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Properties

Label 18.0.17256813706...2464.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 37^{12}$
Root discriminant $22.21$
Ramified primes $2, 37$
Class number $4$
Class group $[2, 2]$
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![1, 0, 29, 0, 136, 0, 477, 0, 1067, 0, 1129, 0, 475, 0, 102, 0, 16, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 16*x^16 + 102*x^14 + 475*x^12 + 1129*x^10 + 1067*x^8 + 477*x^6 + 136*x^4 + 29*x^2 + 1)
 
gp: K = bnfinit(x^18 + 16*x^16 + 102*x^14 + 475*x^12 + 1129*x^10 + 1067*x^8 + 477*x^6 + 136*x^4 + 29*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{18} + 16 x^{16} + 102 x^{14} + 475 x^{12} + 1129 x^{10} + 1067 x^{8} + 477 x^{6} + 136 x^{4} + 29 x^{2} + 1 \)

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:  \(-1725681370618930208702464=-\,2^{18}\cdot 37^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.21$
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:  $2, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{20} a^{12} - \frac{1}{2} a^{11} + \frac{1}{4} a^{10} - \frac{1}{2} a^{9} + \frac{1}{20} a^{8} + \frac{1}{4} a^{6} + \frac{3}{20} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{9}{20}$, $\frac{1}{20} a^{13} + \frac{1}{4} a^{11} + \frac{1}{20} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} - \frac{1}{2} a^{6} + \frac{3}{20} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} + \frac{9}{20} a - \frac{1}{2}$, $\frac{1}{20} a^{14} - \frac{1}{2} a^{11} - \frac{1}{5} a^{10} - \frac{1}{2} a^{7} - \frac{1}{10} a^{6} - \frac{1}{2} a^{5} - \frac{3}{10} a^{2} - \frac{1}{4}$, $\frac{1}{20} a^{15} - \frac{1}{5} a^{11} - \frac{1}{2} a^{10} - \frac{1}{10} a^{7} - \frac{1}{2} a^{4} - \frac{3}{10} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{89484340} a^{16} + \frac{138783}{17896868} a^{14} + \frac{639581}{89484340} a^{12} - \frac{1}{2} a^{11} - \frac{441581}{1626988} a^{10} - \frac{1}{2} a^{9} + \frac{12769643}{89484340} a^{8} - \frac{1}{2} a^{7} + \frac{5420127}{17896868} a^{6} + \frac{18691729}{89484340} a^{4} - \frac{1}{2} a^{3} - \frac{292375}{4474217} a^{2} - \frac{2780903}{8948434}$, $\frac{1}{89484340} a^{17} + \frac{138783}{17896868} a^{15} + \frac{639581}{89484340} a^{13} - \frac{441581}{1626988} a^{11} + \frac{12769643}{89484340} a^{9} + \frac{5420127}{17896868} a^{7} - \frac{1}{2} a^{6} + \frac{18691729}{89484340} a^{5} - \frac{292375}{4474217} a^{3} - \frac{1}{2} a^{2} - \frac{2780903}{8948434} a - \frac{1}{2}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$

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:  \( \frac{15259}{62315} a^{17} + \frac{245612}{62315} a^{15} + \frac{631611}{24926} a^{13} + \frac{1342489}{11330} a^{11} + \frac{35669981}{124630} a^{9} + \frac{35044497}{124630} a^{7} + \frac{15724383}{124630} a^{5} + \frac{3779241}{124630} a^{3} + \frac{744023}{124630} a \) (order $4$)
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:  \( 18639.792218 \)
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

\(\Q(\sqrt{-1}) \), 3.1.5476.1 x3, 3.3.1369.1, 6.0.119946304.2, 6.0.87616.1 x2, 6.0.119946304.1, 9.3.164206490176.1 x3

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

Sibling fields

Degree 6 sibling: 6.0.87616.1
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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
$37$37.9.6.1$x^{9} + 222 x^{6} + 15059 x^{3} + 405224$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
37.9.6.1$x^{9} + 222 x^{6} + 15059 x^{3} + 405224$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$