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Properties

Label 18.0.20803075197...1875.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{33}\cdot 5^{12}\cdot 397^{12}$
Root discriminant $1183.67$
Ramified primes $3, 5, 397$
Class number Not computed
Class group Not computed
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![522764442927, 99972009, -14623618176, 14258110416, -81393138, 541076877, -143151569, -79679565, 36038982, -8088882, 1068837, -404667, 65374, 4122, -588, -63, 39, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 39*x^16 - 63*x^15 - 588*x^14 + 4122*x^13 + 65374*x^12 - 404667*x^11 + 1068837*x^10 - 8088882*x^9 + 36038982*x^8 - 79679565*x^7 - 143151569*x^6 + 541076877*x^5 - 81393138*x^4 + 14258110416*x^3 - 14623618176*x^2 + 99972009*x + 522764442927)
 
gp: K = bnfinit(x^18 - 9*x^17 + 39*x^16 - 63*x^15 - 588*x^14 + 4122*x^13 + 65374*x^12 - 404667*x^11 + 1068837*x^10 - 8088882*x^9 + 36038982*x^8 - 79679565*x^7 - 143151569*x^6 + 541076877*x^5 - 81393138*x^4 + 14258110416*x^3 - 14623618176*x^2 + 99972009*x + 522764442927, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 39 x^{16} - 63 x^{15} - 588 x^{14} + 4122 x^{13} + 65374 x^{12} - 404667 x^{11} + 1068837 x^{10} - 8088882 x^{9} + 36038982 x^{8} - 79679565 x^{7} - 143151569 x^{6} + 541076877 x^{5} - 81393138 x^{4} + 14258110416 x^{3} - 14623618176 x^{2} + 99972009 x + 522764442927 \)

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:  \(-20803075197746806021621278075959514606918210410888671875=-\,3^{33}\cdot 5^{12}\cdot 397^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1183.67$
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:  $3, 5, 397$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{4}$, $\frac{1}{9} a^{11} - \frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{2}{9} a^{5} + \frac{1}{9} a^{4} - \frac{4}{9} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{9} a^{12} + \frac{1}{9} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{9} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{4}{9} a^{3} - \frac{1}{3}$, $\frac{1}{9} a^{13} + \frac{1}{9} a^{10} - \frac{1}{3} a^{8} - \frac{1}{9} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{4}{9} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{10} - \frac{1}{9} a^{9} - \frac{4}{9} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{2}{9} a^{4} + \frac{1}{9} a^{3} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{393347997} a^{15} - \frac{3866983}{131115999} a^{14} - \frac{12919478}{393347997} a^{13} + \frac{4208104}{393347997} a^{12} + \frac{17826125}{393347997} a^{11} - \frac{19842160}{393347997} a^{10} + \frac{43693162}{393347997} a^{9} - \frac{9501855}{43705333} a^{8} - \frac{10914865}{393347997} a^{7} + \frac{86199590}{393347997} a^{6} - \frac{115417115}{393347997} a^{5} - \frac{27888551}{56192571} a^{4} + \frac{185008090}{393347997} a^{3} + \frac{16398454}{131115999} a^{2} + \frac{38500423}{131115999} a - \frac{62894543}{131115999}$, $\frac{1}{11848652701178376565216791} a^{16} + \frac{6152735096631806}{11848652701178376565216791} a^{15} - \frac{456278267252972661225937}{11848652701178376565216791} a^{14} - \frac{97341065509797978569206}{11848652701178376565216791} a^{13} - \frac{608569837436120425288393}{11848652701178376565216791} a^{12} - \frac{168360600932818022015731}{3949550900392792188405597} a^{11} + \frac{475318215519094080552638}{11848652701178376565216791} a^{10} - \frac{381143222972379939707053}{11848652701178376565216791} a^{9} - \frac{4565891589405312726527705}{11848652701178376565216791} a^{8} - \frac{5190493515456572519290205}{11848652701178376565216791} a^{7} + \frac{2099741185726369515351922}{11848652701178376565216791} a^{6} - \frac{64095524305351540655865}{188073852399656770876457} a^{5} + \frac{294317301839868551508919}{1316516966797597396135199} a^{4} + \frac{252854533511964018287939}{11848652701178376565216791} a^{3} + \frac{251317650400029050589823}{1316516966797597396135199} a^{2} + \frac{615794651944366856618863}{1316516966797597396135199} a - \frac{61287989345238375931582}{564221557198970312629371}$, $\frac{1}{768325882757931006134697955264215574963180568338371} a^{17} - \frac{14060651861970082073554295}{768325882757931006134697955264215574963180568338371} a^{16} - \frac{532158964623161088248502009195318556669616}{768325882757931006134697955264215574963180568338371} a^{15} + \frac{4544588649489662735677482952523098392068315227466}{768325882757931006134697955264215574963180568338371} a^{14} + \frac{6832575722914158619139616196873412754885351599132}{768325882757931006134697955264215574963180568338371} a^{13} - \frac{673899233246360995641037543156436616086565068027}{85369542528659000681633106140468397218131174259819} a^{12} - \frac{25443974845174975363823887751098398787315037490406}{768325882757931006134697955264215574963180568338371} a^{11} - \frac{9959582978759551615881638906372581168054636250510}{768325882757931006134697955264215574963180568338371} a^{10} - \frac{22236796139356264014865836044853869792040394611710}{256108627585977002044899318421405191654393522779457} a^{9} + \frac{1210775329554779402967485432218493337953276246908}{17868043785068162933365068727074780813097222519497} a^{8} - \frac{18597032219633063020339224155421009022403872161602}{109760840393990143733528279323459367851882938334053} a^{7} + \frac{21584203188703407839165588708942138976272168119359}{85369542528659000681633106140468397218131174259819} a^{6} + \frac{26833521208909296376844503227246751782812129035419}{85369542528659000681633106140468397218131174259819} a^{5} + \frac{30989965200094350578662228646653764431479261773640}{768325882757931006134697955264215574963180568338371} a^{4} - \frac{299816541023796533314504645820133476461799301066073}{768325882757931006134697955264215574963180568338371} a^{3} + \frac{37671802984747957863532941480716445154657009457960}{85369542528659000681633106140468397218131174259819} a^{2} + \frac{41297209818482321061481233186536330083386407450803}{256108627585977002044899318421405191654393522779457} a - \frac{120507408663287073075498187083747156904523096930572}{256108627585977002044899318421405191654393522779457}$

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

Class group and class number

Not computed

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:  \( \frac{995430876989319305772818}{147831175125307595018122227978412779} a^{17} - \frac{7279577821542634900744858}{147831175125307595018122227978412779} a^{16} + \frac{13128240958953792690541388}{147831175125307595018122227978412779} a^{15} + \frac{75078634467614854097123230}{147831175125307595018122227978412779} a^{14} - \frac{117178517949473319275913086}{16425686125034177224235803108712531} a^{13} + \frac{3942019625963407798280855189}{147831175125307595018122227978412779} a^{12} + \frac{25983251961785236003487323426}{49277058375102531672707409326137593} a^{11} - \frac{334239245348774538887050542826}{147831175125307595018122227978412779} a^{10} - \frac{303363118314146162080516249793}{147831175125307595018122227978412779} a^{9} - \frac{3691055152864462344844257843160}{147831175125307595018122227978412779} a^{8} + \frac{767599888634987695890477472520}{16425686125034177224235803108712531} a^{7} + \frac{83538293253266330808920330292535}{147831175125307595018122227978412779} a^{6} - \frac{373442758283843656417554126978344}{147831175125307595018122227978412779} a^{5} + \frac{859794473804616601933350065338304}{147831175125307595018122227978412779} a^{4} - \frac{200440355643756335995773354040411}{49277058375102531672707409326137593} a^{3} + \frac{3362602184877288172846405165325368}{49277058375102531672707409326137593} a^{2} + \frac{9502814176785031355000327970510032}{49277058375102531672707409326137593} a - \frac{2050412928999706438020829892473435}{16425686125034177224235803108712531} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
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

\(\Q(\sqrt{-3}) \), 3.1.472827.1 x3, Deg 6, 6.0.110716875.3, Deg 6, 6.0.670696115787.1

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

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$3.6.11.18$x^{6} + 21 x^{3} + 12$$6$$1$$11$$S_3\times C_3$$[2, 5/2]_{2}$
3.6.11.18$x^{6} + 21 x^{3} + 12$$6$$1$$11$$S_3\times C_3$$[2, 5/2]_{2}$
3.6.11.18$x^{6} + 21 x^{3} + 12$$6$$1$$11$$S_3\times C_3$$[2, 5/2]_{2}$
$5$5.6.4.2$x^{6} - 5 x^{3} + 50$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
5.6.4.2$x^{6} - 5 x^{3} + 50$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
5.6.4.2$x^{6} - 5 x^{3} + 50$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
397Data not computed