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Properties

Label 18.0.48805253563...0000.3
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{31}\cdot 5^{9}\cdot 61^{14}$
Root discriminant $576.06$
Ramified primes $2, 3, 5, 61$
Class number $2986669260$ (GRH)
Class group $[3, 3, 18, 18436230]$ (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![8983456251904, -13774599094272, 9971179180032, -3757461785088, 512148044928, 158074945248, -70540733496, 972777468, 4765541070, -927212375, -47149131, 31785648, -1669332, -428718, 44394, 2172, -354, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 354*x^16 + 2172*x^15 + 44394*x^14 - 428718*x^13 - 1669332*x^12 + 31785648*x^11 - 47149131*x^10 - 927212375*x^9 + 4765541070*x^8 + 972777468*x^7 - 70540733496*x^6 + 158074945248*x^5 + 512148044928*x^4 - 3757461785088*x^3 + 9971179180032*x^2 - 13774599094272*x + 8983456251904)
 
gp: K = bnfinit(x^18 - 3*x^17 - 354*x^16 + 2172*x^15 + 44394*x^14 - 428718*x^13 - 1669332*x^12 + 31785648*x^11 - 47149131*x^10 - 927212375*x^9 + 4765541070*x^8 + 972777468*x^7 - 70540733496*x^6 + 158074945248*x^5 + 512148044928*x^4 - 3757461785088*x^3 + 9971179180032*x^2 - 13774599094272*x + 8983456251904, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 354 x^{16} + 2172 x^{15} + 44394 x^{14} - 428718 x^{13} - 1669332 x^{12} + 31785648 x^{11} - 47149131 x^{10} - 927212375 x^{9} + 4765541070 x^{8} + 972777468 x^{7} - 70540733496 x^{6} + 158074945248 x^{5} + 512148044928 x^{4} - 3757461785088 x^{3} + 9971179180032 x^{2} - 13774599094272 x + 8983456251904 \)

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:  \(-48805253563272524881557588987780226277016000000000=-\,2^{12}\cdot 3^{31}\cdot 5^{9}\cdot 61^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $576.06$
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:  $2, 3, 5, 61$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{12} a^{5} + \frac{1}{12} a^{4} + \frac{1}{12} a^{3} - \frac{1}{12} a^{2} + \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{48} a^{6} + \frac{1}{48} a^{5} - \frac{5}{48} a^{4} + \frac{11}{48} a^{3} - \frac{1}{12} a^{2} - \frac{1}{12} a$, $\frac{1}{48} a^{7} - \frac{1}{24} a^{5} - \frac{1}{12} a^{4} - \frac{11}{48} a^{3} - \frac{1}{12} a^{2} - \frac{1}{4} a - \frac{1}{3}$, $\frac{1}{96} a^{8} - \frac{1}{48} a^{5} + \frac{1}{32} a^{4} - \frac{1}{16} a^{3} - \frac{5}{24} a^{2} + \frac{1}{4} a$, $\frac{1}{3456} a^{9} - \frac{1}{576} a^{8} - \frac{1}{576} a^{7} - \frac{1}{288} a^{6} + \frac{19}{1152} a^{5} - \frac{1}{576} a^{4} - \frac{59}{288} a^{3} + \frac{35}{144} a^{2} + \frac{17}{36} a + \frac{4}{27}$, $\frac{1}{6912} a^{10} - \frac{1}{6912} a^{9} - \frac{7}{1152} a^{7} + \frac{23}{2304} a^{6} - \frac{1}{768} a^{5} + \frac{25}{384} a^{4} - \frac{19}{192} a^{3} + \frac{23}{96} a^{2} - \frac{53}{216} a + \frac{1}{27}$, $\frac{1}{13824} a^{11} - \frac{1}{13824} a^{10} - \frac{1}{6912} a^{9} + \frac{7}{2304} a^{8} + \frac{3}{512} a^{7} - \frac{43}{4608} a^{6} - \frac{11}{288} a^{5} - \frac{13}{576} a^{4} + \frac{5}{144} a^{3} + \frac{203}{864} a^{2} - \frac{29}{216} a - \frac{11}{27}$, $\frac{1}{27648} a^{12} - \frac{1}{27648} a^{10} + \frac{1}{13824} a^{9} - \frac{31}{9216} a^{8} - \frac{17}{2304} a^{7} + \frac{67}{9216} a^{6} - \frac{5}{4608} a^{5} + \frac{71}{2304} a^{4} - \frac{443}{3456} a^{3} + \frac{7}{72} a^{2} - \frac{103}{216} a + \frac{4}{27}$, $\frac{1}{110592} a^{13} + \frac{1}{36864} a^{11} + \frac{1}{18432} a^{10} + \frac{1}{36864} a^{9} - \frac{11}{9216} a^{8} - \frac{353}{36864} a^{7} + \frac{65}{18432} a^{6} - \frac{11}{9216} a^{5} - \frac{1673}{13824} a^{4} - \frac{247}{1152} a^{3} + \frac{13}{96} a^{2} - \frac{1}{8} a - \frac{1}{3}$, $\frac{1}{663552} a^{14} + \frac{1}{663552} a^{13} + \frac{7}{663552} a^{12} - \frac{7}{663552} a^{11} - \frac{43}{663552} a^{10} - \frac{25}{663552} a^{9} + \frac{407}{221184} a^{8} + \frac{1505}{221184} a^{7} - \frac{215}{110592} a^{6} - \frac{577}{165888} a^{5} + \frac{7933}{82944} a^{4} - \frac{311}{5184} a^{3} + \frac{623}{5184} a^{2} + \frac{97}{648} a + \frac{2}{81}$, $\frac{1}{1327104} a^{15} - \frac{1}{1327104} a^{14} - \frac{1}{1327104} a^{13} + \frac{1}{442368} a^{12} - \frac{47}{1327104} a^{11} + \frac{1}{1327104} a^{10} + \frac{149}{1327104} a^{9} + \frac{211}{442368} a^{8} - \frac{325}{221184} a^{7} - \frac{2767}{331776} a^{6} - \frac{1417}{165888} a^{5} + \frac{913}{20736} a^{4} + \frac{23}{288} a^{3} - \frac{275}{2592} a^{2} - \frac{22}{81} a + \frac{22}{81}$, $\frac{1}{2654208} a^{16} - \frac{1}{1327104} a^{14} + \frac{1}{1327104} a^{13} + \frac{1}{663552} a^{12} - \frac{23}{1327104} a^{11} - \frac{5}{147456} a^{10} - \frac{41}{1327104} a^{9} - \frac{4231}{884736} a^{8} + \frac{9907}{1327104} a^{7} - \frac{1015}{221184} a^{6} - \frac{5741}{331776} a^{5} - \frac{205}{20736} a^{4} + \frac{4069}{20736} a^{3} - \frac{161}{2592} a^{2} + \frac{23}{54} a - \frac{16}{81}$, $\frac{1}{2137029968934953295225744650148334952513352800591413248} a^{17} + \frac{188387545450031423374941242355433782884825072021}{1068514984467476647612872325074167476256676400295706624} a^{16} - \frac{67105163017554388405194790966045244106310401803}{267128746116869161903218081268541869064169100073926656} a^{15} + \frac{100741418055594333752894629507930715597996799863}{178085830744579441268812054179027912709446066715951104} a^{14} + \frac{369560520664174115967753190540887323512670840287}{118723887163052960845874702786018608472964044477300736} a^{13} + \frac{3982213400201729375291154932125995715744761890465}{267128746116869161903218081268541869064169100073926656} a^{12} - \frac{18367232268915899500856344185801078437577654204751}{534257492233738323806436162537083738128338200147853312} a^{11} + \frac{9883857749219631847786766293157202551076313166297}{534257492233738323806436162537083738128338200147853312} a^{10} + \frac{9520897656036085076395462913324643358870605294305}{237447774326105921691749405572037216945928088954601472} a^{9} - \frac{2507773834872811085191674413109356665223740333671137}{1068514984467476647612872325074167476256676400295706624} a^{8} + \frac{4602823381140841904099699551078158392693708441824487}{534257492233738323806436162537083738128338200147853312} a^{7} + \frac{1076766117568781899044739576737734316321379161537251}{267128746116869161903218081268541869064169100073926656} a^{6} - \frac{882654429085759215844645698332613096108951904113191}{22260728843072430158601506772378489088680758339493888} a^{5} + \frac{52767578688861589920012162292110798375770671740707}{5565182210768107539650376693094622272170189584873472} a^{4} - \frac{803640786844559522509339375898268449953096339221449}{4173886658076080654737782519820966704127642188655104} a^{3} + \frac{732911640274905178768978839983191821882671305337}{16304244758109690057569462968050651187998602299434} a^{2} + \frac{14515607990414087052815672978627764748146313780533}{32608489516219380115138925936101302375997204598868} a - \frac{1297223558774355377998128241820746382865210097403}{2717374126351615009594910494675108531333100383239}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{18}\times C_{18436230}$, which has order $2986669260$ (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:  \( 248523426321.35065 \) (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

\(\Q(\sqrt{-15}) \), 3.3.6588.1, 3.3.301401.1, 6.0.16275654000.3, 6.0.34065961050375.6, 9.9.2886075179454821059008.1

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 ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$$0$Trivial$[\ ]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$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$$[\ ]_{2}^{6}$
$61$61.3.2.3$x^{3} - 244$$3$$1$$2$$C_3$$[\ ]_{3}$
61.3.2.3$x^{3} - 244$$3$$1$$2$$C_3$$[\ ]_{3}$
61.6.5.2$x^{6} - 244$$6$$1$$5$$C_6$$[\ ]_{6}$
61.6.5.2$x^{6} - 244$$6$$1$$5$$C_6$$[\ ]_{6}$