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Properties

Label 18.0.55361680899...1923.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{9}\cdot 109^{12}$
Root discriminant $39.52$
Ramified primes $3, 109$
Class number $1$ (GRH)
Class group Trivial (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![432, -3888, 15516, -34668, 45543, -33069, 10297, -1248, 4284, -5256, 3810, -1770, 646, -336, 216, -108, 39, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 39*x^16 - 108*x^15 + 216*x^14 - 336*x^13 + 646*x^12 - 1770*x^11 + 3810*x^10 - 5256*x^9 + 4284*x^8 - 1248*x^7 + 10297*x^6 - 33069*x^5 + 45543*x^4 - 34668*x^3 + 15516*x^2 - 3888*x + 432)
 
gp: K = bnfinit(x^18 - 9*x^17 + 39*x^16 - 108*x^15 + 216*x^14 - 336*x^13 + 646*x^12 - 1770*x^11 + 3810*x^10 - 5256*x^9 + 4284*x^8 - 1248*x^7 + 10297*x^6 - 33069*x^5 + 45543*x^4 - 34668*x^3 + 15516*x^2 - 3888*x + 432, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 39 x^{16} - 108 x^{15} + 216 x^{14} - 336 x^{13} + 646 x^{12} - 1770 x^{11} + 3810 x^{10} - 5256 x^{9} + 4284 x^{8} - 1248 x^{7} + 10297 x^{6} - 33069 x^{5} + 45543 x^{4} - 34668 x^{3} + 15516 x^{2} - 3888 x + 432 \)

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:  \(-55361680899832712162570071923=-\,3^{9}\cdot 109^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.52$
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:  $3, 109$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{6} + \frac{1}{6} a^{4} - \frac{1}{2} a^{3} - \frac{1}{3} a^{2} - \frac{1}{2} a$, $\frac{1}{6} a^{7} + \frac{1}{6} a^{5} - \frac{1}{3} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{12} a^{8} - \frac{1}{12} a^{2}$, $\frac{1}{12} a^{9} - \frac{1}{12} a^{3}$, $\frac{1}{24} a^{10} - \frac{1}{24} a^{9} - \frac{1}{24} a^{8} - \frac{1}{12} a^{7} + \frac{1}{6} a^{5} - \frac{1}{24} a^{4} - \frac{7}{24} a^{3} + \frac{1}{24} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{24} a^{11} - \frac{1}{24} a^{8} - \frac{1}{12} a^{7} + \frac{1}{8} a^{5} + \frac{1}{6} a^{3} - \frac{11}{24} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{72} a^{12} - \frac{1}{72} a^{10} - \frac{1}{24} a^{8} - \frac{1}{12} a^{7} - \frac{1}{72} a^{6} + \frac{1}{6} a^{5} + \frac{1}{72} a^{4} - \frac{1}{3} a^{3} + \frac{1}{24} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{72} a^{13} - \frac{1}{72} a^{11} - \frac{1}{24} a^{9} - \frac{1}{72} a^{7} + \frac{1}{72} a^{5} - \frac{11}{24} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{144} a^{14} - \frac{1}{144} a^{13} - \frac{1}{144} a^{12} + \frac{1}{144} a^{11} - \frac{1}{48} a^{10} - \frac{1}{48} a^{9} + \frac{5}{144} a^{8} + \frac{1}{144} a^{7} - \frac{11}{144} a^{6} + \frac{35}{144} a^{5} - \frac{1}{16} a^{4} - \frac{11}{48} a^{3} + \frac{3}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{144} a^{15} + \frac{1}{72} a^{11} + \frac{1}{72} a^{9} - \frac{1}{24} a^{8} - \frac{1}{12} a^{7} + \frac{11}{72} a^{5} - \frac{17}{48} a^{3} - \frac{11}{24} a^{2} - \frac{1}{4} a$, $\frac{1}{5806368} a^{16} - \frac{1}{725796} a^{15} + \frac{4921}{1451592} a^{14} + \frac{985}{241932} a^{13} + \frac{505}{120966} a^{12} + \frac{7685}{1451592} a^{11} + \frac{3111}{322576} a^{10} - \frac{59833}{1451592} a^{9} + \frac{14273}{362898} a^{8} - \frac{1805}{80644} a^{7} - \frac{1873}{120966} a^{6} - \frac{70049}{1451592} a^{5} - \frac{807679}{5806368} a^{4} + \frac{64091}{161288} a^{3} + \frac{122513}{483864} a^{2} - \frac{9073}{20161} a + \frac{15597}{40322}$, $\frac{1}{1527074784} a^{17} + \frac{41}{509024928} a^{16} - \frac{190753}{63628116} a^{15} + \frac{17716}{5302343} a^{14} - \frac{893093}{381768696} a^{13} - \frac{529081}{381768696} a^{12} - \frac{3566761}{254512464} a^{11} - \frac{11129005}{763537392} a^{10} + \frac{1112401}{63628116} a^{9} + \frac{3443195}{127256232} a^{8} - \frac{8653603}{381768696} a^{7} + \frac{28315789}{381768696} a^{6} + \frac{271514453}{1527074784} a^{5} + \frac{222948239}{1527074784} a^{4} - \frac{1775163}{10604686} a^{3} - \frac{141721}{483864} a^{2} + \frac{2579262}{5302343} a - \frac{2795433}{10604686}$

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:  \( -\frac{2576017}{190884348} a^{17} + \frac{43792289}{381768696} a^{16} - \frac{29863213}{63628116} a^{15} + \frac{467998805}{381768696} a^{14} - \frac{98153857}{42418744} a^{13} + \frac{650375479}{190884348} a^{12} - \frac{2698461893}{381768696} a^{11} + \frac{7799110231}{381768696} a^{10} - \frac{2632595045}{63628116} a^{9} + \frac{19374107713}{381768696} a^{8} - \frac{1416543739}{42418744} a^{7} + \frac{232617023}{190884348} a^{6} - \frac{53034517013}{381768696} a^{5} + \frac{11981669735}{31814058} a^{4} - \frac{2281197834}{5302343} a^{3} + \frac{21070857}{80644} a^{2} - \frac{464285620}{5302343} a + \frac{72369919}{5302343} \) (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:  \( 151363193.213 \) (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

\(\Q(\sqrt{-3}) \), 3.1.35643.1 x3, 3.3.11881.1, 6.0.3811270347.1, 6.0.320787.3 x2, 6.0.3811270347.2, 9.3.45281702992707.2 x3

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

Sibling fields

Degree 6 sibling: 6.0.320787.3
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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$109$109.3.2.1$x^{3} - 109$$3$$1$$2$$C_3$$[\ ]_{3}$
109.3.2.1$x^{3} - 109$$3$$1$$2$$C_3$$[\ ]_{3}$
109.3.2.1$x^{3} - 109$$3$$1$$2$$C_3$$[\ ]_{3}$
109.3.2.1$x^{3} - 109$$3$$1$$2$$C_3$$[\ ]_{3}$
109.3.2.1$x^{3} - 109$$3$$1$$2$$C_3$$[\ ]_{3}$
109.3.2.1$x^{3} - 109$$3$$1$$2$$C_3$$[\ ]_{3}$