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Properties

Label 18.0.30505830462...9328.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{8}\cdot 3^{33}\cdot 11^{8}$
Root discriminant $29.60$
Ramified primes $2, 3, 11$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_3^2:S_3$ (as 18T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, -48, 144, -208, 264, 336, 716, 468, -156, 848, 312, -234, 257, 84, -51, 26, 9, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 9*x^16 + 26*x^15 - 51*x^14 + 84*x^13 + 257*x^12 - 234*x^11 + 312*x^10 + 848*x^9 - 156*x^8 + 468*x^7 + 716*x^6 + 336*x^5 + 264*x^4 - 208*x^3 + 144*x^2 - 48*x + 16)
 
gp: K = bnfinit(x^18 - 3*x^17 + 9*x^16 + 26*x^15 - 51*x^14 + 84*x^13 + 257*x^12 - 234*x^11 + 312*x^10 + 848*x^9 - 156*x^8 + 468*x^7 + 716*x^6 + 336*x^5 + 264*x^4 - 208*x^3 + 144*x^2 - 48*x + 16, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 9 x^{16} + 26 x^{15} - 51 x^{14} + 84 x^{13} + 257 x^{12} - 234 x^{11} + 312 x^{10} + 848 x^{9} - 156 x^{8} + 468 x^{7} + 716 x^{6} + 336 x^{5} + 264 x^{4} - 208 x^{3} + 144 x^{2} - 48 x + 16 \)

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:  \(-305058304629265391270339328=-\,2^{8}\cdot 3^{33}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.60$
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:  $2, 3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{6} - \frac{1}{3}$, $\frac{1}{3} a^{7} - \frac{1}{3} a$, $\frac{1}{9} a^{8} - \frac{1}{9} a^{7} + \frac{1}{9} a^{6} + \frac{1}{3} a^{4} + \frac{2}{9} a^{2} + \frac{1}{9} a + \frac{2}{9}$, $\frac{1}{18} a^{9} - \frac{1}{18} a^{8} + \frac{1}{18} a^{7} - \frac{1}{6} a^{5} - \frac{1}{3} a^{4} + \frac{5}{18} a^{3} + \frac{2}{9} a^{2} - \frac{2}{9} a - \frac{1}{3}$, $\frac{1}{18} a^{10} + \frac{1}{18} a^{7} - \frac{1}{6} a^{6} - \frac{1}{6} a^{5} + \frac{5}{18} a^{4} - \frac{1}{6} a^{3} + \frac{1}{3} a^{2} - \frac{2}{9} a$, $\frac{1}{18} a^{11} - \frac{1}{18} a^{8} - \frac{1}{18} a^{7} + \frac{1}{18} a^{6} - \frac{1}{18} a^{5} + \frac{1}{6} a^{4} + \frac{2}{9} a^{2} - \frac{4}{9} a + \frac{1}{9}$, $\frac{1}{108} a^{12} + \frac{1}{108} a^{11} - \frac{1}{108} a^{10} - \frac{5}{108} a^{8} - \frac{4}{27} a^{7} - \frac{17}{108} a^{6} + \frac{7}{54} a^{5} + \frac{17}{54} a^{4} - \frac{1}{9} a^{3} - \frac{4}{27} a^{2} + \frac{1}{27} a - \frac{8}{27}$, $\frac{1}{108} a^{13} - \frac{1}{54} a^{11} + \frac{1}{108} a^{10} + \frac{1}{108} a^{9} - \frac{5}{108} a^{8} - \frac{7}{108} a^{7} + \frac{7}{108} a^{6} + \frac{1}{54} a^{5} - \frac{23}{54} a^{4} + \frac{13}{54} a^{3} - \frac{10}{27} a^{2} - \frac{4}{9} a - \frac{13}{27}$, $\frac{1}{216} a^{14} - \frac{1}{216} a^{13} - \frac{1}{216} a^{12} - \frac{1}{108} a^{11} - \frac{1}{216} a^{10} + \frac{5}{216} a^{8} - \frac{1}{108} a^{7} + \frac{5}{54} a^{6} - \frac{5}{108} a^{5} - \frac{23}{54} a^{4} - \frac{1}{18} a^{3} + \frac{1}{6} a^{2} + \frac{1}{3} a + \frac{13}{27}$, $\frac{1}{648} a^{15} - \frac{1}{648} a^{14} - \frac{1}{648} a^{13} + \frac{13}{648} a^{11} - \frac{1}{324} a^{10} + \frac{17}{648} a^{9} - \frac{1}{54} a^{8} + \frac{1}{18} a^{7} - \frac{10}{81} a^{6} - \frac{5}{81} a^{5} + \frac{41}{162} a^{4} - \frac{1}{9} a^{3} - \frac{22}{81} a^{2} + \frac{26}{81} a - \frac{17}{81}$, $\frac{1}{648} a^{16} + \frac{1}{648} a^{14} + \frac{1}{324} a^{13} - \frac{1}{324} a^{12} + \frac{17}{648} a^{11} - \frac{1}{108} a^{10} + \frac{11}{648} a^{9} + \frac{11}{216} a^{8} + \frac{7}{162} a^{7} - \frac{17}{108} a^{6} + \frac{5}{324} a^{5} - \frac{73}{162} a^{4} + \frac{31}{162} a^{3} + \frac{5}{162} a^{2} - \frac{8}{27} a + \frac{13}{81}$, $\frac{1}{85459536} a^{17} - \frac{169}{1400976} a^{16} - \frac{17443}{28486512} a^{15} + \frac{18601}{42729768} a^{14} - \frac{2033}{606096} a^{13} + \frac{1201}{464454} a^{12} + \frac{135019}{28486512} a^{11} + \frac{1697}{700488} a^{10} - \frac{487861}{21364884} a^{9} - \frac{1595905}{42729768} a^{8} + \frac{691627}{10682442} a^{7} + \frac{200750}{1780407} a^{6} + \frac{1610909}{21364884} a^{5} + \frac{214595}{1780407} a^{4} - \frac{1575895}{10682442} a^{3} + \frac{75347}{593469} a^{2} - \frac{1561328}{5341221} a - \frac{1656862}{5341221}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( -\frac{529477}{28486512} a^{17} + \frac{26701}{466992} a^{16} - \frac{4967081}{28486512} a^{15} - \frac{6579899}{14243256} a^{14} + \frac{578771}{606096} a^{13} - \frac{1045829}{619272} a^{12} - \frac{130177975}{28486512} a^{11} + \frac{258731}{58374} a^{10} - \frac{93494659}{14243256} a^{9} - \frac{108712801}{7121628} a^{8} + \frac{10139813}{3560814} a^{7} - \frac{69577673}{7121628} a^{6} - \frac{52033853}{3560814} a^{5} - \frac{12144761}{1780407} a^{4} - \frac{18180259}{3560814} a^{3} + \frac{4801165}{3560814} a^{2} - \frac{5028512}{1780407} a + \frac{1675852}{1780407} \) (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:  \( 2153285.052102543 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3^2:S_3$ (as 18T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 10 conjugacy class representatives for $C_3^2:S_3$
Character table for $C_3^2:S_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.243.1 x3, 6.0.177147.2, 9.3.3361317558192.1

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

Sibling fields

Degree 9 siblings: 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
$3$3.6.11.9$x^{6} + 3$$6$$1$$11$$S_3$$[5/2]_{2}$
3.6.11.9$x^{6} + 3$$6$$1$$11$$S_3$$[5/2]_{2}$
3.6.11.9$x^{6} + 3$$6$$1$$11$$S_3$$[5/2]_{2}$
$11$11.6.4.2$x^{6} - 11 x^{3} + 847$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
11.6.4.2$x^{6} - 11 x^{3} + 847$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
11.6.0.1$x^{6} + x^{2} - 2 x + 8$$1$$6$$0$$C_6$$[\ ]^{6}$