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Properties

Label 18.2.21552575879...7104.1
Degree $18$
Signature $[2, 8]$
Discriminant $2^{18}\cdot 3^{20}\cdot 11^{9}$
Root discriminant $22.48$
Ramified primes $2, 3, 11$
Class number $1$
Class group Trivial
Galois group $S_3\wr C_2$ (as 18T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-43, 216, -720, 660, 354, -762, 9, 180, 321, -104, -303, 78, 135, -24, -54, 18, 9, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 9*x^16 + 18*x^15 - 54*x^14 - 24*x^13 + 135*x^12 + 78*x^11 - 303*x^10 - 104*x^9 + 321*x^8 + 180*x^7 + 9*x^6 - 762*x^5 + 354*x^4 + 660*x^3 - 720*x^2 + 216*x - 43)
 
gp: K = bnfinit(x^18 - 6*x^17 + 9*x^16 + 18*x^15 - 54*x^14 - 24*x^13 + 135*x^12 + 78*x^11 - 303*x^10 - 104*x^9 + 321*x^8 + 180*x^7 + 9*x^6 - 762*x^5 + 354*x^4 + 660*x^3 - 720*x^2 + 216*x - 43, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 9 x^{16} + 18 x^{15} - 54 x^{14} - 24 x^{13} + 135 x^{12} + 78 x^{11} - 303 x^{10} - 104 x^{9} + 321 x^{8} + 180 x^{7} + 9 x^{6} - 762 x^{5} + 354 x^{4} + 660 x^{3} - 720 x^{2} + 216 x - 43 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2155257587919164038447104=2^{18}\cdot 3^{20}\cdot 11^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.48$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a$, $\frac{1}{106} a^{16} - \frac{7}{53} a^{15} + \frac{1}{53} a^{14} + \frac{25}{106} a^{13} - \frac{15}{106} a^{12} - \frac{17}{106} a^{11} - \frac{11}{106} a^{10} - \frac{37}{106} a^{9} - \frac{23}{106} a^{8} + \frac{31}{106} a^{7} + \frac{1}{106} a^{6} - \frac{19}{106} a^{5} - \frac{11}{106} a^{4} - \frac{3}{106} a^{3} + \frac{22}{53} a^{2} + \frac{29}{106} a + \frac{13}{106}$, $\frac{1}{134342981078232434} a^{17} + \frac{282384057955819}{134342981078232434} a^{16} - \frac{13229920967874723}{134342981078232434} a^{15} - \frac{3298616753658724}{67171490539116217} a^{14} - \frac{14658689307800105}{134342981078232434} a^{13} - \frac{14487074313885968}{67171490539116217} a^{12} + \frac{50110758135124877}{134342981078232434} a^{11} + \frac{30343987668791816}{67171490539116217} a^{10} - \frac{57487532276775025}{134342981078232434} a^{9} - \frac{24041659055727229}{67171490539116217} a^{8} - \frac{2455352639638403}{134342981078232434} a^{7} + \frac{15102720933686504}{67171490539116217} a^{6} - \frac{56466286759362755}{134342981078232434} a^{5} - \frac{25746367362803726}{67171490539116217} a^{4} + \frac{22032351356572546}{67171490539116217} a^{3} + \frac{567997506724337}{2534773227891178} a^{2} - \frac{26543475044623819}{67171490539116217} a - \frac{20476862579301969}{67171490539116217}$

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

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 171597.947651 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\wr C_2$ (as 18T36):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 9 conjugacy class representatives for $S_3\wr C_2$
Character table for $S_3\wr C_2$

Intermediate fields

\(\Q(\sqrt{11}) \), 9.1.5030030016.1

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

Sibling fields

Degree 6 siblings: 6.2.3881196.1, 6.2.513216.1
Degree 9 sibling: 9.1.5030030016.1
Degree 12 siblings: Deg 12, Deg 12, Deg 12, Deg 12, Deg 12, Deg 12
Degree 18 siblings: Deg 18, Deg 18

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
3Data not computed
$11$11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$