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Properties

Label 18.2.16472982863...0000.1
Degree $18$
Signature $[2, 8]$
Discriminant $2^{12}\cdot 3^{8}\cdot 5^{9}\cdot 11^{12}$
Root discriminant $28.61$
Ramified primes $2, 3, 5, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $S_3^2$ (as 18T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![55, -440, 1166, -1116, 72, -1153, 6027, -10356, 9766, -5914, 2379, -328, -381, 272, -58, -1, 3, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 3*x^16 - x^15 - 58*x^14 + 272*x^13 - 381*x^12 - 328*x^11 + 2379*x^10 - 5914*x^9 + 9766*x^8 - 10356*x^7 + 6027*x^6 - 1153*x^5 + 72*x^4 - 1116*x^3 + 1166*x^2 - 440*x + 55)
 
gp: K = bnfinit(x^18 - 3*x^17 + 3*x^16 - x^15 - 58*x^14 + 272*x^13 - 381*x^12 - 328*x^11 + 2379*x^10 - 5914*x^9 + 9766*x^8 - 10356*x^7 + 6027*x^6 - 1153*x^5 + 72*x^4 - 1116*x^3 + 1166*x^2 - 440*x + 55, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 3 x^{16} - x^{15} - 58 x^{14} + 272 x^{13} - 381 x^{12} - 328 x^{11} + 2379 x^{10} - 5914 x^{9} + 9766 x^{8} - 10356 x^{7} + 6027 x^{6} - 1153 x^{5} + 72 x^{4} - 1116 x^{3} + 1166 x^{2} - 440 x + 55 \)

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:  \(164729828637331848000000000=2^{12}\cdot 3^{8}\cdot 5^{9}\cdot 11^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.61$
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:  $2, 3, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{6} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{39} a^{14} - \frac{2}{39} a^{13} + \frac{1}{39} a^{12} + \frac{1}{39} a^{11} - \frac{2}{13} a^{10} + \frac{5}{39} a^{9} + \frac{1}{13} a^{8} - \frac{5}{39} a^{7} - \frac{2}{13} a^{6} - \frac{10}{39} a^{5} - \frac{3}{13} a^{4} - \frac{2}{39} a^{3} - \frac{10}{39} a^{2} + \frac{1}{39} a - \frac{16}{39}$, $\frac{1}{39} a^{15} - \frac{1}{13} a^{13} + \frac{1}{13} a^{12} - \frac{4}{39} a^{11} + \frac{2}{13} a^{10} + \frac{1}{39} a^{8} - \frac{16}{39} a^{7} + \frac{4}{39} a^{6} + \frac{10}{39} a^{5} + \frac{19}{39} a^{4} + \frac{4}{13} a^{3} - \frac{2}{13} a^{2} - \frac{14}{39} a - \frac{19}{39}$, $\frac{1}{117} a^{16} - \frac{1}{117} a^{14} - \frac{1}{117} a^{13} - \frac{5}{39} a^{12} - \frac{2}{13} a^{11} - \frac{4}{39} a^{10} + \frac{50}{117} a^{9} - \frac{4}{13} a^{8} + \frac{11}{39} a^{7} + \frac{37}{117} a^{6} + \frac{4}{39} a^{5} + \frac{20}{117} a^{4} + \frac{29}{117} a^{3} + \frac{31}{117} a^{2} - \frac{17}{117} a - \frac{32}{117}$, $\frac{1}{4703893673880444381} a^{17} + \frac{460385129780390}{1567964557960148127} a^{16} + \frac{31764979810691384}{4703893673880444381} a^{15} + \frac{46395952195039757}{4703893673880444381} a^{14} - \frac{34443714129451475}{522654852653382709} a^{13} - \frac{1219537265336568}{40204219434875593} a^{12} - \frac{45255017838112408}{522654852653382709} a^{11} + \frac{34991341647296687}{361837974913880337} a^{10} - \frac{581177164053930682}{1567964557960148127} a^{9} + \frac{772049628301138285}{1567964557960148127} a^{8} - \frac{1642108339972768241}{4703893673880444381} a^{7} + \frac{317045116314721163}{1567964557960148127} a^{6} + \frac{1525345175019272489}{4703893673880444381} a^{5} + \frac{1501770362337685295}{4703893673880444381} a^{4} + \frac{2274697765636169221}{4703893673880444381} a^{3} + \frac{133370996116143754}{361837974913880337} a^{2} - \frac{87389196958139708}{4703893673880444381} a + \frac{747921343542329894}{1567964557960148127}$

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:  $9$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1231216.341697647 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3^2$ (as 18T9):

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

\(\Q(\sqrt{5}) \), 3.1.1452.1, 3.1.1815.1, 6.2.2178000.1 x2, 6.2.263538000.1, 6.2.16471125.1, 9.1.1147971528000.1

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 R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
2Data not computed
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$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}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$11$11.3.2.1$x^{3} - 11$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
11.3.2.1$x^{3} - 11$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$