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Properties

Label 18.6.36186392678...1281.1
Degree $18$
Signature $[6, 6]$
Discriminant $3^{9}\cdot 107^{9}$
Root discriminant $17.92$
Ramified primes $3, 107$
Class number $1$
Class group Trivial
Galois group $S_3^2$ (as 18T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, 10, -5, -86, 34, 222, -55, -259, 67, 151, -37, -41, -20, -5, 22, -2, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 - 2*x^16 + 22*x^15 - 5*x^14 - 20*x^13 - 41*x^12 - 37*x^11 + 151*x^10 + 67*x^9 - 259*x^8 - 55*x^7 + 222*x^6 + 34*x^5 - 86*x^4 - 5*x^3 + 10*x^2 - 2*x + 1)
 
gp: K = bnfinit(x^18 - 4*x^17 - 2*x^16 + 22*x^15 - 5*x^14 - 20*x^13 - 41*x^12 - 37*x^11 + 151*x^10 + 67*x^9 - 259*x^8 - 55*x^7 + 222*x^6 + 34*x^5 - 86*x^4 - 5*x^3 + 10*x^2 - 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} - 2 x^{16} + 22 x^{15} - 5 x^{14} - 20 x^{13} - 41 x^{12} - 37 x^{11} + 151 x^{10} + 67 x^{9} - 259 x^{8} - 55 x^{7} + 222 x^{6} + 34 x^{5} - 86 x^{4} - 5 x^{3} + 10 x^{2} - 2 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(36186392678065901161281=3^{9}\cdot 107^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.92$
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:  $3, 107$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{28} a^{14} - \frac{1}{4} a^{13} - \frac{1}{7} a^{12} - \frac{3}{28} a^{11} + \frac{1}{28} a^{10} + \frac{5}{28} a^{9} - \frac{1}{28} a^{8} + \frac{2}{7} a^{7} + \frac{3}{28} a^{6} + \frac{3}{28} a^{5} + \frac{2}{7} a^{4} + \frac{5}{14} a^{3} + \frac{11}{28} a^{2} - \frac{3}{7} a + \frac{9}{28}$, $\frac{1}{28} a^{15} + \frac{3}{28} a^{13} - \frac{3}{28} a^{12} - \frac{3}{14} a^{11} - \frac{1}{14} a^{10} - \frac{2}{7} a^{9} - \frac{13}{28} a^{8} - \frac{11}{28} a^{7} + \frac{5}{14} a^{6} - \frac{13}{28} a^{5} + \frac{5}{14} a^{4} + \frac{11}{28} a^{3} + \frac{9}{28} a^{2} - \frac{5}{28} a - \frac{1}{4}$, $\frac{1}{28} a^{16} + \frac{1}{7} a^{13} + \frac{3}{14} a^{12} - \frac{1}{4} a^{11} + \frac{3}{28} a^{10} - \frac{1}{2} a^{9} + \frac{3}{14} a^{8} + \frac{3}{14} a^{6} + \frac{1}{28} a^{5} + \frac{1}{28} a^{4} + \frac{1}{4} a^{3} - \frac{5}{14} a^{2} + \frac{1}{28} a - \frac{13}{28}$, $\frac{1}{10807372436} a^{17} - \frac{105922253}{10807372436} a^{16} + \frac{93732525}{10807372436} a^{15} - \frac{8955533}{2701843109} a^{14} - \frac{375095551}{10807372436} a^{13} + \frac{272462667}{5403686218} a^{12} + \frac{50778254}{2701843109} a^{11} - \frac{552678957}{10807372436} a^{10} + \frac{1108736400}{2701843109} a^{9} + \frac{2992541231}{10807372436} a^{8} + \frac{468896773}{1543910348} a^{7} + \frac{4173365887}{10807372436} a^{6} - \frac{4832880567}{10807372436} a^{5} + \frac{490817379}{2701843109} a^{4} + \frac{1619382259}{5403686218} a^{3} - \frac{2335986181}{5403686218} a^{2} - \frac{1872017363}{10807372436} a + \frac{742888463}{5403686218}$

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

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $11$
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:  \( 21225.4391343 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3^2$ (as 18T11):

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

\(\Q(\sqrt{321}) \), 3.1.107.1, 3.3.321.1 x3, 6.2.33076161.1, 6.6.33076161.2, 9.3.3539149227.1 x3

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: 6.0.309123.2
Degree 9 sibling: 9.3.3539149227.1
Degree 12 sibling: 12.0.1094032426497921.1
Degree 18 siblings: Deg 18, 18.0.338190585776316833283.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
107Data not computed