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Properties

Label 18.0.59716805240...6347.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{27}\cdot 23^{8}$
Root discriminant $20.94$
Ramified primes $3, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $S_3\times D_9$ (as 18T50)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![991, -3426, 6777, -9323, 10998, -11340, 10122, -7992, 5778, -3813, 2175, -1170, 620, -195, 135, -13, 18, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 18*x^16 - 13*x^15 + 135*x^14 - 195*x^13 + 620*x^12 - 1170*x^11 + 2175*x^10 - 3813*x^9 + 5778*x^8 - 7992*x^7 + 10122*x^6 - 11340*x^5 + 10998*x^4 - 9323*x^3 + 6777*x^2 - 3426*x + 991)
 
gp: K = bnfinit(x^18 + 18*x^16 - 13*x^15 + 135*x^14 - 195*x^13 + 620*x^12 - 1170*x^11 + 2175*x^10 - 3813*x^9 + 5778*x^8 - 7992*x^7 + 10122*x^6 - 11340*x^5 + 10998*x^4 - 9323*x^3 + 6777*x^2 - 3426*x + 991, 1)
 

Normalized defining polynomial

\( x^{18} + 18 x^{16} - 13 x^{15} + 135 x^{14} - 195 x^{13} + 620 x^{12} - 1170 x^{11} + 2175 x^{10} - 3813 x^{9} + 5778 x^{8} - 7992 x^{7} + 10122 x^{6} - 11340 x^{5} + 10998 x^{4} - 9323 x^{3} + 6777 x^{2} - 3426 x + 991 \)

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:  \(-597168052405647575476347=-\,3^{27}\cdot 23^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.94$
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:  $3, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{35} a^{12} + \frac{12}{35} a^{10} - \frac{3}{35} a^{9} - \frac{16}{35} a^{8} + \frac{8}{35} a^{7} + \frac{3}{7} a^{6} - \frac{11}{35} a^{5} + \frac{13}{35} a^{4} - \frac{2}{7} a^{3} + \frac{3}{35} a^{2} + \frac{3}{35} a - \frac{8}{35}$, $\frac{1}{35} a^{13} + \frac{12}{35} a^{11} - \frac{3}{35} a^{10} - \frac{16}{35} a^{9} + \frac{8}{35} a^{8} + \frac{3}{7} a^{7} - \frac{11}{35} a^{6} + \frac{13}{35} a^{5} - \frac{2}{7} a^{4} + \frac{3}{35} a^{3} + \frac{3}{35} a^{2} - \frac{8}{35} a$, $\frac{1}{35} a^{14} - \frac{3}{35} a^{11} + \frac{3}{7} a^{10} + \frac{9}{35} a^{9} - \frac{3}{35} a^{8} - \frac{2}{35} a^{7} + \frac{8}{35} a^{6} + \frac{17}{35} a^{5} - \frac{13}{35} a^{4} - \frac{17}{35} a^{3} - \frac{9}{35} a^{2} - \frac{1}{35} a - \frac{9}{35}$, $\frac{1}{2905} a^{15} + \frac{3}{581} a^{13} + \frac{3}{2905} a^{12} + \frac{18}{581} a^{11} + \frac{36}{2905} a^{10} - \frac{681}{2905} a^{9} + \frac{162}{2905} a^{8} + \frac{561}{2905} a^{7} - \frac{338}{2905} a^{6} + \frac{711}{2905} a^{5} - \frac{824}{2905} a^{4} + \frac{921}{2905} a^{3} - \frac{148}{2905} a^{2} + \frac{1359}{2905} a - \frac{783}{2905}$, $\frac{1}{2905} a^{16} + \frac{3}{581} a^{14} + \frac{3}{2905} a^{13} + \frac{1}{415} a^{12} + \frac{36}{2905} a^{11} + \frac{1228}{2905} a^{10} + \frac{411}{2905} a^{9} - \frac{1016}{2905} a^{8} - \frac{1002}{2905} a^{7} - \frac{534}{2905} a^{6} + \frac{89}{2905} a^{5} - \frac{158}{2905} a^{4} + \frac{682}{2905} a^{3} + \frac{222}{581} a^{2} - \frac{1032}{2905} a + \frac{8}{35}$, $\frac{1}{107485} a^{17} - \frac{6}{107485} a^{16} + \frac{17}{107485} a^{15} - \frac{4}{107485} a^{14} - \frac{396}{107485} a^{13} + \frac{8}{1295} a^{12} + \frac{36633}{107485} a^{11} - \frac{34192}{107485} a^{10} + \frac{9266}{107485} a^{9} - \frac{1174}{21497} a^{8} - \frac{29339}{107485} a^{7} + \frac{3281}{107485} a^{6} - \frac{39608}{107485} a^{5} + \frac{7566}{21497} a^{4} + \frac{33139}{107485} a^{3} - \frac{2178}{107485} a^{2} - \frac{34582}{107485} a + \frac{11}{107485}$

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:  \( -\frac{4}{581} a^{15} - \frac{60}{581} a^{13} + \frac{71}{581} a^{12} - \frac{360}{581} a^{11} + \frac{852}{581} a^{10} - \frac{1592}{581} a^{9} + \frac{3834}{581} a^{8} - \frac{6228}{581} a^{7} + \frac{1367}{83} a^{6} - \frac{14796}{581} a^{5} + \frac{2451}{83} a^{4} - \frac{17877}{581} a^{3} + \frac{17109}{581} a^{2} - \frac{1737}{83} a + \frac{4792}{581} \) (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:  \( 84776.5081683 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times D_9$ (as 18T50):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 18 conjugacy class representatives for $S_3\times D_9$
Character table for $S_3\times D_9$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.23.1, 6.0.14283.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $18$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ R $18$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
3Data not computed
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$