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Properties

Label 18.2.67990593154...2672.4
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![-48, 0, 432, 0, -576, 0, 936, 0, -180, 0, 180, 0, -39, 0, 27, 0, -9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^16 + 27*x^14 - 39*x^12 + 180*x^10 - 180*x^8 + 936*x^6 - 576*x^4 + 432*x^2 - 48)
 
gp: K = bnfinit(x^18 - 9*x^16 + 27*x^14 - 39*x^12 + 180*x^10 - 180*x^8 + 936*x^6 - 576*x^4 + 432*x^2 - 48, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{16} + 27 x^{14} - 39 x^{12} + 180 x^{10} - 180 x^{8} + 936 x^{6} - 576 x^{4} + 432 x^{2} - 48 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{56} a^{12} - \frac{1}{4} a^{11} - \frac{1}{8} a^{10} - \frac{1}{4} a^{9} + \frac{5}{56} a^{8} - \frac{1}{4} a^{7} + \frac{11}{56} a^{6} - \frac{1}{4} a^{5} - \frac{3}{28} a^{4} + \frac{1}{14} a^{2} + \frac{1}{14}$, $\frac{1}{112} a^{13} + \frac{3}{16} a^{11} - \frac{1}{4} a^{10} - \frac{23}{112} a^{9} - \frac{1}{4} a^{8} + \frac{11}{112} a^{7} - \frac{1}{4} a^{6} + \frac{25}{56} a^{5} - \frac{1}{4} a^{4} - \frac{13}{28} a^{3} - \frac{1}{2} a^{2} + \frac{1}{28} a - \frac{1}{2}$, $\frac{1}{112} a^{14} - \frac{1}{112} a^{12} + \frac{19}{112} a^{10} + \frac{13}{112} a^{8} - \frac{3}{14} a^{6} - \frac{1}{2} a^{5} + \frac{3}{14} a^{4} + \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{3}{14}$, $\frac{1}{112} a^{15} - \frac{1}{7} a^{11} - \frac{1}{4} a^{10} - \frac{5}{56} a^{9} - \frac{1}{4} a^{8} - \frac{13}{112} a^{7} - \frac{1}{4} a^{6} + \frac{9}{56} a^{5} + \frac{1}{4} a^{4} - \frac{3}{14} a^{3} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{570940832} a^{16} + \frac{834259}{570940832} a^{14} - \frac{1677513}{570940832} a^{12} - \frac{88850175}{570940832} a^{10} - \frac{1}{4} a^{9} - \frac{17092475}{142735208} a^{8} - \frac{1}{4} a^{7} - \frac{2901655}{35683802} a^{6} - \frac{1}{4} a^{5} - \frac{10231885}{142735208} a^{4} - \frac{1}{4} a^{3} - \frac{32063005}{71367604} a^{2} + \frac{3740608}{17841901}$, $\frac{1}{570940832} a^{17} + \frac{834259}{570940832} a^{15} - \frac{1677513}{570940832} a^{13} - \frac{88850175}{570940832} a^{11} - \frac{1}{4} a^{10} - \frac{17092475}{142735208} a^{9} - \frac{1}{4} a^{8} - \frac{2901655}{35683802} a^{7} - \frac{1}{4} a^{6} - \frac{10231885}{142735208} a^{5} - \frac{1}{4} a^{4} - \frac{32063005}{71367604} a^{3} + \frac{3740608}{17841901} a$

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:  \( 1565563.8981764752 \) (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, 3.1.972.2, 6.2.1259712.2, 6.2.45349632.4 x2, 6.2.45349632.3, 9.1.1190155742208.6

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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