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Properties

Label 18.2.67990593154...2672.3
Degree $18$
Signature $[2, 8]$
Discriminant $2^{24}\cdot 3^{39}$
Root discriminant $27.24$
Ramified primes $2, 3$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $S_3^2$ (as 18T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, -72, 108, -336, 540, -144, -156, 144, -72, -260, 216, 36, -123, 18, 45, -6, -9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^16 - 6*x^15 + 45*x^14 + 18*x^13 - 123*x^12 + 36*x^11 + 216*x^10 - 260*x^9 - 72*x^8 + 144*x^7 - 156*x^6 - 144*x^5 + 540*x^4 - 336*x^3 + 108*x^2 - 72*x + 4)
 
gp: K = bnfinit(x^18 - 9*x^16 - 6*x^15 + 45*x^14 + 18*x^13 - 123*x^12 + 36*x^11 + 216*x^10 - 260*x^9 - 72*x^8 + 144*x^7 - 156*x^6 - 144*x^5 + 540*x^4 - 336*x^3 + 108*x^2 - 72*x + 4, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{16} - 6 x^{15} + 45 x^{14} + 18 x^{13} - 123 x^{12} + 36 x^{11} + 216 x^{10} - 260 x^{9} - 72 x^{8} + 144 x^{7} - 156 x^{6} - 144 x^{5} + 540 x^{4} - 336 x^{3} + 108 x^{2} - 72 x + 4 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(67990593154112416930332672=2^{24}\cdot 3^{39}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.24$
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:  $2, 3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{4} a^{12} - \frac{1}{2} a^{11} + \frac{1}{4} a^{10} - \frac{1}{2} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{13} + \frac{1}{4} a^{11} - \frac{1}{4} a^{9} + \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{4} a^{15} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{266713528} a^{16} + \frac{1032697}{66678382} a^{15} + \frac{2004747}{266713528} a^{14} - \frac{4005595}{66678382} a^{13} + \frac{20629491}{266713528} a^{12} + \frac{1147397}{66678382} a^{11} + \frac{30581653}{266713528} a^{10} - \frac{58422277}{133356764} a^{9} - \frac{17449031}{66678382} a^{8} + \frac{53432107}{133356764} a^{7} - \frac{12614899}{66678382} a^{6} - \frac{6376393}{33339191} a^{5} + \frac{47555501}{133356764} a^{4} - \frac{10656063}{33339191} a^{3} - \frac{1692006}{33339191} a^{2} - \frac{5915375}{33339191} a - \frac{12269511}{66678382}$, $\frac{1}{8977310638952} a^{17} + \frac{1561}{1122163829869} a^{16} + \frac{685753899641}{8977310638952} a^{15} + \frac{200844459155}{4488655319476} a^{14} - \frac{295057162441}{8977310638952} a^{13} + \frac{371881468781}{4488655319476} a^{12} - \frac{1887880441155}{8977310638952} a^{11} - \frac{502514547533}{1122163829869} a^{10} + \frac{816487716057}{2244327659738} a^{9} + \frac{1061948529169}{2244327659738} a^{8} - \frac{67920922081}{4488655319476} a^{7} - \frac{403973086677}{2244327659738} a^{6} + \frac{2061684919065}{4488655319476} a^{5} - \frac{431338809505}{1122163829869} a^{4} - \frac{149707607861}{1122163829869} a^{3} + \frac{537043873231}{2244327659738} a^{2} - \frac{391849709284}{1122163829869} a - \frac{100126346633}{1122163829869}$

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:  $9$
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:  \( 1450943.530132557 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3^2$ (as 18T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 9 conjugacy class representatives for $S_3^2$
Character table for $S_3^2$

Intermediate fields

\(\Q(\sqrt{3}) \), 3.1.324.1, 3.1.972.1, 6.2.45349632.5 x2, 6.2.1259712.2, 6.2.45349632.2, 9.1.1190155742208.4

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 6 sibling: data not computed
Degree 9 sibling: data not computed
Degree 12 sibling: 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.6.8.3$x^{6} + 2 x^{3} + 6$$6$$1$$8$$D_{6}$$[2]_{3}^{2}$
2.12.16.13$x^{12} + 12 x^{10} + 12 x^{8} + 8 x^{6} + 32 x^{4} - 16 x^{2} + 16$$6$$2$$16$$D_6$$[2]_{3}^{2}$
3Data not computed