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Properties

Label 18.0.97748081397...5007.3
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{30}\cdot 7^{15}$
Root discriminant $31.58$
Ramified primes $3, 7$
Class number $21$ (GRH)
Class group $[21]$ (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![512, -1728, 1056, 2892, 756, -4479, -3579, 828, 4896, 1160, -1764, -390, 498, -108, -72, 36, 0, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 36*x^15 - 72*x^14 - 108*x^13 + 498*x^12 - 390*x^11 - 1764*x^10 + 1160*x^9 + 4896*x^8 + 828*x^7 - 3579*x^6 - 4479*x^5 + 756*x^4 + 2892*x^3 + 1056*x^2 - 1728*x + 512)
 
gp: K = bnfinit(x^18 - 3*x^17 + 36*x^15 - 72*x^14 - 108*x^13 + 498*x^12 - 390*x^11 - 1764*x^10 + 1160*x^9 + 4896*x^8 + 828*x^7 - 3579*x^6 - 4479*x^5 + 756*x^4 + 2892*x^3 + 1056*x^2 - 1728*x + 512, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 36 x^{15} - 72 x^{14} - 108 x^{13} + 498 x^{12} - 390 x^{11} - 1764 x^{10} + 1160 x^{9} + 4896 x^{8} + 828 x^{7} - 3579 x^{6} - 4479 x^{5} + 756 x^{4} + 2892 x^{3} + 1056 x^{2} - 1728 x + 512 \)

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:  \(-977480813971145474830595007=-\,3^{30}\cdot 7^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.58$
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:  $3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{3}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{16} a^{9} - \frac{1}{8} a^{7} + \frac{1}{8} a^{5} - \frac{1}{8} a^{4} - \frac{1}{16} a^{3} + \frac{1}{8} a^{2}$, $\frac{1}{16} a^{10} - \frac{1}{8} a^{6} - \frac{1}{8} a^{5} + \frac{3}{16} a^{4} - \frac{1}{8} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{16} a^{11} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} + \frac{3}{16} a^{5} - \frac{1}{8} a^{4} - \frac{1}{8} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{16} a^{12} - \frac{1}{8} a^{7} - \frac{1}{16} a^{6} - \frac{1}{8} a^{5} + \frac{1}{8} a^{4} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{64} a^{13} - \frac{1}{32} a^{12} - \frac{1}{32} a^{11} + \frac{1}{64} a^{10} + \frac{7}{64} a^{7} - \frac{1}{32} a^{6} + \frac{7}{32} a^{5} + \frac{7}{64} a^{4} - \frac{1}{16} a^{3} + \frac{3}{16} a^{2}$, $\frac{1}{64} a^{14} - \frac{1}{32} a^{12} + \frac{1}{64} a^{11} - \frac{1}{32} a^{10} - \frac{1}{64} a^{8} - \frac{1}{16} a^{7} + \frac{3}{32} a^{6} - \frac{1}{64} a^{5} + \frac{7}{32} a^{4} + \frac{1}{16} a^{3} + \frac{1}{4} a$, $\frac{1}{128} a^{15} - \frac{1}{128} a^{14} - \frac{1}{128} a^{13} - \frac{3}{128} a^{12} - \frac{1}{128} a^{11} - \frac{1}{128} a^{10} - \frac{1}{128} a^{9} - \frac{3}{128} a^{8} + \frac{1}{128} a^{7} - \frac{5}{128} a^{6} + \frac{9}{128} a^{5} + \frac{17}{128} a^{4} - \frac{1}{16} a^{3} + \frac{3}{32} a^{2} + \frac{1}{8} a$, $\frac{1}{128} a^{16} - \frac{1}{64} a^{10} - \frac{1}{32} a^{9} - \frac{1}{32} a^{8} + \frac{1}{16} a^{6} + \frac{1}{16} a^{5} - \frac{15}{128} a^{4} + \frac{3}{32} a^{3} + \frac{3}{32} a^{2} - \frac{1}{8} a$, $\frac{1}{18020469535911876608} a^{17} + \frac{12571137473967989}{4505117383977969152} a^{16} + \frac{17072549849943847}{4505117383977969152} a^{15} - \frac{8507659972985439}{2252558691988984576} a^{14} - \frac{941619596268447}{1126279345994492288} a^{13} - \frac{96594722674660271}{4505117383977969152} a^{12} + \frac{200362037784964055}{9010234767955938304} a^{11} - \frac{61922181660701473}{4505117383977969152} a^{10} - \frac{9864432295664869}{563139672997246144} a^{9} + \frac{80921493536077377}{2252558691988984576} a^{8} - \frac{137544272569241841}{2252558691988984576} a^{7} + \frac{86301813267200689}{4505117383977969152} a^{6} + \frac{704391421862768385}{18020469535911876608} a^{5} + \frac{482759124879405935}{2252558691988984576} a^{4} + \frac{192284796392871555}{4505117383977969152} a^{3} + \frac{73844851244440941}{563139672997246144} a^{2} + \frac{105349588656038665}{281569836498623072} a - \frac{2921903719375231}{35196229562327884}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{21}$, which has order $21$ (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:  \( -1 \) (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:  \( 9447907.468229258 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{-7}) \), 3.3.3969.1, 3.1.1323.1, 6.0.110270727.2, 6.0.12252303.1, 9.3.1688134559643.5

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 12 sibling: data not computed
Degree 18 sibling: 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{6}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
3Data not computed
7Data not computed