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Properties

Label 18.0.55485880903...9568.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{33}\cdot 7^{12}\cdot 127^{12}$
Root discriminant $1099.88$
Ramified primes $2, 3, 7, 127$
Class number $2125764$ (GRH)
Class group $[3, 3, 3, 3, 3, 3, 9, 18, 18]$ (GRH)
Galois group $C_3\times C_3:S_3$ (as 18T23)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![256690812, -427567140, 983394720, -969139794, 831229938, -633662316, 227790487, -18695511, 7983021, -1046958, -382071, 110313, -17408, 3465, -795, 18, 39, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 39*x^16 + 18*x^15 - 795*x^14 + 3465*x^13 - 17408*x^12 + 110313*x^11 - 382071*x^10 - 1046958*x^9 + 7983021*x^8 - 18695511*x^7 + 227790487*x^6 - 633662316*x^5 + 831229938*x^4 - 969139794*x^3 + 983394720*x^2 - 427567140*x + 256690812)
 
gp: K = bnfinit(x^18 - 9*x^17 + 39*x^16 + 18*x^15 - 795*x^14 + 3465*x^13 - 17408*x^12 + 110313*x^11 - 382071*x^10 - 1046958*x^9 + 7983021*x^8 - 18695511*x^7 + 227790487*x^6 - 633662316*x^5 + 831229938*x^4 - 969139794*x^3 + 983394720*x^2 - 427567140*x + 256690812, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 39 x^{16} + 18 x^{15} - 795 x^{14} + 3465 x^{13} - 17408 x^{12} + 110313 x^{11} - 382071 x^{10} - 1046958 x^{9} + 7983021 x^{8} - 18695511 x^{7} + 227790487 x^{6} - 633662316 x^{5} + 831229938 x^{4} - 969139794 x^{3} + 983394720 x^{2} - 427567140 x + 256690812 \)

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:  \(-5548588090332456647311228183246490588513332762983149568=-\,2^{12}\cdot 3^{33}\cdot 7^{12}\cdot 127^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1099.88$
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:  $2, 3, 7, 127$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{14} a^{12} - \frac{3}{7} a^{11} + \frac{2}{7} a^{10} - \frac{1}{2} a^{9} + \frac{3}{7} a^{8} - \frac{3}{7} a^{7} - \frac{1}{2} a^{6} - \frac{2}{7} a^{5} + \frac{3}{7} a^{4} - \frac{1}{14} a^{3}$, $\frac{1}{14} a^{13} - \frac{2}{7} a^{11} + \frac{3}{14} a^{10} + \frac{3}{7} a^{9} + \frac{1}{7} a^{8} - \frac{1}{14} a^{7} - \frac{2}{7} a^{6} - \frac{2}{7} a^{5} - \frac{1}{2} a^{4} - \frac{3}{7} a^{3}$, $\frac{1}{14} a^{14} - \frac{1}{2} a^{11} - \frac{3}{7} a^{10} + \frac{1}{7} a^{9} - \frac{5}{14} a^{8} - \frac{2}{7} a^{6} + \frac{5}{14} a^{5} + \frac{2}{7} a^{4} - \frac{2}{7} a^{3}$, $\frac{1}{672098} a^{15} + \frac{186}{336049} a^{14} - \frac{1315}{336049} a^{13} - \frac{6207}{336049} a^{12} + \frac{70233}{336049} a^{11} + \frac{50010}{336049} a^{10} + \frac{100969}{336049} a^{9} + \frac{20980}{336049} a^{8} + \frac{41152}{336049} a^{7} + \frac{3109}{48007} a^{6} + \frac{69037}{336049} a^{5} - \frac{142536}{336049} a^{4} + \frac{256397}{672098} a^{3} - \frac{19136}{48007} a^{2} + \frac{16790}{48007} a + \frac{19687}{48007}$, $\frac{1}{15552191763305015906542} a^{16} + \frac{4118080389598547}{7776095881652507953271} a^{15} - \frac{13567120457774397915}{1196322443331155069734} a^{14} + \frac{184238042477849030366}{7776095881652507953271} a^{13} - \frac{145071864705899292234}{7776095881652507953271} a^{12} + \frac{4274552693732290451}{1196322443331155069734} a^{11} - \frac{1779376364731453768883}{7776095881652507953271} a^{10} + \frac{867050788614995302886}{7776095881652507953271} a^{9} - \frac{5661956298906216620607}{15552191763305015906542} a^{8} + \frac{119596324343264228445}{7776095881652507953271} a^{7} - \frac{1185564729504659433036}{7776095881652507953271} a^{6} + \frac{3802944812213672366791}{15552191763305015906542} a^{5} - \frac{494052271207134284633}{2221741680472145129506} a^{4} - \frac{283788141479813170243}{7776095881652507953271} a^{3} + \frac{19060074685282399073}{1110870840236072564753} a^{2} + \frac{115020443868917254912}{1110870840236072564753} a + \frac{500992502450978703468}{1110870840236072564753}$, $\frac{1}{1690804350459956564354818208264827609144301746} a^{17} + \frac{21180368645129384384483}{1690804350459956564354818208264827609144301746} a^{16} - \frac{59864112403223229581893829919528966075}{120771739318568326025344157733201972081735839} a^{15} + \frac{17657238348537851876143420752602308830776673}{845402175229978282177409104132413804572150873} a^{14} + \frac{46485456694737998284099001103842433639523633}{1690804350459956564354818208264827609144301746} a^{13} + \frac{42278004240033227640199666399923132528813731}{1690804350459956564354818208264827609144301746} a^{12} - \frac{33247384784863027097888400318859961580600827}{120771739318568326025344157733201972081735839} a^{11} - \frac{37976719385841473076555886274391526619094839}{130061873112304351104216785251140585318792442} a^{10} - \frac{28632977314622435069884639223386788695669257}{1690804350459956564354818208264827609144301746} a^{9} + \frac{251072928050238733050811236314709190102289311}{845402175229978282177409104132413804572150873} a^{8} - \frac{534154663237587119945863977956677049011206421}{1690804350459956564354818208264827609144301746} a^{7} - \frac{22075035470503290561947994320462154461674619}{1690804350459956564354818208264827609144301746} a^{6} - \frac{557953337083641183397268973950110222813073261}{1690804350459956564354818208264827609144301746} a^{5} + \frac{339390909596550859839583568023539163325059090}{845402175229978282177409104132413804572150873} a^{4} + \frac{542335202147393112942137256895764156055014269}{1690804350459956564354818208264827609144301746} a^{3} + \frac{2181708669248333444824204220408998698104632}{9290133793736025078872627517938613237056603} a^{2} - \frac{17877742279629866058246928065729305079575581}{120771739318568326025344157733201972081735839} a + \frac{50832221350273784390277534083165289438808860}{120771739318568326025344157733201972081735839}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{9}\times C_{18}\times C_{18}$, which has order $2125764$ (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:  \( \frac{248575508395646864762733}{68763184808619385477832551024871} a^{17} - \frac{4745671693900053292333515}{137526369617238770955665102049742} a^{16} + \frac{10573618999428194259707025}{68763184808619385477832551024871} a^{15} + \frac{4080115281650398274911551}{137526369617238770955665102049742} a^{14} - \frac{60483620652759446643676767}{19646624231034110136523586007106} a^{13} + \frac{273186848203298673813157257}{19646624231034110136523586007106} a^{12} - \frac{9067396642055229212995961637}{137526369617238770955665102049742} a^{11} + \frac{57483571580613314769199265787}{137526369617238770955665102049742} a^{10} - \frac{209159393145993043138211695243}{137526369617238770955665102049742} a^{9} - \frac{485124869094432395633529466839}{137526369617238770955665102049742} a^{8} + \frac{4470316177276284587375478027585}{137526369617238770955665102049742} a^{7} - \frac{1518014058230107416101734022175}{19646624231034110136523586007106} a^{6} + \frac{113424777331342335196215451112991}{137526369617238770955665102049742} a^{5} - \frac{26185175899508708998841694701646}{9823312115517055068261793003553} a^{4} + \frac{435084009617110519706370450882061}{137526369617238770955665102049742} a^{3} - \frac{26749221280661828360592736547427}{9823312115517055068261793003553} a^{2} + \frac{25167097559457566390697297323976}{9823312115517055068261793003553} a - \frac{1566442817071621965390882831011}{9823312115517055068261793003553} \) (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:  \( 1521039359576000.2 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_3:S_3$ (as 18T23):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 18 conjugacy class representatives for $C_3\times C_3:S_3$
Character table for $C_3\times C_3:S_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.48387.1 x3, Deg 6, Deg 6, 6.0.6805279152.6, 6.0.7023905307.1

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

Sibling fields

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 ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{9}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
$3$3.6.11.14$x^{6} + 12 x^{3} + 18 x^{2} + 3$$6$$1$$11$$S_3\times C_3$$[2, 5/2]_{2}$
3.6.11.14$x^{6} + 12 x^{3} + 18 x^{2} + 3$$6$$1$$11$$S_3\times C_3$$[2, 5/2]_{2}$
3.6.11.14$x^{6} + 12 x^{3} + 18 x^{2} + 3$$6$$1$$11$$S_3\times C_3$$[2, 5/2]_{2}$
$7$7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
$127$127.3.2.1$x^{3} - 127$$3$$1$$2$$C_3$$[\ ]_{3}$
127.3.2.1$x^{3} - 127$$3$$1$$2$$C_3$$[\ ]_{3}$
127.3.2.1$x^{3} - 127$$3$$1$$2$$C_3$$[\ ]_{3}$
127.3.2.1$x^{3} - 127$$3$$1$$2$$C_3$$[\ ]_{3}$
127.3.2.1$x^{3} - 127$$3$$1$$2$$C_3$$[\ ]_{3}$
127.3.2.1$x^{3} - 127$$3$$1$$2$$C_3$$[\ ]_{3}$