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Properties

Label 18.0.87223376826...2243.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{33}\cdot 271^{12}$
Root discriminant $313.84$
Ramified primes $3, 271$
Class number $59049$ (GRH)
Class group $[3, 9, 9, 9, 27]$ (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![21867232251, 2120234994, -2708142030, 498275931, 104639490, -19419525, 4368925, -1186992, 2059689, -946308, 155823, -46008, 4363, 1845, -129, -99, 39, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 39*x^16 - 99*x^15 - 129*x^14 + 1845*x^13 + 4363*x^12 - 46008*x^11 + 155823*x^10 - 946308*x^9 + 2059689*x^8 - 1186992*x^7 + 4368925*x^6 - 19419525*x^5 + 104639490*x^4 + 498275931*x^3 - 2708142030*x^2 + 2120234994*x + 21867232251)
 
gp: K = bnfinit(x^18 - 9*x^17 + 39*x^16 - 99*x^15 - 129*x^14 + 1845*x^13 + 4363*x^12 - 46008*x^11 + 155823*x^10 - 946308*x^9 + 2059689*x^8 - 1186992*x^7 + 4368925*x^6 - 19419525*x^5 + 104639490*x^4 + 498275931*x^3 - 2708142030*x^2 + 2120234994*x + 21867232251, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 39 x^{16} - 99 x^{15} - 129 x^{14} + 1845 x^{13} + 4363 x^{12} - 46008 x^{11} + 155823 x^{10} - 946308 x^{9} + 2059689 x^{8} - 1186992 x^{7} + 4368925 x^{6} - 19419525 x^{5} + 104639490 x^{4} + 498275931 x^{3} - 2708142030 x^{2} + 2120234994 x + 21867232251 \)

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:  \(-872233768265190335916797362984151904157842243=-\,3^{33}\cdot 271^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $313.84$
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:  $3, 271$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{71083} a^{15} + \frac{7861}{71083} a^{14} + \frac{16030}{71083} a^{13} + \frac{8888}{71083} a^{12} - \frac{1629}{71083} a^{11} + \frac{20566}{71083} a^{10} + \frac{19122}{71083} a^{9} + \frac{24274}{71083} a^{8} + \frac{13443}{71083} a^{7} + \frac{1196}{71083} a^{6} + \frac{21138}{71083} a^{5} - \frac{6538}{71083} a^{4} - \frac{34}{2293} a^{3} - \frac{12752}{71083} a^{2} + \frac{1020}{2293} a + \frac{14612}{71083}$, $\frac{1}{1837134490219097730708743} a^{16} + \frac{7472253269088982689}{1837134490219097730708743} a^{15} - \frac{57141099157306590015957}{1837134490219097730708743} a^{14} + \frac{310492156189845186379718}{1837134490219097730708743} a^{13} + \frac{143057586664914299060468}{1837134490219097730708743} a^{12} + \frac{331650153547982833771461}{1837134490219097730708743} a^{11} - \frac{433374400801484500341218}{1837134490219097730708743} a^{10} + \frac{909887713451475419853300}{1837134490219097730708743} a^{9} + \frac{246969737691486178542119}{1837134490219097730708743} a^{8} + \frac{400486279968495313111009}{1837134490219097730708743} a^{7} - \frac{225719815554073537545021}{1837134490219097730708743} a^{6} + \frac{19735603067631607560084}{1837134490219097730708743} a^{5} + \frac{295295566779334453149334}{1837134490219097730708743} a^{4} + \frac{171766318098771179215963}{1837134490219097730708743} a^{3} + \frac{589350772274760271582888}{1837134490219097730708743} a^{2} - \frac{237885103497004901812303}{1837134490219097730708743} a + \frac{830115608724168813187284}{1837134490219097730708743}$, $\frac{1}{10709851026474611241157175365328422696980804628117361} a^{17} + \frac{935581426971596931341304453}{10709851026474611241157175365328422696980804628117361} a^{16} + \frac{1851495077382379711245494247928097309344345872}{345479065370148749714747592429949119257445310584431} a^{15} + \frac{4565613366188535026285790620103281600345904993744631}{10709851026474611241157175365328422696980804628117361} a^{14} + \frac{2731102229985096239560331023187150659414549075473921}{10709851026474611241157175365328422696980804628117361} a^{13} + \frac{2394830809921540563003739304627939953586848937930712}{10709851026474611241157175365328422696980804628117361} a^{12} + \frac{1895498396284924650746386610168954008879525463584478}{10709851026474611241157175365328422696980804628117361} a^{11} - \frac{890909921847510888050876495218455458735901695521235}{10709851026474611241157175365328422696980804628117361} a^{10} + \frac{1144682151917086723880200842022426834889376144201297}{10709851026474611241157175365328422696980804628117361} a^{9} - \frac{2625603957999198029498064741506167654420937031163011}{10709851026474611241157175365328422696980804628117361} a^{8} + \frac{2762717571946565884434369038378218540365017413624293}{10709851026474611241157175365328422696980804628117361} a^{7} + \frac{2461325217300835987280084692791347877358888360976117}{10709851026474611241157175365328422696980804628117361} a^{6} - \frac{3442511711800973523767324054002317217201606512074042}{10709851026474611241157175365328422696980804628117361} a^{5} - \frac{1504688725105304752384775825005546069413457704453247}{10709851026474611241157175365328422696980804628117361} a^{4} - \frac{2435631798016868829506228338218136613596939802051930}{10709851026474611241157175365328422696980804628117361} a^{3} - \frac{5307155003693981011142631479894283411739398386183714}{10709851026474611241157175365328422696980804628117361} a^{2} + \frac{1813158822706573234349380089055364874731126496208768}{10709851026474611241157175365328422696980804628117361} a + \frac{725537892168696059457594883953321017944728971862751}{10709851026474611241157175365328422696980804628117361}$

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

Class group and class number

$C_{3}\times C_{9}\times C_{9}\times C_{9}\times C_{27}$, which has order $59049$ (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:  \( \frac{2274083489486187949907684580}{26111771365972209393201946511342978627} a^{17} - \frac{20166101670959196556235218728}{26111771365972209393201946511342978627} a^{16} + \frac{113506098668031272891558836800}{26111771365972209393201946511342978627} a^{15} - \frac{426300385125157382371471981278}{26111771365972209393201946511342978627} a^{14} + \frac{533597676028520720105362409676}{26111771365972209393201946511342978627} a^{13} + \frac{2114777606439632367215865710007}{26111771365972209393201946511342978627} a^{12} + \frac{7249680870294097949654728385088}{26111771365972209393201946511342978627} a^{11} - \frac{68871487566633642148553032362870}{26111771365972209393201946511342978627} a^{10} + \frac{547179414515073363428679669909457}{26111771365972209393201946511342978627} a^{9} - \frac{3205014335322444622354579352376978}{26111771365972209393201946511342978627} a^{8} + \frac{7683270916760636618901895225125288}{26111771365972209393201946511342978627} a^{7} - \frac{24504152119512500450963066475478437}{26111771365972209393201946511342978627} a^{6} + \frac{50552996847591787804369555975652346}{26111771365972209393201946511342978627} a^{5} - \frac{91032147372229840711109808769539960}{26111771365972209393201946511342978627} a^{4} + \frac{839819815583821407743989405351392857}{26111771365972209393201946511342978627} a^{3} - \frac{428117539991850364612598402558227020}{26111771365972209393201946511342978627} a^{2} - \frac{3065042088677030637908159655622343262}{26111771365972209393201946511342978627} a + \frac{23871366434474215020142379222327668276}{26111771365972209393201946511342978627} \) (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:  \( 165741048476.84244 \) (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

\(\Q(\sqrt{-3}) \), 3.1.220323.1 x3, Deg 6, 6.0.177147.1, Deg 6, 6.0.145626672987.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{9}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$3$3.6.11.15$x^{6} + 3 x^{3} + 3$$6$$1$$11$$S_3\times C_3$$[2, 5/2]_{2}$
3.6.11.15$x^{6} + 3 x^{3} + 3$$6$$1$$11$$S_3\times C_3$$[2, 5/2]_{2}$
3.6.11.15$x^{6} + 3 x^{3} + 3$$6$$1$$11$$S_3\times C_3$$[2, 5/2]_{2}$
271Data not computed