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Properties

Label 18.0.13985883359...1875.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{33}\cdot 5^{12}\cdot 683^{12}$
Root discriminant $1699.48$
Ramified primes $3, 5, 683$
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![9167365520487, -1368271656, -132534531936, 134538470331, -4922138088, 6138070497, -1758315869, -160156980, 89645457, -25029222, 3126507, -1036062, 170809, -4383, 627, -63, 39, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 39*x^16 - 63*x^15 + 627*x^14 - 4383*x^13 + 170809*x^12 - 1036062*x^11 + 3126507*x^10 - 25029222*x^9 + 89645457*x^8 - 160156980*x^7 - 1758315869*x^6 + 6138070497*x^5 - 4922138088*x^4 + 134538470331*x^3 - 132534531936*x^2 - 1368271656*x + 9167365520487)
 
gp: K = bnfinit(x^18 - 9*x^17 + 39*x^16 - 63*x^15 + 627*x^14 - 4383*x^13 + 170809*x^12 - 1036062*x^11 + 3126507*x^10 - 25029222*x^9 + 89645457*x^8 - 160156980*x^7 - 1758315869*x^6 + 6138070497*x^5 - 4922138088*x^4 + 134538470331*x^3 - 132534531936*x^2 - 1368271656*x + 9167365520487, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 39 x^{16} - 63 x^{15} + 627 x^{14} - 4383 x^{13} + 170809 x^{12} - 1036062 x^{11} + 3126507 x^{10} - 25029222 x^{9} + 89645457 x^{8} - 160156980 x^{7} - 1758315869 x^{6} + 6138070497 x^{5} - 4922138088 x^{4} + 134538470331 x^{3} - 132534531936 x^{2} - 1368271656 x + 9167365520487 \)

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:  \(-13985883359540449487500333263200539051545952247168701171875=-\,3^{33}\cdot 5^{12}\cdot 683^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1699.48$
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:  $3, 5, 683$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{5}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{6}$, $\frac{1}{9} a^{13} + \frac{1}{9} a^{12} + \frac{1}{9} a^{11} + \frac{1}{3} a^{8} + \frac{2}{9} a^{7} - \frac{1}{9} a^{6} - \frac{4}{9} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{14} - \frac{1}{9} a^{11} - \frac{1}{9} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{2}{9} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{2338066773} a^{15} + \frac{33804056}{779355591} a^{14} + \frac{45888131}{2338066773} a^{13} + \frac{357243610}{2338066773} a^{12} - \frac{22173244}{334009539} a^{11} + \frac{71849668}{779355591} a^{10} - \frac{96308827}{2338066773} a^{9} - \frac{370091011}{779355591} a^{8} + \frac{128382526}{2338066773} a^{7} - \frac{122865616}{334009539} a^{6} + \frac{869653900}{2338066773} a^{5} + \frac{224248571}{779355591} a^{4} - \frac{15810860}{779355591} a^{3} + \frac{83137456}{779355591} a^{2} - \frac{118435206}{259785197} a + \frac{100767374}{259785197}$, $\frac{1}{12650927008407430340090340729} a^{16} - \frac{385864627498942336}{1807275286915347191441477247} a^{15} + \frac{32346305543232963349204840}{602425095638449063813825749} a^{14} + \frac{218450933442435669416594506}{4216975669469143446696780243} a^{13} - \frac{1614238322348976813050095966}{12650927008407430340090340729} a^{12} + \frac{1714089416325910007887198912}{12650927008407430340090340729} a^{11} - \frac{1337557294122135516053346088}{12650927008407430340090340729} a^{10} - \frac{259257119810804316339196253}{12650927008407430340090340729} a^{9} + \frac{73352409069828587382454119}{200808365212816354604608583} a^{8} + \frac{520825918866233307264995066}{1405658556489714482232260081} a^{7} + \frac{1390466309531766191709236290}{12650927008407430340090340729} a^{6} - \frac{3604264816782853140672205537}{12650927008407430340090340729} a^{5} - \frac{991052932718204815856192627}{4216975669469143446696780243} a^{4} + \frac{689970476545781503050908586}{1405658556489714482232260081} a^{3} + \frac{483674481558175610858396303}{4216975669469143446696780243} a^{2} + \frac{422321138636544609512719177}{1405658556489714482232260081} a + \frac{610158804476797227389146062}{1405658556489714482232260081}$, $\frac{1}{15953707410425663835644240807655768208507295377081952331} a^{17} + \frac{201484661652088389367835503}{15953707410425663835644240807655768208507295377081952331} a^{16} + \frac{41838414936051693357953418747057284504914550}{5317902470141887945214746935885256069502431792360650777} a^{15} - \frac{678784624768284645561893421935008388563926854445467911}{15953707410425663835644240807655768208507295377081952331} a^{14} - \frac{163219010593534040181032578535105012011540127042418992}{5317902470141887945214746935885256069502431792360650777} a^{13} + \frac{2408773919979334425996002020970754074140660390986703690}{15953707410425663835644240807655768208507295377081952331} a^{12} + \frac{227143461548150917856825711029986904894088119294496363}{5317902470141887945214746935885256069502431792360650777} a^{11} + \frac{1294363273095576730553435778909403259452624863359636378}{15953707410425663835644240807655768208507295377081952331} a^{10} + \frac{726517801395291483361544361658754748179229053736820033}{5317902470141887945214746935885256069502431792360650777} a^{9} + \frac{7758968556140213834992860988048514332232187471022970125}{15953707410425663835644240807655768208507295377081952331} a^{8} - \frac{111987665082781184922363071398342080471170093505998186}{253233450959137521200702235042155050928687228207650037} a^{7} - \frac{2740737688927723566321654596828571489954174557397301570}{15953707410425663835644240807655768208507295377081952331} a^{6} - \frac{4696025488321598693586214534478419338974930537846359990}{15953707410425663835644240807655768208507295377081952331} a^{5} - \frac{622729681768490428065081748431393651903010953356422721}{1772634156713962648404915645295085356500810597453550259} a^{4} - \frac{742000804478308877707719945298947745510838021297085398}{1772634156713962648404915645295085356500810597453550259} a^{3} + \frac{2849050887509917444158464268223469764240602464314343}{79371678658834148436040999043063523425409429736726131} a^{2} + \frac{240987971231705085128252054457064582704837199576234565}{1772634156713962648404915645295085356500810597453550259} a - \frac{523083536338037793343127185652710644533509617649695507}{1772634156713962648404915645295085356500810597453550259}$

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:  \( -\frac{56744133131062217564761238}{131500143009066625244837744165969943069} a^{17} + \frac{285409833296647338008297428}{131500143009066625244837744165969943069} a^{16} - \frac{76882217704286489981480696}{43833381003022208414945914721989981023} a^{15} - \frac{4142015980203335019447613630}{131500143009066625244837744165969943069} a^{14} + \frac{15270644554960253468288322014}{131500143009066625244837744165969943069} a^{13} - \frac{51201253167769160277053915173}{43833381003022208414945914721989981023} a^{12} - \frac{8172179496650214041082607442218}{131500143009066625244837744165969943069} a^{11} + \frac{23541618524880997660828379767316}{131500143009066625244837744165969943069} a^{10} + \frac{21015613618265038046733577135471}{43833381003022208414945914721989981023} a^{9} + \frac{739123920258260113946062376014030}{131500143009066625244837744165969943069} a^{8} + \frac{7140735941905987175103489185022100}{131500143009066625244837744165969943069} a^{7} - \frac{4001896246719433568047019066505775}{14611127001007402804981971573996660341} a^{6} + \frac{12905732856800859054768389986768666}{14611127001007402804981971573996660341} a^{5} - \frac{164611935831534728953199630506306438}{43833381003022208414945914721989981023} a^{4} + \frac{129653880459374347479286852125844841}{43833381003022208414945914721989981023} a^{3} - \frac{576385472077221590782817071488530986}{14611127001007402804981971573996660341} a^{2} - \frac{7095571454607828516232488738433660124}{14611127001007402804981971573996660341} a + \frac{12090920363200742605579534183897200411}{14611127001007402804981971573996660341} \) (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

\(\Q(\sqrt{-3}) \), 3.1.1399467.1 x3, 6.0.110716875.3, Deg 6, Deg 6, 6.0.5875523652267.1

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}$
683Data not computed