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Properties

Label 18.2.76924344897...7808.1
Degree $18$
Signature $[2, 8]$
Discriminant $2^{12}\cdot 3^{8}\cdot 17^{15}$
Root discriminant $27.42$
Ramified primes $2, 3, 17$
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![-2197, 0, 0, -2290, 0, 0, -2701, 0, 0, -44, 0, 0, 145, 0, 0, -22, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 22*x^15 + 145*x^12 - 44*x^9 - 2701*x^6 - 2290*x^3 - 2197)
 
gp: K = bnfinit(x^18 - 22*x^15 + 145*x^12 - 44*x^9 - 2701*x^6 - 2290*x^3 - 2197, 1)
 

Normalized defining polynomial

\( x^{18} - 22 x^{15} + 145 x^{12} - 44 x^{9} - 2701 x^{6} - 2290 x^{3} - 2197 \)

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:  \(76924344897355372207607808=2^{12}\cdot 3^{8}\cdot 17^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.42$
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:  $2, 3, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{6} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{6}$, $\frac{1}{6} a^{10} - \frac{1}{2} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{6} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{6} a$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{6} a^{5} - \frac{1}{3} a^{4} + \frac{1}{6} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{18} a^{12} - \frac{1}{18} a^{9} - \frac{1}{2} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{7}{18} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{2}{9}$, $\frac{1}{18} a^{13} - \frac{1}{18} a^{10} - \frac{1}{2} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{7}{18} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{2}{9} a$, $\frac{1}{54} a^{14} + \frac{1}{54} a^{12} - \frac{2}{27} a^{11} - \frac{2}{27} a^{9} + \frac{1}{3} a^{7} + \frac{7}{27} a^{5} - \frac{11}{27} a^{3} + \frac{17}{54} a^{2} - \frac{1}{3} a + \frac{17}{54}$, $\frac{1}{67299444} a^{15} - \frac{1}{54} a^{13} + \frac{689}{3542076} a^{12} + \frac{2}{27} a^{10} - \frac{26089}{623143} a^{9} + \frac{1}{3} a^{7} - \frac{15554297}{33649722} a^{6} + \frac{1}{3} a^{5} + \frac{11}{27} a^{4} + \frac{8583983}{67299444} a^{3} - \frac{1}{3} a^{2} + \frac{19}{54} a + \frac{3152491}{7477716}$, $\frac{1}{874892772} a^{16} + \frac{984599}{46046988} a^{13} + \frac{1}{54} a^{12} - \frac{4596802}{72907731} a^{10} - \frac{2}{27} a^{9} - \frac{82853741}{437446386} a^{7} - \frac{1}{3} a^{6} + \frac{315170339}{874892772} a^{4} - \frac{2}{27} a^{3} - \frac{138850561}{291630924} a - \frac{1}{54}$, $\frac{1}{11373606036} a^{17} + \frac{896681}{199536948} a^{14} + \frac{1}{54} a^{13} - \frac{1}{54} a^{12} - \frac{119101573}{2843401509} a^{11} - \frac{2}{27} a^{10} + \frac{2}{27} a^{9} + \frac{937854493}{5686803018} a^{8} - \frac{1}{3} a^{6} + \frac{3976758607}{11373606036} a^{5} - \frac{11}{27} a^{4} + \frac{11}{27} a^{3} + \frac{5092032437}{11373606036} a^{2} - \frac{1}{54} a + \frac{1}{54}$

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:  \( 3505188.609604665 \) (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

\(\Q(\sqrt{17}) \), 3.1.867.1, 3.1.204.1, 6.2.204459408.1 x2, 6.2.12778713.1, 6.2.707472.1, 9.1.2127195680832.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{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}$
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$17$17.6.5.1$x^{6} - 17$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
17.12.10.1$x^{12} - 170 x^{6} + 210681$$6$$2$$10$$D_6$$[\ ]_{6}^{2}$