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Properties

Label 18.0.52172811665...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{24}\cdot 5^{9}\cdot 877^{14}$
Root discriminant $1096.12$
Ramified primes $2, 5, 877$
Class number $13262411232$ (GRH)
Class group $[2, 2, 54, 61400052]$ (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![9154324841549, 1982246556352, 1226187715060, -488373903890, 43144941139, -6887580648, 30254061027, -3685397192, -628744217, -258684356, 148150274, 5188354, -5376324, 34920, 96179, 1678, -517, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 8*x^17 - 517*x^16 + 1678*x^15 + 96179*x^14 + 34920*x^13 - 5376324*x^12 + 5188354*x^11 + 148150274*x^10 - 258684356*x^9 - 628744217*x^8 - 3685397192*x^7 + 30254061027*x^6 - 6887580648*x^5 + 43144941139*x^4 - 488373903890*x^3 + 1226187715060*x^2 + 1982246556352*x + 9154324841549)
 
gp: K = bnfinit(x^18 - 8*x^17 - 517*x^16 + 1678*x^15 + 96179*x^14 + 34920*x^13 - 5376324*x^12 + 5188354*x^11 + 148150274*x^10 - 258684356*x^9 - 628744217*x^8 - 3685397192*x^7 + 30254061027*x^6 - 6887580648*x^5 + 43144941139*x^4 - 488373903890*x^3 + 1226187715060*x^2 + 1982246556352*x + 9154324841549, 1)
 

Normalized defining polynomial

\( x^{18} - 8 x^{17} - 517 x^{16} + 1678 x^{15} + 96179 x^{14} + 34920 x^{13} - 5376324 x^{12} + 5188354 x^{11} + 148150274 x^{10} - 258684356 x^{9} - 628744217 x^{8} - 3685397192 x^{7} + 30254061027 x^{6} - 6887580648 x^{5} + 43144941139 x^{4} - 488373903890 x^{3} + 1226187715060 x^{2} + 1982246556352 x + 9154324841549 \)

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:  \(-5217281166509238670480233915527947833639206912000000000=-\,2^{24}\cdot 5^{9}\cdot 877^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1096.12$
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:  $2, 5, 877$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{6} a^{12} + \frac{1}{6} a^{10} - \frac{1}{6} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} + \frac{1}{3} a^{4} + \frac{1}{6} a^{2} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{6} a^{13} + \frac{1}{6} a^{11} - \frac{1}{6} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} + \frac{1}{3} a^{5} + \frac{1}{6} a^{3} - \frac{1}{2} a^{2} - \frac{1}{3} a$, $\frac{1}{174} a^{14} + \frac{1}{29} a^{13} - \frac{3}{58} a^{12} + \frac{9}{58} a^{11} + \frac{5}{87} a^{10} - \frac{5}{29} a^{9} - \frac{19}{87} a^{8} - \frac{27}{58} a^{7} + \frac{5}{174} a^{6} - \frac{5}{58} a^{5} + \frac{40}{87} a^{4} - \frac{25}{58} a^{3} - \frac{1}{2} a^{2} + \frac{5}{29} a + \frac{1}{87}$, $\frac{1}{440742} a^{15} - \frac{229}{440742} a^{14} - \frac{11918}{220371} a^{13} + \frac{6949}{146914} a^{12} + \frac{34235}{146914} a^{11} + \frac{34567}{220371} a^{10} + \frac{16553}{440742} a^{9} + \frac{105301}{220371} a^{8} - \frac{32327}{220371} a^{7} + \frac{2450}{220371} a^{6} - \frac{21863}{146914} a^{5} + \frac{105013}{440742} a^{4} + \frac{39538}{220371} a^{3} + \frac{65521}{146914} a^{2} - \frac{123599}{440742} a - \frac{112613}{440742}$, $\frac{1}{96424909224134982} a^{16} - \frac{80889517669}{96424909224134982} a^{15} + \frac{50531810349}{16070818204022497} a^{14} + \frac{922295423153966}{16070818204022497} a^{13} + \frac{939095979304694}{48212454612067491} a^{12} - \frac{820379175552599}{48212454612067491} a^{11} - \frac{5447020476660220}{48212454612067491} a^{10} + \frac{7761321099920789}{96424909224134982} a^{9} - \frac{23932528102157407}{96424909224134982} a^{8} + \frac{46551769906444105}{96424909224134982} a^{7} - \frac{38184039151898545}{96424909224134982} a^{6} - \frac{19752798732460954}{48212454612067491} a^{5} + \frac{4783747694371504}{16070818204022497} a^{4} + \frac{14579447544577061}{32141636408044994} a^{3} - \frac{1873666190540000}{48212454612067491} a^{2} - \frac{22892944387677245}{96424909224134982} a - \frac{217731905813}{16070818204022497}$, $\frac{1}{64070425775160038783733570567724171225331734651523164706129621457096321189256662} a^{17} - \frac{90086745563063495240974612387013062690109326382749917355054749}{21356808591720012927911190189241390408443911550507721568709873819032107063085554} a^{16} - \frac{14607851986233027491263730621641908622135045074029107564022508042230149883}{32035212887580019391866785283862085612665867325761582353064810728548160594628331} a^{15} - \frac{10728876812810666456790409246349872090331328142191517408858724031073618664877}{21356808591720012927911190189241390408443911550507721568709873819032107063085554} a^{14} + \frac{3272493076657453965684516104804371687830738659913823910377200680842353533616019}{64070425775160038783733570567724171225331734651523164706129621457096321189256662} a^{13} + \frac{1206558313820739132680800034813265734404030013907959932475018715983102901246791}{64070425775160038783733570567724171225331734651523164706129621457096321189256662} a^{12} + \frac{2211688395292753174208597615088152738400996492335639336006232271327893210057739}{21356808591720012927911190189241390408443911550507721568709873819032107063085554} a^{11} + \frac{12630133064729875278879111965488177545612833071037813274154457372156848369497287}{64070425775160038783733570567724171225331734651523164706129621457096321189256662} a^{10} + \frac{7345243690255554927688123848988804815168792194104604256015867212599486364207696}{32035212887580019391866785283862085612665867325761582353064810728548160594628331} a^{9} + \frac{10642551841176644458541999908831299389091265605254789980871501280988676542510927}{64070425775160038783733570567724171225331734651523164706129621457096321189256662} a^{8} - \frac{102840405629175911042775805954074614938388184870454099137695664118143472299293}{628141429168235674350329123212982070836585633838462399079702171148003148914281} a^{7} - \frac{9361618930524224087982011366638061607666131776002655599666669606625459665846033}{21356808591720012927911190189241390408443911550507721568709873819032107063085554} a^{6} - \frac{26235328625543703621105376628956738418697210726872164693851309646087970302171085}{64070425775160038783733570567724171225331734651523164706129621457096321189256662} a^{5} + \frac{26296261092875914327556811862202282166340144025932591642237002343015967478653539}{64070425775160038783733570567724171225331734651523164706129621457096321189256662} a^{4} + \frac{73506047657094560039849866229959464302171402691414907649171565896912838351569}{64070425775160038783733570567724171225331734651523164706129621457096321189256662} a^{3} - \frac{89119458633967897186662608605748810526674904925661326426345260558238158687197}{32035212887580019391866785283862085612665867325761582353064810728548160594628331} a^{2} + \frac{24096210380606928402272533600896177207337047832267793310072412097657497832891065}{64070425775160038783733570567724171225331734651523164706129621457096321189256662} a - \frac{11298618085976876849144608244228289822073336000895633674664262161695403363938988}{32035212887580019391866785283862085612665867325761582353064810728548160594628331}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{54}\times C_{61400052}$, which has order $13262411232$ (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:  \( 11318340440.066559 \) (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

\(\Q(\sqrt{-5}) \), 3.3.3508.1, 3.3.769129.1, 6.0.24612128000.1, 6.0.4732475349128000.1, 9.9.25537426374121078096192.1

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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
2Data not computed
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$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}$
877Data not computed