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Properties

Label 18.0.26777612762...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{31}\cdot 5^{9}\cdot 7^{14}\cdot 19^{14}$
Root discriminant $1056.25$
Ramified primes $2, 3, 5, 7, 19$
Class number $47659588704$ (GRH)
Class group $[3, 6, 12, 220646244]$ (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![3159025422781024, 2932106044957716, 286839979293732, -419376128500893, -81820212046617, 26905061270919, 6760093363683, -759018551721, -240910122405, 10471753955, 4474849215, -66738279, -46410555, 73797, 269049, 1005, -807, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 807*x^16 + 1005*x^15 + 269049*x^14 + 73797*x^13 - 46410555*x^12 - 66738279*x^11 + 4474849215*x^10 + 10471753955*x^9 - 240910122405*x^8 - 759018551721*x^7 + 6760093363683*x^6 + 26905061270919*x^5 - 81820212046617*x^4 - 419376128500893*x^3 + 286839979293732*x^2 + 2932106044957716*x + 3159025422781024)
 
gp: K = bnfinit(x^18 - 3*x^17 - 807*x^16 + 1005*x^15 + 269049*x^14 + 73797*x^13 - 46410555*x^12 - 66738279*x^11 + 4474849215*x^10 + 10471753955*x^9 - 240910122405*x^8 - 759018551721*x^7 + 6760093363683*x^6 + 26905061270919*x^5 - 81820212046617*x^4 - 419376128500893*x^3 + 286839979293732*x^2 + 2932106044957716*x + 3159025422781024, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 807 x^{16} + 1005 x^{15} + 269049 x^{14} + 73797 x^{13} - 46410555 x^{12} - 66738279 x^{11} + 4474849215 x^{10} + 10471753955 x^{9} - 240910122405 x^{8} - 759018551721 x^{7} + 6760093363683 x^{6} + 26905061270919 x^{5} - 81820212046617 x^{4} - 419376128500893 x^{3} + 286839979293732 x^{2} + 2932106044957716 x + 3159025422781024 \)

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:  \(-2677761276299797163442203648730856426156058904000000000=-\,2^{12}\cdot 3^{31}\cdot 5^{9}\cdot 7^{14}\cdot 19^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1056.25$
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:  $2, 3, 5, 7, 19$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{4} + \frac{1}{8} a^{3} + \frac{1}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{48} a^{6} - \frac{1}{16} a^{5} + \frac{1}{16} a^{4} - \frac{1}{48} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{1}{3}$, $\frac{1}{192} a^{7} - \frac{1}{32} a^{5} - \frac{1}{12} a^{4} + \frac{3}{64} a^{3} - \frac{5}{48} a - \frac{1}{2}$, $\frac{1}{768} a^{8} + \frac{1}{768} a^{7} - \frac{1}{128} a^{6} - \frac{11}{384} a^{5} - \frac{7}{768} a^{4} - \frac{29}{256} a^{3} - \frac{5}{192} a^{2} - \frac{5}{192} a - \frac{1}{8}$, $\frac{1}{1536} a^{9} + \frac{1}{1536} a^{7} - \frac{1}{96} a^{6} - \frac{11}{512} a^{5} + \frac{11}{96} a^{4} - \frac{245}{1536} a^{3} - \frac{1}{4} a^{2} + \frac{37}{384} a - \frac{7}{16}$, $\frac{1}{6144} a^{10} + \frac{1}{6144} a^{9} + \frac{1}{6144} a^{8} + \frac{1}{6144} a^{7} + \frac{5}{2048} a^{6} + \frac{47}{6144} a^{5} + \frac{443}{6144} a^{4} + \frac{393}{2048} a^{3} - \frac{11}{1536} a^{2} + \frac{269}{1536} a - \frac{85}{192}$, $\frac{1}{73728} a^{11} - \frac{1}{18432} a^{10} + \frac{1}{18432} a^{9} - \frac{1}{2048} a^{8} + \frac{19}{12288} a^{7} - \frac{61}{6144} a^{6} + \frac{59}{3072} a^{5} - \frac{83}{2048} a^{4} - \frac{4393}{24576} a^{3} - \frac{695}{4608} a^{2} + \frac{7391}{18432} a + \frac{161}{2304}$, $\frac{1}{294912} a^{12} + \frac{1}{294912} a^{11} + \frac{1}{36864} a^{10} - \frac{11}{36864} a^{9} - \frac{7}{49152} a^{8} + \frac{89}{49152} a^{7} - \frac{29}{24576} a^{6} + \frac{105}{8192} a^{5} - \frac{3269}{98304} a^{4} + \frac{18577}{294912} a^{3} - \frac{6773}{73728} a^{2} - \frac{27205}{73728} a + \frac{781}{9216}$, $\frac{1}{1179648} a^{13} - \frac{1}{589824} a^{12} - \frac{1}{393216} a^{11} - \frac{5}{73728} a^{10} + \frac{95}{589824} a^{9} - \frac{49}{98304} a^{8} + \frac{97}{65536} a^{7} - \frac{65}{16384} a^{6} - \frac{24137}{393216} a^{5} + \frac{10607}{589824} a^{4} - \frac{169391}{1179648} a^{3} + \frac{11519}{49152} a^{2} - \frac{12737}{294912} a + \frac{16337}{36864}$, $\frac{1}{4718592} a^{14} + \frac{1}{4718592} a^{13} + \frac{7}{4718592} a^{12} + \frac{23}{4718592} a^{11} + \frac{13}{786432} a^{10} + \frac{439}{2359296} a^{9} + \frac{397}{786432} a^{8} + \frac{589}{786432} a^{7} - \frac{7121}{1572864} a^{6} - \frac{19955}{4718592} a^{5} + \frac{241147}{4718592} a^{4} - \frac{895925}{4718592} a^{3} - \frac{215587}{1179648} a^{2} + \frac{52397}{131072} a + \frac{55075}{147456}$, $\frac{1}{75497472} a^{15} + \frac{1}{12582912} a^{13} + \frac{5}{4718592} a^{12} + \frac{247}{75497472} a^{11} - \frac{119}{2359296} a^{10} - \frac{641}{2359296} a^{9} + \frac{25}{49152} a^{8} - \frac{14059}{25165824} a^{7} + \frac{2405}{589824} a^{6} + \frac{653437}{12582912} a^{5} + \frac{16163}{1572864} a^{4} - \frac{17484823}{75497472} a^{3} + \frac{349919}{2359296} a^{2} - \frac{799933}{18874368} a + \frac{665245}{2359296}$, $\frac{1}{7247757312} a^{16} + \frac{19}{7247757312} a^{15} - \frac{31}{1207959552} a^{14} + \frac{385}{3623878656} a^{13} + \frac{2471}{7247757312} a^{12} + \frac{4301}{805306368} a^{11} + \frac{6379}{113246208} a^{10} - \frac{64807}{226492416} a^{9} + \frac{441991}{805306368} a^{8} - \frac{8558675}{7247757312} a^{7} + \frac{3912791}{3623878656} a^{6} - \frac{68247745}{1207959552} a^{5} + \frac{871369817}{7247757312} a^{4} + \frac{1175494379}{7247757312} a^{3} + \frac{48375635}{201326592} a^{2} - \frac{402853343}{1811939328} a - \frac{12293017}{226492416}$, $\frac{1}{330006961902117628528638606167822212334914202382886857790961221632} a^{17} - \frac{3560735664200639055628390982553693674115480714058147895}{55001160317019604754773101027970368722485700397147809631826870272} a^{16} - \frac{398021044890747451758994268864090502814658649615684434977}{330006961902117628528638606167822212334914202382886857790961221632} a^{15} + \frac{5210388372594237196234257719907293709906752474859045028901}{82501740475529407132159651541955553083728550595721714447740305408} a^{14} + \frac{37322320147479483681917685599975560518874001982241573671503}{110002320634039209509546202055940737444971400794295619263653740544} a^{13} + \frac{134464089805600845609768439326542511458687464538911787842373}{165003480951058814264319303083911106167457101191443428895480610816} a^{12} - \frac{1126579697542601259527521024816445211526475238497634931886849}{330006961902117628528638606167822212334914202382886857790961221632} a^{11} + \frac{9445339842954783322627700673289462206169731885443250621569}{381952502201525033019257646027572005017261808313526455776575488} a^{10} + \frac{25533296769648378446689927460408171404035604349846964900825119}{330006961902117628528638606167822212334914202382886857790961221632} a^{9} - \frac{21582237567982203341345543287157033510674357954250060945627995}{165003480951058814264319303083911106167457101191443428895480610816} a^{8} + \frac{54869628096630389099978567295788847168054962150448537385106375}{110002320634039209509546202055940737444971400794295619263653740544} a^{7} + \frac{852549507800622433661957679031640340264419996421124267765417361}{82501740475529407132159651541955553083728550595721714447740305408} a^{6} + \frac{3605634159748696100466647381938571923036285586356476788949426695}{330006961902117628528638606167822212334914202382886857790961221632} a^{5} + \frac{4665351677637353105474781338298423670359512703123371017996422553}{55001160317019604754773101027970368722485700397147809631826870272} a^{4} - \frac{37561097137868727700467349042309773535683203391494556178563722291}{330006961902117628528638606167822212334914202382886857790961221632} a^{3} + \frac{4174373578689112411225061814050832790336184630092905900342095969}{41250870237764703566079825770977776541864275297860857223870152704} a^{2} + \frac{955670439684150322694538617220905898347601020405138239142183587}{9166860052836600792462183504661728120414283399524634938637811712} a - \frac{2382697876525771445418323557738829164243728353623403193357280235}{10312717559441175891519956442744444135466068824465214305967538176}$

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

Class group and class number

$C_{3}\times C_{6}\times C_{12}\times C_{220646244}$, which has order $47659588704$ (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:  \( 5963727865967.527 \) (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

\(\Q(\sqrt{-15}) \), 3.3.1432809.2, 3.3.71820.1, 6.0.769853111430375.4, 6.0.77371686000.1, 9.9.84502643371668907233912000.2

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 R R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{6}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
3Data not computed
$5$5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$7$7.6.4.2$x^{6} - 7 x^{3} + 147$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.12.10.3$x^{12} - 49 x^{6} + 3969$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$19$19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
19.6.5.6$x^{6} + 19456$$6$$1$$5$$C_6$$[\ ]_{6}$
19.6.5.6$x^{6} + 19456$$6$$1$$5$$C_6$$[\ ]_{6}$