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Properties

Label 18.0.51906361512...5191.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{9}\cdot 13^{15}\cdot 61^{6}$
Root discriminant $57.80$
Ramified primes $3, 13, 61$
Class number $1120$ (GRH)
Class group $[2, 2, 2, 140]$ (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![352312, -11032, -55738, 55577, 216263, -81955, -43679, 61786, 21415, -19194, 1722, 4448, 554, -201, 181, 12, 8, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 8*x^16 + 12*x^15 + 181*x^14 - 201*x^13 + 554*x^12 + 4448*x^11 + 1722*x^10 - 19194*x^9 + 21415*x^8 + 61786*x^7 - 43679*x^6 - 81955*x^5 + 216263*x^4 + 55577*x^3 - 55738*x^2 - 11032*x + 352312)
 
gp: K = bnfinit(x^18 - 3*x^17 + 8*x^16 + 12*x^15 + 181*x^14 - 201*x^13 + 554*x^12 + 4448*x^11 + 1722*x^10 - 19194*x^9 + 21415*x^8 + 61786*x^7 - 43679*x^6 - 81955*x^5 + 216263*x^4 + 55577*x^3 - 55738*x^2 - 11032*x + 352312, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 8 x^{16} + 12 x^{15} + 181 x^{14} - 201 x^{13} + 554 x^{12} + 4448 x^{11} + 1722 x^{10} - 19194 x^{9} + 21415 x^{8} + 61786 x^{7} - 43679 x^{6} - 81955 x^{5} + 216263 x^{4} + 55577 x^{3} - 55738 x^{2} - 11032 x + 352312 \)

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:  \(-51906361512443096981169273175191=-\,3^{9}\cdot 13^{15}\cdot 61^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.80$
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:  $3, 13, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{47} a^{13} + \frac{13}{47} a^{12} - \frac{11}{47} a^{11} + \frac{12}{47} a^{10} - \frac{18}{47} a^{9} - \frac{19}{47} a^{8} + \frac{12}{47} a^{7} - \frac{2}{47} a^{6} - \frac{22}{47} a^{5} - \frac{7}{47} a^{4} + \frac{19}{47} a^{3} + \frac{1}{47} a^{2} + \frac{21}{47} a$, $\frac{1}{47} a^{14} + \frac{8}{47} a^{12} + \frac{14}{47} a^{11} + \frac{14}{47} a^{10} - \frac{20}{47} a^{9} - \frac{23}{47} a^{8} - \frac{17}{47} a^{7} + \frac{4}{47} a^{6} - \frac{3}{47} a^{5} + \frac{16}{47} a^{4} - \frac{11}{47} a^{3} + \frac{8}{47} a^{2} + \frac{9}{47} a$, $\frac{1}{47} a^{15} + \frac{4}{47} a^{12} + \frac{8}{47} a^{11} - \frac{22}{47} a^{10} - \frac{20}{47} a^{9} - \frac{6}{47} a^{8} + \frac{2}{47} a^{7} + \frac{13}{47} a^{6} + \frac{4}{47} a^{5} - \frac{2}{47} a^{4} - \frac{3}{47} a^{3} + \frac{1}{47} a^{2} + \frac{20}{47} a$, $\frac{1}{94} a^{16} - \frac{1}{94} a^{15} - \frac{1}{94} a^{12} - \frac{33}{94} a^{11} - \frac{23}{47} a^{10} - \frac{4}{47} a^{9} - \frac{5}{47} a^{8} + \frac{5}{47} a^{7} - \frac{1}{94} a^{6} - \frac{6}{47} a^{5} + \frac{27}{94} a^{4} - \frac{25}{94} a^{3} + \frac{15}{94} a^{2} + \frac{37}{94} a$, $\frac{1}{5383098276789006553285369986174309553259982908} a^{17} + \frac{28540105743666204797345427249299781414511073}{5383098276789006553285369986174309553259982908} a^{16} + \frac{9137274400875898826907956291029266837521185}{1345774569197251638321342496543577388314995727} a^{15} - \frac{11268670507660461499209854358483581505751184}{1345774569197251638321342496543577388314995727} a^{14} - \frac{51739995302539952529602181464641025118319083}{5383098276789006553285369986174309553259982908} a^{13} - \frac{1257453765698272118052199238648461796326954365}{5383098276789006553285369986174309553259982908} a^{12} + \frac{967738057773997645689268734795865132722023517}{2691549138394503276642684993087154776629991454} a^{11} + \frac{635872871603797584009682351454338461314507969}{1345774569197251638321342496543577388314995727} a^{10} + \frac{1254473918465680964862853097545552012281616117}{2691549138394503276642684993087154776629991454} a^{9} - \frac{336734591916316123504550037067353313184755141}{2691549138394503276642684993087154776629991454} a^{8} - \frac{1971884082239688483392962099446799680549360633}{5383098276789006553285369986174309553259982908} a^{7} + \frac{144432758364156566664575852049911330285326983}{2691549138394503276642684993087154776629991454} a^{6} - \frac{1113471865129389959965754872788431433274748775}{5383098276789006553285369986174309553259982908} a^{5} + \frac{324516247816408040257933547559918150531083217}{5383098276789006553285369986174309553259982908} a^{4} + \frac{1492225352421051078423654383637298700023349963}{5383098276789006553285369986174309553259982908} a^{3} - \frac{162274246025705554520791702795925603918999331}{5383098276789006553285369986174309553259982908} a^{2} - \frac{534617439306005087132622051709940209734467581}{2691549138394503276642684993087154776629991454} a + \frac{289276906678428064826713176343088144711834}{28633501472281949751517925458373986985425441}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{140}$, which has order $1120$ (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:  \( 230602.64598601736 \) (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

\(\Q(\sqrt{-39}) \), 3.3.169.1, 3.3.10309.1, 6.0.37302693831.1, 6.0.10024911.1, 9.9.1095593933629.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
13Data not computed
$61$61.3.0.1$x^{3} - x + 10$$1$$3$$0$$C_3$$[\ ]^{3}$
61.3.0.1$x^{3} - x + 10$$1$$3$$0$$C_3$$[\ ]^{3}$
61.6.3.1$x^{6} - 122 x^{4} + 3721 x^{2} - 22698100$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
61.6.3.1$x^{6} - 122 x^{4} + 3721 x^{2} - 22698100$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$