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Properties

Label 18.0.23402250812...3536.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{33}\cdot 3^{9}\cdot 7^{12}$
Root discriminant $22.59$
Ramified primes $2, 3, 7$
Class number $6$
Class group $[6]$
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![6, 12, 34, -8, 94, -40, 55, 112, -210, 248, -153, -16, 180, -212, 175, -88, 34, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 8*x^17 + 34*x^16 - 88*x^15 + 175*x^14 - 212*x^13 + 180*x^12 - 16*x^11 - 153*x^10 + 248*x^9 - 210*x^8 + 112*x^7 + 55*x^6 - 40*x^5 + 94*x^4 - 8*x^3 + 34*x^2 + 12*x + 6)
 
gp: K = bnfinit(x^18 - 8*x^17 + 34*x^16 - 88*x^15 + 175*x^14 - 212*x^13 + 180*x^12 - 16*x^11 - 153*x^10 + 248*x^9 - 210*x^8 + 112*x^7 + 55*x^6 - 40*x^5 + 94*x^4 - 8*x^3 + 34*x^2 + 12*x + 6, 1)
 

Normalized defining polynomial

\( x^{18} - 8 x^{17} + 34 x^{16} - 88 x^{15} + 175 x^{14} - 212 x^{13} + 180 x^{12} - 16 x^{11} - 153 x^{10} + 248 x^{9} - 210 x^{8} + 112 x^{7} + 55 x^{6} - 40 x^{5} + 94 x^{4} - 8 x^{3} + 34 x^{2} + 12 x + 6 \)

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:  \(-2340225081216495607873536=-\,2^{33}\cdot 3^{9}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.59$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6}$, $\frac{1}{6} a^{13} + \frac{1}{6} a^{12} - \frac{1}{6} a^{11} - \frac{1}{6} a^{10} - \frac{1}{6} a^{9} - \frac{1}{6} a^{8} + \frac{1}{6} a^{7} + \frac{1}{6} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{6} a^{14} + \frac{1}{6} a^{12} - \frac{1}{2} a^{10} - \frac{1}{6} a^{8} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{36} a^{15} + \frac{1}{36} a^{14} - \frac{1}{18} a^{13} - \frac{2}{9} a^{12} + \frac{1}{3} a^{11} + \frac{1}{6} a^{10} - \frac{5}{18} a^{9} + \frac{2}{9} a^{8} + \frac{1}{12} a^{7} - \frac{1}{12} a^{6} + \frac{4}{9} a^{5} - \frac{2}{9} a^{4} - \frac{7}{18} a^{3} - \frac{1}{18} a^{2} - \frac{1}{6} a + \frac{1}{6}$, $\frac{1}{108} a^{16} - \frac{1}{108} a^{15} + \frac{1}{54} a^{14} - \frac{1}{27} a^{13} + \frac{4}{27} a^{12} - \frac{1}{6} a^{11} + \frac{25}{54} a^{10} - \frac{11}{27} a^{9} - \frac{1}{108} a^{8} - \frac{5}{12} a^{7} - \frac{8}{27} a^{6} + \frac{8}{27} a^{5} - \frac{5}{54} a^{4} - \frac{5}{54} a^{3} + \frac{11}{54} a^{2} + \frac{1}{6} a + \frac{2}{9}$, $\frac{1}{97013231789004} a^{17} + \frac{316672096175}{97013231789004} a^{16} + \frac{322300213513}{48506615894502} a^{15} - \frac{4001968275611}{48506615894502} a^{14} - \frac{1979209230842}{24253307947251} a^{13} - \frac{123092124842}{8084435982417} a^{12} - \frac{8559165091295}{48506615894502} a^{11} - \frac{155943779711}{545018156118} a^{10} - \frac{22071994947445}{97013231789004} a^{9} - \frac{2751245254633}{32337743929668} a^{8} - \frac{9135409083335}{24253307947251} a^{7} - \frac{4736580850000}{24253307947251} a^{6} - \frac{20976567670979}{48506615894502} a^{5} - \frac{158416298839}{545018156118} a^{4} + \frac{20598615017531}{48506615894502} a^{3} + \frac{621431531465}{16168871964834} a^{2} + \frac{2126124233348}{8084435982417} a + \frac{794499137825}{2694811994139}$

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:  \( 168976.3636822919 \)
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

\(\Q(\sqrt{-6}) \), 3.1.1176.1 x3, 3.1.588.1, 6.0.132765696.1, 6.0.33191424.2, 9.1.78066229248.1 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: 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$2.6.11.5$x^{6} + 6$$6$$1$$11$$D_{6}$$[3]_{3}^{2}$
2.6.11.5$x^{6} + 6$$6$$1$$11$$D_{6}$$[3]_{3}^{2}$
2.6.11.5$x^{6} + 6$$6$$1$$11$$D_{6}$$[3]_{3}^{2}$
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{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$$[\ ]_{2}^{2}$
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$