095812377617052124795210687}{4223933870573809842514110384207048909820661537199800202525622984467758191742678796} a^{11} + \frac{18815090436676173768192098459824839359716313676547951271221436685173042374668026}{150854781091921780089789656578823175350737912042850007233057963730991363990809957} a^{10} - \frac{971037603130890314338914057379695544892570596909135331331335332709697498230845825}{4223933870573809842514110384207048909820661537199800202525622984467758191742678796} a^{9} + \frac{442104164705157392079566186645228302351513493907816473747937053162026158618290573}{2111966935286904921257055192103524454910330768599900101262811492233879095871339398} a^{8} + \frac{1477591083262211278056821989141036734057486361650852966359381584320940391319943775}{4223933870573809842514110384207048909820661537199800202525622984467758191742678796} a^{7} + \frac{22323768449586535447218174196285628912071167938040531415068412321993177251448273}{50284927030640593363263218859607725116912637347616669077685987910330454663603319} a^{6} - \frac{1204149110442660833401420128384615065892798058157337598881752886853792107531716681}{4223933870573809842514110384207048909820661537199800202525622984467758191742678796} a^{5} - \frac{142982544228092243490776862919346447959107421588577135818088910058260200759755912}{351994489214484153542842532017254075818388461433316683543801915372313182645223233} a^{4} + \frac{945294082895441512184593189151676314175287440308457986228575310922086463422517465}{2111966935286904921257055192103524454910330768599900101262811492233879095871339398} a^{3} + \frac{1474704022167821670963084612920860811546465342955305544631073891704897872326050}{21550683013131682869969950939831882192962558863264286747579709104427337712972851} a^{2} - \frac{40936218293652692811277419087812170131329113805101722820609065156890516997298697}{86202732052526731479879803759327528771850235453057146990318836417709350851891404} a - \frac{78689998711086501648182272819815755124870793892732079472050164989058642729651498}{1055983467643452460628527596051762227455165384299950050631405746116939547935669699}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{84}\times C_{5460}$, which has order $14676480$ (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:  \( -1 \) (order $2$)
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:  \( 9044146.559729666 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{-17}) \), 3.3.6241.1, 3.3.316.1, 6.0.12247151868992.2, 6.0.1962370112.2, 9.9.1229050175114176.1

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 12 sibling: 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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$$[2]^{3}$
$17$17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{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$$[\ ]_{2}^{2}$
$79$79.3.2.1$x^{3} - 79$$3$$1$$2$$C_3$$[\ ]_{3}$
79.3.2.1$x^{3} - 79$$3$$1$$2$$C_3$$[\ ]_{3}$
79.6.5.1$x^{6} - 79$$6$$1$$5$$C_6$$[\ ]_{6}$
79.6.5.1$x^{6} - 79$$6$$1$$5$$C_6$$[\ ]_{6}$