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Properties

Label 18.0.85141600462...1744.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 7^{12}\cdot 31^{15}$
Root discriminant $128.01$
Ramified primes $2, 7, 31$
Class number $1051218$ (GRH)
Class group $[3, 3, 3, 38934]$ (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![30224384, 46538240, 30682880, 6357024, 3205568, 7108056, 6943544, 3675820, 1397556, 406326, 39670, -31897, -10513, 1445, 1049, 29, -43, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 43*x^16 + 29*x^15 + 1049*x^14 + 1445*x^13 - 10513*x^12 - 31897*x^11 + 39670*x^10 + 406326*x^9 + 1397556*x^8 + 3675820*x^7 + 6943544*x^6 + 7108056*x^5 + 3205568*x^4 + 6357024*x^3 + 30682880*x^2 + 46538240*x + 30224384)
 
gp: K = bnfinit(x^18 - 3*x^17 - 43*x^16 + 29*x^15 + 1049*x^14 + 1445*x^13 - 10513*x^12 - 31897*x^11 + 39670*x^10 + 406326*x^9 + 1397556*x^8 + 3675820*x^7 + 6943544*x^6 + 7108056*x^5 + 3205568*x^4 + 6357024*x^3 + 30682880*x^2 + 46538240*x + 30224384, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 43 x^{16} + 29 x^{15} + 1049 x^{14} + 1445 x^{13} - 10513 x^{12} - 31897 x^{11} + 39670 x^{10} + 406326 x^{9} + 1397556 x^{8} + 3675820 x^{7} + 6943544 x^{6} + 7108056 x^{5} + 3205568 x^{4} + 6357024 x^{3} + 30682880 x^{2} + 46538240 x + 30224384 \)

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:  \(-85141600462656403601383898181048991744=-\,2^{18}\cdot 7^{12}\cdot 31^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $128.01$
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:  $2, 7, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{10} - \frac{1}{8} a^{7} + \frac{1}{16} a^{6} + \frac{1}{8} a^{5} - \frac{1}{8} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{32} a^{11} - \frac{1}{32} a^{10} - \frac{1}{16} a^{9} - \frac{1}{32} a^{7} - \frac{3}{32} a^{6} + \frac{3}{16} a^{5} - \frac{1}{4} a^{4} + \frac{1}{8} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{32} a^{12} - \frac{1}{32} a^{10} - \frac{1}{16} a^{9} - \frac{1}{32} a^{8} - \frac{3}{32} a^{6} - \frac{3}{16} a^{5} - \frac{3}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{128} a^{13} + \frac{1}{128} a^{11} + \frac{1}{64} a^{10} - \frac{1}{128} a^{9} + \frac{1}{32} a^{8} - \frac{1}{128} a^{7} + \frac{5}{64} a^{6} - \frac{1}{32} a^{5} - \frac{3}{16} a^{4} + \frac{1}{32} a^{3} - \frac{7}{16} a^{2} - \frac{1}{2} a$, $\frac{1}{128} a^{14} + \frac{1}{128} a^{12} - \frac{1}{64} a^{11} + \frac{3}{128} a^{10} - \frac{1}{32} a^{9} - \frac{1}{128} a^{8} + \frac{7}{64} a^{7} + \frac{1}{16} a^{6} - \frac{1}{4} a^{5} - \frac{7}{32} a^{4} + \frac{7}{16} a^{3} - \frac{3}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{256} a^{15} - \frac{1}{256} a^{14} - \frac{1}{256} a^{13} + \frac{1}{256} a^{12} + \frac{3}{256} a^{11} - \frac{7}{256} a^{10} + \frac{13}{256} a^{9} + \frac{3}{256} a^{8} - \frac{5}{64} a^{7} - \frac{3}{64} a^{5} - \frac{3}{64} a^{4} + \frac{5}{16} a^{3} + \frac{1}{16} a^{2} - \frac{1}{4} a$, $\frac{1}{512} a^{16} - \frac{1}{512} a^{15} - \frac{1}{512} a^{14} - \frac{1}{512} a^{13} - \frac{5}{512} a^{12} + \frac{7}{512} a^{11} - \frac{15}{512} a^{10} - \frac{11}{512} a^{9} - \frac{5}{128} a^{8} - \frac{23}{256} a^{7} + \frac{7}{64} a^{6} + \frac{27}{128} a^{5} + \frac{1}{16} a^{4} + \frac{17}{64} a^{3} - \frac{3}{32} a^{2} - \frac{3}{8} a$, $\frac{1}{579358617633327965685147397716671672114476033024} a^{17} - \frac{270071981491486493179134915545306367137392177}{579358617633327965685147397716671672114476033024} a^{16} + \frac{1042433983915008386346632742349182943895385123}{579358617633327965685147397716671672114476033024} a^{15} + \frac{48584457395079430322049703004915877316750443}{579358617633327965685147397716671672114476033024} a^{14} + \frac{533649966627309969458073246539220186275199895}{579358617633327965685147397716671672114476033024} a^{13} - \frac{4073977344722323186264543086361480293690458981}{579358617633327965685147397716671672114476033024} a^{12} + \frac{6666532048848390622513612362739664606543509149}{579358617633327965685147397716671672114476033024} a^{11} - \frac{6569384198353657599604277505596187312544447895}{579358617633327965685147397716671672114476033024} a^{10} - \frac{163922506264543080489389337693967579865209125}{4526239200260374731915214044661497438394344008} a^{9} + \frac{864772318231293228523761198866815422127470335}{289679308816663982842573698858335836057238016512} a^{8} - \frac{8015400716351116310668610048785396706798139435}{72419827204165995710643424714583959014309504128} a^{7} + \frac{13463291151939958293751955636578498661975011675}{144839654408331991421286849429167918028619008256} a^{6} + \frac{358831751178556639218808009831252597709252135}{36209913602082997855321712357291979507154752064} a^{5} - \frac{7272411866006808083611342797783812410963997381}{72419827204165995710643424714583959014309504128} a^{4} - \frac{9755824722808291581305117108485687069725765231}{36209913602082997855321712357291979507154752064} a^{3} - \frac{617611302382253042309412417658529143247847639}{4526239200260374731915214044661497438394344008} a^{2} - \frac{860639738126686142473929075181574862401668797}{2263119600130187365957607022330748719197172004} a + \frac{4483209575445904108041797460764158928278798}{12037870213458443435944718203886961272325383}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{3}\times C_{38934}$, which has order $1051218$ (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:  \( 845253452.5841897 \) (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

\(\Q(\sqrt{-31}) \), 3.3.961.1, 3.3.376712.1, 6.0.28629151.1, 6.0.4399269859264.1, 9.9.53459927329776128.4

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$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$$0$Trivial$[\ ]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
31Data not computed