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Properties

Label 18.0.16517057975...8703.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,13^{15}\cdot 19^{9}$
Root discriminant $36.95$
Ramified primes $13, 19$
Class number $6$ (GRH)
Class group $[6]$ (GRH)
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![51193, -64737, 106613, -6038, -93131, 99218, -21032, -33814, 28560, -7848, 1426, -2746, 2546, -892, -3, 85, -13, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 - 13*x^16 + 85*x^15 - 3*x^14 - 892*x^13 + 2546*x^12 - 2746*x^11 + 1426*x^10 - 7848*x^9 + 28560*x^8 - 33814*x^7 - 21032*x^6 + 99218*x^5 - 93131*x^4 - 6038*x^3 + 106613*x^2 - 64737*x + 51193)
 
gp: K = bnfinit(x^18 - 4*x^17 - 13*x^16 + 85*x^15 - 3*x^14 - 892*x^13 + 2546*x^12 - 2746*x^11 + 1426*x^10 - 7848*x^9 + 28560*x^8 - 33814*x^7 - 21032*x^6 + 99218*x^5 - 93131*x^4 - 6038*x^3 + 106613*x^2 - 64737*x + 51193, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} - 13 x^{16} + 85 x^{15} - 3 x^{14} - 892 x^{13} + 2546 x^{12} - 2746 x^{11} + 1426 x^{10} - 7848 x^{9} + 28560 x^{8} - 33814 x^{7} - 21032 x^{6} + 99218 x^{5} - 93131 x^{4} - 6038 x^{3} + 106613 x^{2} - 64737 x + 51193 \)

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:  \(-16517057975479145583293328703=-\,13^{15}\cdot 19^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.95$
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:  $13, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{6}$, $\frac{1}{20} a^{14} - \frac{1}{5} a^{13} - \frac{1}{10} a^{12} - \frac{1}{5} a^{11} + \frac{1}{10} a^{10} - \frac{1}{5} a^{9} + \frac{1}{5} a^{8} + \frac{1}{10} a^{7} + \frac{1}{5} a^{6} + \frac{3}{10} a^{5} - \frac{2}{5} a^{4} + \frac{3}{10} a^{3} + \frac{2}{5} a^{2} - \frac{1}{5} a + \frac{1}{20}$, $\frac{1}{260} a^{15} + \frac{3}{65} a^{13} + \frac{7}{65} a^{12} + \frac{3}{130} a^{11} + \frac{27}{130} a^{10} - \frac{1}{130} a^{9} - \frac{8}{65} a^{8} - \frac{7}{65} a^{7} - \frac{12}{65} a^{6} + \frac{9}{65} a^{5} - \frac{53}{130} a^{4} - \frac{49}{130} a^{3} + \frac{9}{130} a^{2} + \frac{19}{52} a + \frac{16}{65}$, $\frac{1}{260} a^{16} - \frac{1}{260} a^{14} - \frac{5}{26} a^{13} + \frac{8}{65} a^{12} - \frac{6}{65} a^{11} - \frac{7}{65} a^{10} + \frac{1}{13} a^{9} + \frac{5}{26} a^{8} + \frac{14}{65} a^{7} + \frac{57}{130} a^{6} + \frac{19}{65} a^{5} - \frac{31}{65} a^{4} - \frac{3}{13} a^{3} - \frac{9}{260} a^{2} - \frac{7}{130} a + \frac{9}{20}$, $\frac{1}{222090626248033653008929780749495288260} a^{17} + \frac{62308924644109558101049104436497372}{55522656562008413252232445187373822065} a^{16} + \frac{27458535985366064464942870392691663}{17083894326771819462225367749961176020} a^{15} - \frac{147842228636571579679630395077433433}{222090626248033653008929780749495288260} a^{14} + \frac{3402832591379939276662054162759486413}{55522656562008413252232445187373822065} a^{13} + \frac{4865567173279470999767850437880221773}{55522656562008413252232445187373822065} a^{12} - \frac{2267085278654659140674663621507328813}{11104531312401682650446489037474764413} a^{11} - \frac{5621925821211423684910714766602257027}{111045313124016826504464890374747644130} a^{10} - \frac{1253576690876425708527464616759295166}{11104531312401682650446489037474764413} a^{9} + \frac{6570310666924173582555218427318905503}{111045313124016826504464890374747644130} a^{8} - \frac{5297848562831106487567709775255208842}{55522656562008413252232445187373822065} a^{7} + \frac{363401040901983695860176307456556429}{4270973581692954865556341937490294005} a^{6} + \frac{16568207961640406215789084700205555436}{55522656562008413252232445187373822065} a^{5} - \frac{24813261812039940479132782866136470383}{55522656562008413252232445187373822065} a^{4} - \frac{20266036922147615275618350262828603139}{222090626248033653008929780749495288260} a^{3} - \frac{14464105206570521397340199176074068839}{55522656562008413252232445187373822065} a^{2} - \frac{6413240376559102562923892806789955925}{44418125249606730601785956149899057652} a - \frac{97786414907284793348490280878106096719}{222090626248033653008929780749495288260}$

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

Class group and class number

$C_{6}$, which has order $6$ (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:  \( 1209775.91379 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 18T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-247}) \), 3.1.247.1 x3, 3.3.169.1, 6.0.15069223.1, 6.0.2546698687.1, 6.0.2546698687.2 x2, 9.3.430392078103.1 x3

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

Sibling fields

Degree 6 sibling: 6.0.2546698687.2
Degree 9 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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$13$13.6.5.5$x^{6} + 104$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.5$x^{6} + 104$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.5$x^{6} + 104$$6$$1$$5$$C_6$$[\ ]_{6}$
$19$19.6.3.1$x^{6} - 38 x^{4} + 361 x^{2} - 109744$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
19.6.3.1$x^{6} - 38 x^{4} + 361 x^{2} - 109744$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
19.6.3.1$x^{6} - 38 x^{4} + 361 x^{2} - 109744$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$