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Properties

Label 18.0.27084263512...8912.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{21}\cdot 43^{6}$
Root discriminant $20.04$
Ramified primes $2, 3, 43$
Class number $1$ (GRH)
Class group Trivial (GRH)
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, -6, 12, -28, 60, -12, 163, 84, 120, 44, -132, -60, -37, 54, 60, -12, -12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 12*x^16 - 12*x^15 + 60*x^14 + 54*x^13 - 37*x^12 - 60*x^11 - 132*x^10 + 44*x^9 + 120*x^8 + 84*x^7 + 163*x^6 - 12*x^5 + 60*x^4 - 28*x^3 + 12*x^2 - 6*x + 1)
 
gp: K = bnfinit(x^18 - 12*x^16 - 12*x^15 + 60*x^14 + 54*x^13 - 37*x^12 - 60*x^11 - 132*x^10 + 44*x^9 + 120*x^8 + 84*x^7 + 163*x^6 - 12*x^5 + 60*x^4 - 28*x^3 + 12*x^2 - 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 12 x^{16} - 12 x^{15} + 60 x^{14} + 54 x^{13} - 37 x^{12} - 60 x^{11} - 132 x^{10} + 44 x^{9} + 120 x^{8} + 84 x^{7} + 163 x^{6} - 12 x^{5} + 60 x^{4} - 28 x^{3} + 12 x^{2} - 6 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-270842635128557551398912=-\,2^{12}\cdot 3^{21}\cdot 43^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.04$
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:  $2, 3, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{10} a^{12} + \frac{1}{10} a^{10} - \frac{1}{5} a^{9} - \frac{2}{5} a^{8} - \frac{3}{10} a^{7} + \frac{2}{5} a^{5} - \frac{1}{10} a^{4} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2} - \frac{1}{2} a - \frac{1}{10}$, $\frac{1}{20} a^{13} + \frac{1}{20} a^{11} + \frac{3}{20} a^{10} + \frac{1}{20} a^{9} + \frac{7}{20} a^{8} + \frac{1}{4} a^{7} - \frac{1}{20} a^{6} + \frac{9}{20} a^{5} + \frac{9}{20} a^{4} + \frac{1}{20} a^{3} + \frac{1}{4} a^{2} + \frac{1}{5} a + \frac{1}{4}$, $\frac{1}{20} a^{14} - \frac{1}{20} a^{12} + \frac{3}{20} a^{11} - \frac{1}{20} a^{10} + \frac{1}{20} a^{9} - \frac{7}{20} a^{8} + \frac{1}{4} a^{7} - \frac{1}{20} a^{6} + \frac{1}{20} a^{5} + \frac{3}{20} a^{4} + \frac{7}{20} a^{3} - \frac{2}{5} a^{2} - \frac{1}{4} a - \frac{2}{5}$, $\frac{1}{20} a^{15} - \frac{1}{20} a^{12} + \frac{1}{10} a^{9} + \frac{2}{5} a^{8} - \frac{1}{5} a^{7} - \frac{1}{5} a^{5} - \frac{3}{20} a^{3} - \frac{1}{5} a^{2} - \frac{1}{5} a + \frac{9}{20}$, $\frac{1}{100} a^{16} - \frac{1}{100} a^{15} + \frac{1}{50} a^{13} + \frac{3}{100} a^{12} + \frac{13}{100} a^{11} - \frac{7}{100} a^{10} + \frac{3}{20} a^{9} + \frac{31}{100} a^{8} - \frac{47}{100} a^{7} + \frac{3}{100} a^{6} + \frac{49}{100} a^{5} - \frac{19}{50} a^{4} + \frac{2}{5} a^{3} + \frac{17}{100} a^{2} - \frac{1}{20} a - \frac{13}{50}$, $\frac{1}{5353973000} a^{17} - \frac{5462233}{5353973000} a^{16} - \frac{95990823}{5353973000} a^{15} + \frac{28046347}{5353973000} a^{14} - \frac{43375941}{5353973000} a^{13} + \frac{129167307}{5353973000} a^{12} + \frac{309821841}{2676986500} a^{11} - \frac{23954627}{669246625} a^{10} - \frac{102029027}{2676986500} a^{9} - \frac{27131328}{669246625} a^{8} - \frac{680771619}{2676986500} a^{7} - \frac{575237803}{1338493250} a^{6} - \frac{1920818791}{5353973000} a^{5} + \frac{821301141}{5353973000} a^{4} - \frac{48330543}{5353973000} a^{3} + \frac{324691741}{5353973000} a^{2} - \frac{1979251241}{5353973000} a + \frac{196142697}{5353973000}$

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:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{42603}{89000} a^{17} + \frac{26551}{89000} a^{16} - \frac{507919}{89000} a^{15} - \frac{832759}{89000} a^{14} + \frac{2200477}{89000} a^{13} + \frac{3889071}{89000} a^{12} + \frac{7212}{11125} a^{11} - \frac{1785199}{44500} a^{10} - \frac{1830653}{22250} a^{9} - \frac{772361}{44500} a^{8} + \frac{731242}{11125} a^{7} + \frac{3531757}{44500} a^{6} + \frac{9193877}{89000} a^{5} + \frac{3877123}{89000} a^{4} + \frac{2991971}{89000} a^{3} + \frac{69273}{89000} a^{2} + \frac{222027}{89000} a - \frac{93609}{89000} \) (order $6$)
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:  \( 166897.8979387987 \) (assuming GRH)
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

\(\Q(\sqrt{-3}) \), 3.1.172.1, 3.1.108.1 x3, 6.0.798768.3, 6.0.34992.1, 9.1.100155921984.1 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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$2.6.4.1$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}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$3$3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
3.12.14.6$x^{12} + 3 x^{11} + 3 x^{10} - 6 x^{9} + 3 x^{8} + 9 x^{7} + 9 x^{4} + 9 x^{3} + 9$$6$$2$$14$$D_6$$[3/2]_{2}^{2}$
$43$$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$