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Properties

Label 18.6.21725661348...1601.1
Degree $18$
Signature $[6, 6]$
Discriminant $3^{6}\cdot 1129^{9}$
Root discriminant $48.46$
Ramified primes $3, 1129$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_2^2:D_9$ (as 18T39)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7883, -26871, 5918, 80053, -117880, 41206, 46314, -55783, 25356, -5955, 2547, -1905, 318, 426, -308, 68, 5, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 5*x^16 + 68*x^15 - 308*x^14 + 426*x^13 + 318*x^12 - 1905*x^11 + 2547*x^10 - 5955*x^9 + 25356*x^8 - 55783*x^7 + 46314*x^6 + 41206*x^5 - 117880*x^4 + 80053*x^3 + 5918*x^2 - 26871*x + 7883)
 
gp: K = bnfinit(x^18 - 6*x^17 + 5*x^16 + 68*x^15 - 308*x^14 + 426*x^13 + 318*x^12 - 1905*x^11 + 2547*x^10 - 5955*x^9 + 25356*x^8 - 55783*x^7 + 46314*x^6 + 41206*x^5 - 117880*x^4 + 80053*x^3 + 5918*x^2 - 26871*x + 7883, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 5 x^{16} + 68 x^{15} - 308 x^{14} + 426 x^{13} + 318 x^{12} - 1905 x^{11} + 2547 x^{10} - 5955 x^{9} + 25356 x^{8} - 55783 x^{7} + 46314 x^{6} + 41206 x^{5} - 117880 x^{4} + 80053 x^{3} + 5918 x^{2} - 26871 x + 7883 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2172566134897789192777842401601=3^{6}\cdot 1129^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.46$
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:  $3, 1129$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{11} a^{12} + \frac{2}{11} a^{11} + \frac{3}{11} a^{10} - \frac{1}{11} a^{9} + \frac{3}{11} a^{8} + \frac{5}{11} a^{7} - \frac{5}{11} a^{6} + \frac{1}{11} a^{5} - \frac{3}{11} a^{4} - \frac{3}{11} a^{3} + \frac{1}{11} a^{2} + \frac{2}{11} a + \frac{4}{11}$, $\frac{1}{11} a^{13} - \frac{1}{11} a^{11} + \frac{4}{11} a^{10} + \frac{5}{11} a^{9} - \frac{1}{11} a^{8} - \frac{4}{11} a^{7} - \frac{5}{11} a^{5} + \frac{3}{11} a^{4} - \frac{4}{11} a^{3} + \frac{3}{11}$, $\frac{1}{11} a^{14} - \frac{5}{11} a^{11} - \frac{3}{11} a^{10} - \frac{2}{11} a^{9} - \frac{1}{11} a^{8} + \frac{5}{11} a^{7} + \frac{1}{11} a^{6} + \frac{4}{11} a^{5} + \frac{4}{11} a^{4} - \frac{3}{11} a^{3} + \frac{1}{11} a^{2} + \frac{5}{11} a + \frac{4}{11}$, $\frac{1}{11} a^{15} - \frac{4}{11} a^{11} + \frac{2}{11} a^{10} + \frac{5}{11} a^{9} - \frac{2}{11} a^{8} + \frac{4}{11} a^{7} + \frac{1}{11} a^{6} - \frac{2}{11} a^{5} + \frac{4}{11} a^{4} - \frac{3}{11} a^{3} - \frac{1}{11} a^{2} + \frac{3}{11} a - \frac{2}{11}$, $\frac{1}{3031831} a^{16} + \frac{78224}{3031831} a^{15} + \frac{58680}{3031831} a^{14} - \frac{9950}{275621} a^{13} + \frac{121488}{3031831} a^{12} + \frac{36532}{97801} a^{11} - \frac{720516}{3031831} a^{10} + \frac{608557}{3031831} a^{9} + \frac{196696}{3031831} a^{8} + \frac{53928}{3031831} a^{7} + \frac{41295}{275621} a^{6} - \frac{1406683}{3031831} a^{5} + \frac{109976}{3031831} a^{4} + \frac{708799}{3031831} a^{3} - \frac{21532}{178343} a^{2} + \frac{791250}{3031831} a - \frac{874571}{3031831}$, $\frac{1}{855721450920351145103631184549} a^{17} + \frac{82470982885304275290692}{855721450920351145103631184549} a^{16} - \frac{21101472867207931870934929846}{855721450920351145103631184549} a^{15} + \frac{15771475461402007613022458133}{855721450920351145103631184549} a^{14} - \frac{28918188029887269883460209384}{855721450920351145103631184549} a^{13} - \frac{1713385251533646771814001308}{50336555936491243829625363797} a^{12} - \frac{235862564400534768325504406287}{855721450920351145103631184549} a^{11} + \frac{403772817875814057082365313868}{855721450920351145103631184549} a^{10} + \frac{223207063213070329306139443057}{855721450920351145103631184549} a^{9} - \frac{67256137046411465788038705516}{855721450920351145103631184549} a^{8} + \frac{357313037824183619396611788144}{855721450920351145103631184549} a^{7} + \frac{38486779011451607752199224417}{855721450920351145103631184549} a^{6} - \frac{338208011620976196780442934285}{855721450920351145103631184549} a^{5} - \frac{16261310956299732662372267967}{855721450920351145103631184549} a^{4} + \frac{29384496754695062760590177266}{77792859174577376827602834959} a^{3} + \frac{133888454077374628646411273579}{855721450920351145103631184549} a^{2} - \frac{53065895457581070979071865205}{855721450920351145103631184549} a + \frac{87245113170487772904280644772}{855721450920351145103631184549}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  $11$
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:  \( 42390273.4365 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:D_9$ (as 18T39):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 9 conjugacy class representatives for $C_2^2:D_9$
Character table for $C_2^2:D_9$

Intermediate fields

3.3.1129.1, 6.2.12951627201.5, 9.9.1624709678881.1

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

Sibling fields

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$$[\ ]_{2}^{3}$
3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
1129Data not computed