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Properties

Label 18.0.68700542660...0096.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{27}\cdot 13^{15}$
Root discriminant $23.98$
Ramified primes $2, 13$
Class number $6$
Class group $[6]$
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![104, 0, 1352, 0, 2522, 0, -883, 0, 1084, 0, -203, 0, -110, 0, 6, 0, 2, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 2*x^16 + 6*x^14 - 110*x^12 - 203*x^10 + 1084*x^8 - 883*x^6 + 2522*x^4 + 1352*x^2 + 104)
 
gp: K = bnfinit(x^18 + 2*x^16 + 6*x^14 - 110*x^12 - 203*x^10 + 1084*x^8 - 883*x^6 + 2522*x^4 + 1352*x^2 + 104, 1)
 

Normalized defining polynomial

\( x^{18} + 2 x^{16} + 6 x^{14} - 110 x^{12} - 203 x^{10} + 1084 x^{8} - 883 x^{6} + 2522 x^{4} + 1352 x^{2} + 104 \)

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:  \(-6870054266002333390340096=-\,2^{27}\cdot 13^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.98$
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:  $2, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{372} a^{12} - \frac{5}{372} a^{10} + \frac{47}{372} a^{8} - \frac{27}{62} a^{6} + \frac{143}{372} a^{4} - \frac{1}{2} a^{3} + \frac{37}{93} a^{2} + \frac{19}{93}$, $\frac{1}{372} a^{13} - \frac{5}{372} a^{11} + \frac{47}{372} a^{9} - \frac{27}{62} a^{7} + \frac{143}{372} a^{5} - \frac{1}{2} a^{4} + \frac{37}{93} a^{3} + \frac{19}{93} a$, $\frac{1}{1860} a^{14} - \frac{1}{1860} a^{12} + \frac{71}{620} a^{10} + \frac{53}{465} a^{8} + \frac{53}{1860} a^{6} - \frac{1}{2} a^{5} + \frac{29}{155} a^{4} + \frac{11}{186} a^{2} + \frac{76}{465}$, $\frac{1}{1860} a^{15} - \frac{1}{1860} a^{13} + \frac{71}{620} a^{11} + \frac{53}{465} a^{9} + \frac{53}{1860} a^{7} - \frac{1}{2} a^{6} + \frac{29}{155} a^{5} + \frac{11}{186} a^{3} + \frac{76}{465} a$, $\frac{1}{6272543100} a^{16} + \frac{420332}{1568135775} a^{14} + \frac{166796}{1568135775} a^{12} - \frac{131389694}{1568135775} a^{10} + \frac{473911657}{2090847700} a^{8} + \frac{36941641}{209084770} a^{6} + \frac{499788749}{2090847700} a^{4} - \frac{153126038}{522711925} a^{2} - \frac{184878001}{1568135775}$, $\frac{1}{6272543100} a^{17} + \frac{420332}{1568135775} a^{15} + \frac{166796}{1568135775} a^{13} - \frac{131389694}{1568135775} a^{11} + \frac{473911657}{2090847700} a^{9} + \frac{36941641}{209084770} a^{7} + \frac{499788749}{2090847700} a^{5} - \frac{153126038}{522711925} a^{3} - \frac{184878001}{1568135775} a$

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

Class group and class number

$C_{6}$, which has order $6$

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:  \( -1 \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 26737.5967738 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 18T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-26}) \), 3.1.104.1 x3, 3.3.169.1, 6.0.1124864.1, 6.0.190102016.3 x2, 6.0.190102016.1, 9.3.32127240704.1 x3

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

Sibling fields

Degree 6 sibling: 6.0.190102016.3
Degree 9 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 R ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.9.7$x^{6} + 4 x^{4} + 4 x^{2} - 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.7$x^{6} + 4 x^{4} + 4 x^{2} - 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.7$x^{6} + 4 x^{4} + 4 x^{2} - 24$$2$$3$$9$$C_6$$[3]^{3}$
$13$13.6.5.5$x^{6} + 104$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.5$x^{6} + 104$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.5$x^{6} + 104$$6$$1$$5$$C_6$$[\ ]_{6}$