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Properties

Label 18.0.24335159153...2163.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{9}\cdot 181^{12}$
Root discriminant $55.42$
Ramified primes $3, 181$
Class number $7$ (GRH)
Class group $[7]$ (GRH)
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1323, -11151, 42606, -95049, 128958, -100956, 39985, -8421, 8343, -8037, 5508, -2229, 586, -273, 207, -108, 39, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 39*x^16 - 108*x^15 + 207*x^14 - 273*x^13 + 586*x^12 - 2229*x^11 + 5508*x^10 - 8037*x^9 + 8343*x^8 - 8421*x^7 + 39985*x^6 - 100956*x^5 + 128958*x^4 - 95049*x^3 + 42606*x^2 - 11151*x + 1323)
 
gp: K = bnfinit(x^18 - 9*x^17 + 39*x^16 - 108*x^15 + 207*x^14 - 273*x^13 + 586*x^12 - 2229*x^11 + 5508*x^10 - 8037*x^9 + 8343*x^8 - 8421*x^7 + 39985*x^6 - 100956*x^5 + 128958*x^4 - 95049*x^3 + 42606*x^2 - 11151*x + 1323, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 39 x^{16} - 108 x^{15} + 207 x^{14} - 273 x^{13} + 586 x^{12} - 2229 x^{11} + 5508 x^{10} - 8037 x^{9} + 8343 x^{8} - 8421 x^{7} + 39985 x^{6} - 100956 x^{5} + 128958 x^{4} - 95049 x^{3} + 42606 x^{2} - 11151 x + 1323 \)

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:  \(-24335159153759228982627211382163=-\,3^{9}\cdot 181^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.42$
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:  $3, 181$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{21} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{2}{7} a$, $\frac{1}{21} a^{8} - \frac{1}{21} a^{2}$, $\frac{1}{21} a^{9} - \frac{1}{21} a^{3}$, $\frac{1}{21} a^{10} - \frac{1}{21} a^{4}$, $\frac{1}{105} a^{11} + \frac{2}{105} a^{10} - \frac{2}{105} a^{9} - \frac{1}{105} a^{8} - \frac{2}{105} a^{7} + \frac{2}{15} a^{6} + \frac{16}{35} a^{5} - \frac{3}{35} a^{4} + \frac{3}{35} a^{3} - \frac{16}{35} a^{2} + \frac{3}{35} a + \frac{2}{5}$, $\frac{1}{315} a^{12} - \frac{1}{315} a^{10} + \frac{1}{105} a^{9} + \frac{1}{105} a^{7} + \frac{4}{63} a^{6} - \frac{1}{3} a^{5} + \frac{22}{315} a^{4} + \frac{16}{35} a^{3} - \frac{1}{3} a^{2} + \frac{16}{35} a + \frac{2}{5}$, $\frac{1}{315} a^{13} - \frac{1}{315} a^{11} + \frac{1}{105} a^{10} + \frac{1}{105} a^{8} + \frac{1}{63} a^{7} - \frac{83}{315} a^{5} - \frac{22}{105} a^{4} + \frac{1}{3} a^{3} - \frac{22}{105} a^{2} + \frac{4}{35} a$, $\frac{1}{2205} a^{14} - \frac{1}{315} a^{10} + \frac{2}{105} a^{9} - \frac{37}{2205} a^{8} - \frac{1}{105} a^{7} + \frac{1}{3} a^{5} + \frac{127}{315} a^{4} - \frac{3}{35} a^{3} - \frac{9}{49} a^{2} - \frac{16}{35} a$, $\frac{1}{2205} a^{15} - \frac{1}{315} a^{11} + \frac{2}{105} a^{10} - \frac{37}{2205} a^{9} - \frac{1}{105} a^{8} + \frac{127}{315} a^{5} - \frac{44}{105} a^{4} - \frac{9}{49} a^{3} + \frac{22}{105} a^{2}$, $\frac{1}{241912755} a^{16} - \frac{8}{241912755} a^{15} - \frac{8336}{48382551} a^{14} + \frac{2780}{2303931} a^{13} + \frac{1693}{3839885} a^{12} + \frac{24692}{34558965} a^{11} - \frac{5165197}{241912755} a^{10} - \frac{4895434}{241912755} a^{9} + \frac{2358262}{241912755} a^{8} + \frac{47637}{3839885} a^{7} + \frac{341916}{3839885} a^{6} + \frac{3416011}{34558965} a^{5} - \frac{6784007}{16127517} a^{4} - \frac{6198928}{16127517} a^{3} - \frac{1125286}{26879195} a^{2} + \frac{292741}{3839885} a - \frac{154607}{548555}$, $\frac{1}{1209563775} a^{17} - \frac{1}{1209563775} a^{16} + \frac{177686}{1209563775} a^{15} + \frac{4}{34558965} a^{14} + \frac{197426}{172794825} a^{13} - \frac{29357}{57598275} a^{12} - \frac{53763}{26879195} a^{11} + \frac{6027472}{403187925} a^{10} + \frac{19873816}{1209563775} a^{9} - \frac{175202}{172794825} a^{8} - \frac{4015037}{172794825} a^{7} + \frac{2797349}{19199425} a^{6} + \frac{435021371}{1209563775} a^{5} + \frac{151594687}{1209563775} a^{4} + \frac{63374308}{403187925} a^{3} - \frac{730182}{2742775} a^{2} + \frac{206299}{548555} a - \frac{107588}{391825}$

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

Class group and class number

$C_{7}$, which has order $7$ (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:  \( -\frac{14104942}{1209563775} a^{17} + \frac{119892007}{1209563775} a^{16} - \frac{163435924}{403187925} a^{15} + \frac{51178726}{48382551} a^{14} - \frac{108809614}{57598275} a^{13} + \frac{388670542}{172794825} a^{12} - \frac{1385312318}{241912755} a^{11} + \frac{28000634948}{1209563775} a^{10} - \frac{21254125954}{403187925} a^{9} + \frac{81769084148}{1209563775} a^{8} - \frac{526514606}{8228325} a^{7} + \frac{11541737738}{172794825} a^{6} - \frac{524248456912}{1209563775} a^{5} + \frac{129169716769}{134395975} a^{4} - \frac{414472775746}{403187925} a^{3} + \frac{81062935346}{134395975} a^{2} - \frac{782641443}{3839885} a + \frac{90145802}{2742775} \) (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:  \( 782508303.126 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 18T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.98283.1 x3, 3.3.32761.1, 6.0.28978644267.1, 6.0.884547.1 x2, 6.0.28978644267.2, 9.3.949369364831187.1 x3

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

Sibling fields

Degree 6 sibling: 6.0.884547.1
Degree 9 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/7.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$$[\ ]_{2}^{3}$
$181$181.3.2.1$x^{3} - 181$$3$$1$$2$$C_3$$[\ ]_{3}$
181.3.2.1$x^{3} - 181$$3$$1$$2$$C_3$$[\ ]_{3}$
181.3.2.1$x^{3} - 181$$3$$1$$2$$C_3$$[\ ]_{3}$
181.3.2.1$x^{3} - 181$$3$$1$$2$$C_3$$[\ ]_{3}$
181.3.2.1$x^{3} - 181$$3$$1$$2$$C_3$$[\ ]_{3}$
181.3.2.1$x^{3} - 181$$3$$1$$2$$C_3$$[\ ]_{3}$