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Properties

Label 18.2.86331506811...2304.1
Degree $18$
Signature $[2, 8]$
Discriminant $2^{12}\cdot 11^{9}\cdot 19^{7}$
Root discriminant $16.55$
Ramified primes $2, 11, 19$
Class number $1$
Class group Trivial
Galois group $S_3\times S_4$ (as 18T69)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2, -4, -8, -9, -4, -19, -35, -33, 35, 68, 3, -45, 46, 51, -1, -9, -3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^16 - 9*x^15 - x^14 + 51*x^13 + 46*x^12 - 45*x^11 + 3*x^10 + 68*x^9 + 35*x^8 - 33*x^7 - 35*x^6 - 19*x^5 - 4*x^4 - 9*x^3 - 8*x^2 - 4*x - 2)
 
gp: K = bnfinit(x^18 - 3*x^16 - 9*x^15 - x^14 + 51*x^13 + 46*x^12 - 45*x^11 + 3*x^10 + 68*x^9 + 35*x^8 - 33*x^7 - 35*x^6 - 19*x^5 - 4*x^4 - 9*x^3 - 8*x^2 - 4*x - 2, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{16} - 9 x^{15} - x^{14} + 51 x^{13} + 46 x^{12} - 45 x^{11} + 3 x^{10} + 68 x^{9} + 35 x^{8} - 33 x^{7} - 35 x^{6} - 19 x^{5} - 4 x^{4} - 9 x^{3} - 8 x^{2} - 4 x - 2 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(8633150681191238242304=2^{12}\cdot 11^{9}\cdot 19^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.55$
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:  $2, 11, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} + \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2}$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{14} - \frac{1}{8} a^{13} - \frac{3}{8} a^{11} - \frac{3}{8} a^{10} - \frac{1}{4} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{7} + \frac{1}{4} a^{6} - \frac{1}{8} a^{5} + \frac{1}{8} a^{4} - \frac{3}{8} a^{3} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{16} a^{16} - \frac{1}{8} a^{14} + \frac{3}{16} a^{13} + \frac{1}{16} a^{12} - \frac{3}{8} a^{11} - \frac{1}{16} a^{10} - \frac{7}{16} a^{9} - \frac{1}{8} a^{8} + \frac{5}{16} a^{7} + \frac{5}{16} a^{6} - \frac{1}{2} a^{5} - \frac{3}{8} a^{4} - \frac{7}{16} a^{3} + \frac{1}{8} a^{2} - \frac{1}{4} a - \frac{3}{8}$, $\frac{1}{114147570616336} a^{17} + \frac{796339947157}{57073785308168} a^{16} - \frac{522442731691}{28536892654084} a^{15} + \frac{5036975048565}{114147570616336} a^{14} - \frac{8054404461943}{114147570616336} a^{13} - \frac{2733232217295}{14268446327042} a^{12} + \frac{4412513770181}{114147570616336} a^{11} + \frac{38617257970349}{114147570616336} a^{10} + \frac{4714846493}{750970859318} a^{9} + \frac{37102639571679}{114147570616336} a^{8} + \frac{22452736631401}{114147570616336} a^{7} + \frac{9482178984473}{57073785308168} a^{6} - \frac{2939705744915}{14268446327042} a^{5} + \frac{41389360089643}{114147570616336} a^{4} + \frac{14319640173175}{57073785308168} a^{3} - \frac{1242596031262}{7134223163521} a^{2} - \frac{1274876812371}{3003883437272} a + \frac{1968879357887}{14268446327042}$

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:  $9$
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:  \( 19984.3522922 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times S_4$ (as 18T69):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 144
The 15 conjugacy class representatives for $S_3\times S_4$
Character table for $S_3\times S_4$

Intermediate fields

3.1.76.1, 3.1.44.1, 6.2.404624.1, 9.1.584277056.1

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

Sibling fields

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{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}$
$11$11.6.0.1$x^{6} + x^{2} - 2 x + 8$$1$$6$$0$$C_6$$[\ ]^{6}$
11.12.9.1$x^{12} - 121 x^{4} + 3993$$4$$3$$9$$D_4 \times C_3$$[\ ]_{4}^{6}$
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$