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Properties

Label 18.0.96873331012...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{24}\cdot 5^{12}\cdot 7^{3}$
Root discriminant $27.78$
Ramified primes $2, 3, 5, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $S_3\times S_3\wr C_2$ (as 18T150)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![79, -564, 1716, -2679, 2007, -387, -6, -597, 822, -901, 849, -399, 147, -72, 18, -15, 15, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 15*x^16 - 15*x^15 + 18*x^14 - 72*x^13 + 147*x^12 - 399*x^11 + 849*x^10 - 901*x^9 + 822*x^8 - 597*x^7 - 6*x^6 - 387*x^5 + 2007*x^4 - 2679*x^3 + 1716*x^2 - 564*x + 79)
 
gp: K = bnfinit(x^18 - 6*x^17 + 15*x^16 - 15*x^15 + 18*x^14 - 72*x^13 + 147*x^12 - 399*x^11 + 849*x^10 - 901*x^9 + 822*x^8 - 597*x^7 - 6*x^6 - 387*x^5 + 2007*x^4 - 2679*x^3 + 1716*x^2 - 564*x + 79, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 15 x^{16} - 15 x^{15} + 18 x^{14} - 72 x^{13} + 147 x^{12} - 399 x^{11} + 849 x^{10} - 901 x^{9} + 822 x^{8} - 597 x^{7} - 6 x^{6} - 387 x^{5} + 2007 x^{4} - 2679 x^{3} + 1716 x^{2} - 564 x + 79 \)

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:  \(-96873331012983000000000000=-\,2^{12}\cdot 3^{24}\cdot 5^{12}\cdot 7^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.78$
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:  $2, 3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} + \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{5}$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{6}$, $\frac{1}{42} a^{16} - \frac{1}{21} a^{15} - \frac{2}{21} a^{14} + \frac{5}{42} a^{13} - \frac{1}{21} a^{12} + \frac{5}{42} a^{11} - \frac{1}{6} a^{10} - \frac{1}{7} a^{9} + \frac{1}{7} a^{8} + \frac{1}{42} a^{7} + \frac{2}{21} a^{6} - \frac{2}{21} a^{5} - \frac{5}{21} a^{4} - \frac{5}{42} a^{3} + \frac{11}{42} a^{2} - \frac{2}{21} a - \frac{1}{14}$, $\frac{1}{4536201588088340142} a^{17} - \frac{51429753114479737}{4536201588088340142} a^{16} + \frac{264773413727213}{46287771307023879} a^{15} + \frac{213288673314140171}{4536201588088340142} a^{14} - \frac{546696043367044373}{4536201588088340142} a^{13} + \frac{375027440660541419}{4536201588088340142} a^{12} - \frac{220350411129428515}{2268100794044170071} a^{11} - \frac{537166321130698321}{4536201588088340142} a^{10} + \frac{169957155916468769}{2268100794044170071} a^{9} - \frac{654651046764304223}{4536201588088340142} a^{8} + \frac{728560973256065567}{4536201588088340142} a^{7} - \frac{930547599337020979}{2268100794044170071} a^{6} - \frac{846335804582537228}{2268100794044170071} a^{5} - \frac{1162773077845085699}{4536201588088340142} a^{4} + \frac{9299343734710729}{46287771307023879} a^{3} + \frac{21773863431584119}{4536201588088340142} a^{2} + \frac{193422493010072891}{4536201588088340142} a + \frac{63945137618871403}{4536201588088340142}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( \frac{7175307659}{86086833617} a^{17} - \frac{54990751003}{172173667234} a^{16} + \frac{94009736090}{258260500851} a^{15} + \frac{115443947921}{258260500851} a^{14} + \frac{436988127035}{516521001702} a^{13} - \frac{984760271117}{258260500851} a^{12} + \frac{882010689997}{516521001702} a^{11} - \frac{9684114095023}{516521001702} a^{10} + \frac{4363504130924}{258260500851} a^{9} + \frac{1422033579589}{86086833617} a^{8} + \frac{3099828783115}{172173667234} a^{7} + \frac{6441605277155}{258260500851} a^{6} - \frac{4974310642765}{258260500851} a^{5} - \frac{18007061712191}{258260500851} a^{4} + \frac{29147965749503}{516521001702} a^{3} + \frac{14051257079617}{516521001702} a^{2} - \frac{10080468143732}{258260500851} a + \frac{6556947869473}{516521001702} \) (order $6$)
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:  \( 3435407.66129 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times S_3\wr C_2$ (as 18T150):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 432
The 27 conjugacy class representatives for $S_3\times S_3\wr C_2$
Character table for $S_3\times S_3\wr C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.3.2700.1, 6.0.21870000.1, 6.0.3189375.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 12 siblings: 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 R ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
3Data not computed
5Data not computed
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$