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Properties

Label 18.0.13126401193...0608.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{33}\cdot 7^{8}$
Root discriminant $28.25$
Ramified primes $2, 3, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3^2:S_3$ (as 18T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1372, 8232, 12348, -4900, -12348, 7056, 4823, -7056, 5061, -3246, 1896, -924, 573, -306, 168, -68, 21, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 21*x^16 - 68*x^15 + 168*x^14 - 306*x^13 + 573*x^12 - 924*x^11 + 1896*x^10 - 3246*x^9 + 5061*x^8 - 7056*x^7 + 4823*x^6 + 7056*x^5 - 12348*x^4 - 4900*x^3 + 12348*x^2 + 8232*x + 1372)
 
gp: K = bnfinit(x^18 - 6*x^17 + 21*x^16 - 68*x^15 + 168*x^14 - 306*x^13 + 573*x^12 - 924*x^11 + 1896*x^10 - 3246*x^9 + 5061*x^8 - 7056*x^7 + 4823*x^6 + 7056*x^5 - 12348*x^4 - 4900*x^3 + 12348*x^2 + 8232*x + 1372, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 21 x^{16} - 68 x^{15} + 168 x^{14} - 306 x^{13} + 573 x^{12} - 924 x^{11} + 1896 x^{10} - 3246 x^{9} + 5061 x^{8} - 7056 x^{7} + 4823 x^{6} + 7056 x^{5} - 12348 x^{4} - 4900 x^{3} + 12348 x^{2} + 8232 x + 1372 \)

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:  \(-131264011932220807356100608=-\,2^{12}\cdot 3^{33}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.25$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{14} a^{12} + \frac{1}{14} a^{11} + \frac{1}{7} a^{9} - \frac{1}{2} a^{8} - \frac{5}{14} a^{7} + \frac{3}{7} a^{6} - \frac{1}{14} a^{4} - \frac{5}{14} a^{3}$, $\frac{1}{28} a^{13} - \frac{1}{28} a^{12} + \frac{5}{28} a^{11} + \frac{1}{14} a^{10} + \frac{3}{28} a^{9} + \frac{9}{28} a^{8} - \frac{5}{28} a^{7} + \frac{1}{14} a^{6} + \frac{13}{28} a^{5} - \frac{3}{28} a^{4} + \frac{3}{28} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{28} a^{14} + \frac{3}{28} a^{11} + \frac{5}{28} a^{10} + \frac{1}{7} a^{9} + \frac{1}{7} a^{8} - \frac{11}{28} a^{7} - \frac{9}{28} a^{6} + \frac{5}{14} a^{5} + \frac{1}{7} a^{4} + \frac{9}{28} a^{3} - \frac{1}{2}$, $\frac{1}{196} a^{15} + \frac{1}{196} a^{14} - \frac{5}{196} a^{12} + \frac{1}{7} a^{11} - \frac{19}{196} a^{10} + \frac{17}{98} a^{9} + \frac{13}{28} a^{8} + \frac{12}{49} a^{7} - \frac{61}{196} a^{6} - \frac{3}{14} a^{5} - \frac{1}{4} a^{4} + \frac{1}{28} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{31948} a^{16} - \frac{2}{1141} a^{15} + \frac{143}{15974} a^{14} + \frac{289}{31948} a^{13} + \frac{565}{31948} a^{12} + \frac{3361}{15974} a^{11} + \frac{1002}{7987} a^{10} - \frac{2225}{31948} a^{9} + \frac{349}{31948} a^{8} - \frac{4111}{15974} a^{7} + \frac{955}{7987} a^{6} + \frac{1483}{4564} a^{5} - \frac{69}{163} a^{4} - \frac{82}{1141} a^{3} - \frac{17}{326} a^{2} - \frac{36}{163} a + \frac{20}{163}$, $\frac{1}{980584081194588505065363688} a^{17} - \frac{367853687506062244036}{122573010149323563133170461} a^{16} + \frac{106415003651315076310661}{140083440170655500723623384} a^{15} + \frac{15222228664692590836829}{5002980006094839311557978} a^{14} - \frac{7483710123919120161437715}{490292040597294252532681844} a^{13} - \frac{2498800831746315074478054}{122573010149323563133170461} a^{12} - \frac{864677341388379196114447}{5417591608809881243455048} a^{11} - \frac{101886778450877396268814889}{490292040597294252532681844} a^{10} + \frac{64962168084763848860472347}{490292040597294252532681844} a^{9} + \frac{6958269853048507395400710}{17510430021331937590452923} a^{8} - \frac{282473475509893000894519435}{980584081194588505065363688} a^{7} + \frac{87172524433926562408530443}{490292040597294252532681844} a^{6} + \frac{43114323391105855956016315}{140083440170655500723623384} a^{5} - \frac{841095733620337189836985}{5002980006094839311557978} a^{4} - \frac{9908876712394916243821305}{70041720085327750361811692} a^{3} - \frac{1596877992069099563519559}{10005960012189678623115956} a^{2} - \frac{367098204251151289109413}{5002980006094839311557978} a - \frac{1614815662685031860760543}{5002980006094839311557978}$

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:  \( \frac{13415990153565}{2938871985251944} a^{17} - \frac{10655322773073}{367358998156493} a^{16} + \frac{311774412341603}{2938871985251944} a^{15} - \frac{510958467829857}{1469435992625972} a^{14} + \frac{653173771452777}{734717996312986} a^{13} - \frac{2510073095473015}{1469435992625972} a^{12} + \frac{9438538895883753}{2938871985251944} a^{11} - \frac{3919935831974499}{734717996312986} a^{10} + \frac{3858834660259749}{367358998156493} a^{9} - \frac{27139813341638307}{1469435992625972} a^{8} + \frac{86690412720139233}{2938871985251944} a^{7} - \frac{31185671999204279}{734717996312986} a^{6} + \frac{15372947144017737}{419838855035992} a^{5} + \frac{4159700811160893}{209919427517996} a^{4} - \frac{13455334429467517}{209919427517996} a^{3} + \frac{21073636829091}{29988489645428} a^{2} + \frac{422367163755117}{7497122411357} a + \frac{266138791120197}{14994244822714} \) (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:  \( 2990563.699821467 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3^2:S_3$ (as 18T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 10 conjugacy class representatives for $C_3^2:S_3$
Character table for $C_3^2:S_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.972.2 x3, 6.0.2834352.2, 9.3.2204910445248.1

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

Sibling fields

Degree 9 siblings: data not computed
Degree 18 siblings: 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
$2$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
3Data not computed
$7$7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$