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Properties

Label 18.0.27526280777...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{24}\cdot 3^{6}\cdot 5^{9}\cdot 271^{14}$
Root discriminant $634.17$
Ramified primes $2, 3, 5, 271$
Class number $1827033880$ (GRH)
Class group $[86, 21244580]$ (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![245644707669, 282909781276, 210708993481, 87252708550, 20797548331, -146306990, -1540727786, -328077626, 56319898, 37744642, 3702763, -1389596, -295241, 23272, 9349, -50, -140, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 - 140*x^16 - 50*x^15 + 9349*x^14 + 23272*x^13 - 295241*x^12 - 1389596*x^11 + 3702763*x^10 + 37744642*x^9 + 56319898*x^8 - 328077626*x^7 - 1540727786*x^6 - 146306990*x^5 + 20797548331*x^4 + 87252708550*x^3 + 210708993481*x^2 + 282909781276*x + 245644707669)
 
gp: K = bnfinit(x^18 - 2*x^17 - 140*x^16 - 50*x^15 + 9349*x^14 + 23272*x^13 - 295241*x^12 - 1389596*x^11 + 3702763*x^10 + 37744642*x^9 + 56319898*x^8 - 328077626*x^7 - 1540727786*x^6 - 146306990*x^5 + 20797548331*x^4 + 87252708550*x^3 + 210708993481*x^2 + 282909781276*x + 245644707669, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} - 140 x^{16} - 50 x^{15} + 9349 x^{14} + 23272 x^{13} - 295241 x^{12} - 1389596 x^{11} + 3702763 x^{10} + 37744642 x^{9} + 56319898 x^{8} - 328077626 x^{7} - 1540727786 x^{6} - 146306990 x^{5} + 20797548331 x^{4} + 87252708550 x^{3} + 210708993481 x^{2} + 282909781276 x + 245644707669 \)

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:  \(-275262807777657994100187725010026740088832000000000=-\,2^{24}\cdot 3^{6}\cdot 5^{9}\cdot 271^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $634.17$
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:  $2, 3, 5, 271$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{9} a^{7} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{4}{9} a + \frac{1}{3}$, $\frac{1}{9} a^{8} + \frac{1}{3} a^{4} - \frac{4}{9} a^{2}$, $\frac{1}{54} a^{9} - \frac{1}{54} a^{7} + \frac{1}{18} a^{6} + \frac{1}{18} a^{5} + \frac{1}{27} a^{3} + \frac{1}{6} a^{2} - \frac{5}{54} a + \frac{5}{18}$, $\frac{1}{54} a^{10} - \frac{1}{54} a^{8} - \frac{1}{18} a^{7} + \frac{1}{18} a^{6} - \frac{8}{27} a^{4} - \frac{1}{6} a^{3} - \frac{23}{54} a^{2} - \frac{5}{18} a - \frac{1}{3}$, $\frac{1}{162} a^{11} + \frac{1}{162} a^{9} + \frac{1}{54} a^{8} + \frac{1}{162} a^{7} + \frac{1}{27} a^{6} + \frac{4}{81} a^{5} - \frac{5}{18} a^{4} + \frac{53}{162} a^{3} - \frac{7}{54} a^{2} - \frac{32}{81} a - \frac{4}{27}$, $\frac{1}{162} a^{12} + \frac{1}{162} a^{10} + \frac{1}{162} a^{8} - \frac{1}{18} a^{7} - \frac{1}{162} a^{6} - \frac{1}{162} a^{4} - \frac{1}{6} a^{3} + \frac{17}{162} a^{2} - \frac{5}{18} a + \frac{7}{18}$, $\frac{1}{1458} a^{13} - \frac{1}{486} a^{11} + \frac{2}{243} a^{10} + \frac{1}{243} a^{9} + \frac{13}{486} a^{8} + \frac{38}{729} a^{7} - \frac{5}{486} a^{6} + \frac{35}{243} a^{5} - \frac{157}{486} a^{4} - \frac{221}{486} a^{3} + \frac{7}{243} a^{2} + \frac{308}{729} a - \frac{31}{486}$, $\frac{1}{1458} a^{14} - \frac{1}{486} a^{12} + \frac{1}{486} a^{11} + \frac{1}{243} a^{10} + \frac{1}{486} a^{9} + \frac{49}{1458} a^{8} + \frac{1}{486} a^{7} + \frac{25}{486} a^{6} - \frac{23}{243} a^{5} - \frac{43}{243} a^{4} - \frac{1}{486} a^{3} + \frac{281}{729} a^{2} - \frac{59}{243} a - \frac{7}{54}$, $\frac{1}{4374} a^{15} - \frac{1}{4374} a^{13} + \frac{2}{729} a^{12} - \frac{1}{243} a^{10} + \frac{7}{4374} a^{9} + \frac{8}{243} a^{8} + \frac{100}{2187} a^{7} + \frac{157}{1458} a^{6} - \frac{1}{9} a^{5} - \frac{59}{243} a^{4} - \frac{1601}{4374} a^{3} - \frac{199}{486} a^{2} + \frac{211}{2187} a - \frac{467}{1458}$, $\frac{1}{56862} a^{16} - \frac{1}{18954} a^{15} + \frac{17}{56862} a^{14} - \frac{1}{6318} a^{13} - \frac{2}{3159} a^{12} + \frac{2}{1053} a^{11} + \frac{421}{56862} a^{10} + \frac{5}{1458} a^{9} - \frac{116}{28431} a^{8} - \frac{175}{6318} a^{7} + \frac{1}{234} a^{6} + \frac{22}{351} a^{5} - \frac{22913}{56862} a^{4} + \frac{1498}{9477} a^{3} + \frac{23111}{56862} a^{2} + \frac{47}{486} a - \frac{635}{6318}$, $\frac{1}{75990708950021984042852281670031701843186230709460881417202} a^{17} + \frac{2125932385922053666819481848586116713595754783289187}{12665118158336997340475380278338616973864371784910146902867} a^{16} - \frac{2566241775456137558139285285616813922557691710615377959}{75990708950021984042852281670031701843186230709460881417202} a^{15} - \frac{93711548230392269289597791266722476883503844048296929}{436728212356448184154323457873745412891874889134832651823} a^{14} - \frac{7018242228929480022665911625916044887346718241176981037}{37995354475010992021426140835015850921593115354730440708601} a^{13} - \frac{35225204669867802126497599806175699182761153395022677346}{12665118158336997340475380278338616973864371784910146902867} a^{12} + \frac{129545479339740453500539038667934588157365361299165987173}{75990708950021984042852281670031701843186230709460881417202} a^{11} + \frac{172554273319435682453505322958312223871226383230012829}{45971390774362966753086679776183727672828935698403437034} a^{10} + \frac{258112966786828162890784796025860204912221585984808515909}{75990708950021984042852281670031701843186230709460881417202} a^{9} - \frac{504240956534943111340807995689781935213318379867051863465}{12665118158336997340475380278338616973864371784910146902867} a^{8} + \frac{96397083093704110995286523724729416156967444479281266989}{1999755498684789053759270570263992153768058702880549510979} a^{7} - \frac{367379213354073839362888590594805200398411428381844703152}{12665118158336997340475380278338616973864371784910146902867} a^{6} + \frac{5022026173699257542434496726821033407694606942852688439503}{37995354475010992021426140835015850921593115354730440708601} a^{5} + \frac{1954303519378047208692217882342182506521641326328490769045}{25330236316673994680950760556677233947728743569820293805734} a^{4} - \frac{29781981696196956816813072700533221488579526278337290469645}{75990708950021984042852281670031701843186230709460881417202} a^{3} + \frac{4744354443959166015096908151798435335504808641208238061117}{25330236316673994680950760556677233947728743569820293805734} a^{2} - \frac{8924867507691742881551102802186554057168113837497078757807}{37995354475010992021426140835015850921593115354730440708601} a + \frac{250708806402920946778171410040963176163800672412995215185}{873456424712896368308646915747490825783749778269665303646}$

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

Class group and class number

$C_{86}\times C_{21244580}$, which has order $1827033880$ (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:  \( 48005482170.92574 \) (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

\(\Q(\sqrt{-5}) \), 3.3.73441.1, 3.3.3252.1, 6.0.21151008000.1, 6.0.43148643848000.1, 9.9.185493533505098902848.1

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 R R ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
2Data not computed
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$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}$
271Data not computed