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Properties

Label 18.18.2877002654...5744.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{33}\cdot 3^{24}\cdot 17^{9}$
Root discriminant $63.57$
Ramified primes $2, 3, 17$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
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![-100, 2568, -9012, -5076, 54420, -48144, -51087, 77838, 10971, -47130, 4938, 14238, -2829, -2286, 516, 186, -39, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 - 39*x^16 + 186*x^15 + 516*x^14 - 2286*x^13 - 2829*x^12 + 14238*x^11 + 4938*x^10 - 47130*x^9 + 10971*x^8 + 77838*x^7 - 51087*x^6 - 48144*x^5 + 54420*x^4 - 5076*x^3 - 9012*x^2 + 2568*x - 100)
 
gp: K = bnfinit(x^18 - 6*x^17 - 39*x^16 + 186*x^15 + 516*x^14 - 2286*x^13 - 2829*x^12 + 14238*x^11 + 4938*x^10 - 47130*x^9 + 10971*x^8 + 77838*x^7 - 51087*x^6 - 48144*x^5 + 54420*x^4 - 5076*x^3 - 9012*x^2 + 2568*x - 100, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} - 39 x^{16} + 186 x^{15} + 516 x^{14} - 2286 x^{13} - 2829 x^{12} + 14238 x^{11} + 4938 x^{10} - 47130 x^{9} + 10971 x^{8} + 77838 x^{7} - 51087 x^{6} - 48144 x^{5} + 54420 x^{4} - 5076 x^{3} - 9012 x^{2} + 2568 x - 100 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(287700265443621620596836604575744=2^{33}\cdot 3^{24}\cdot 17^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $63.57$
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:  $2, 3, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{5} a^{14} - \frac{2}{5} a^{13} + \frac{2}{5} a^{12} - \frac{1}{5} a^{11} - \frac{1}{5} a^{9} + \frac{1}{5} a^{8} + \frac{2}{5} a^{7} + \frac{2}{5} a^{6} - \frac{2}{5} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{2} + \frac{2}{5} a$, $\frac{1}{10} a^{15} + \frac{3}{10} a^{13} - \frac{1}{5} a^{12} - \frac{1}{5} a^{11} + \frac{2}{5} a^{10} - \frac{1}{10} a^{9} + \frac{2}{5} a^{8} - \frac{2}{5} a^{7} + \frac{1}{5} a^{6} - \frac{1}{10} a^{5} - \frac{2}{5} a^{4} + \frac{1}{10} a^{3} + \frac{2}{5} a^{2} + \frac{2}{5} a$, $\frac{1}{10} a^{16} - \frac{1}{10} a^{14} - \frac{2}{5} a^{13} - \frac{1}{5} a^{11} - \frac{1}{10} a^{10} - \frac{1}{5} a^{9} + \frac{1}{5} a^{8} + \frac{2}{5} a^{7} + \frac{1}{10} a^{6} + \frac{2}{5} a^{5} - \frac{1}{10} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a$, $\frac{1}{101972559057102021588980} a^{17} + \frac{474712814216595795983}{25493139764275505397245} a^{16} + \frac{2384532871065996051847}{101972559057102021588980} a^{15} - \frac{507699098694212616149}{50986279528551010794490} a^{14} - \frac{711120030076755202461}{10197255905710202158898} a^{13} - \frac{2828199720069018401998}{25493139764275505397245} a^{12} - \frac{1998270225071552082289}{101972559057102021588980} a^{11} - \frac{7353877100597123875738}{25493139764275505397245} a^{10} - \frac{8364736528841231650782}{25493139764275505397245} a^{9} + \frac{12062594028760148385827}{25493139764275505397245} a^{8} + \frac{14803810208682777812083}{101972559057102021588980} a^{7} - \frac{6356435249955572538898}{25493139764275505397245} a^{6} + \frac{25807069484350851465443}{101972559057102021588980} a^{5} - \frac{7416018056476645099444}{25493139764275505397245} a^{4} - \frac{1424328743585817190957}{50986279528551010794490} a^{3} - \frac{268121649160284889053}{614292524440373624030} a^{2} - \frac{2572328934017770089352}{25493139764275505397245} a - \frac{1032317548442566962625}{5098627952855101079449}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $17$
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:  \( 105925015490 \) (assuming GRH)
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

\(\Q(\sqrt{34}) \), 9.9.21389063233536.1

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

Sibling fields

Degree 6 siblings: 6.6.114610464.1, 6.6.25380864.1
Degree 9 sibling: 9.9.21389063233536.1
Degree 12 siblings: Deg 12, Deg 12, 12.12.13450811861294383104.1, 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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.6.11.1$x^{6} + 14$$6$$1$$11$$D_{6}$$[3]_{3}^{2}$
2.12.22.60$x^{12} - 84 x^{10} + 444 x^{8} + 32 x^{6} - 272 x^{4} - 320 x^{2} + 64$$6$$2$$22$$D_6$$[3]_{3}^{2}$
$3$3.9.12.21$x^{9} + 3 x^{4} + 6$$9$$1$$12$$C_3^2:C_4$$[3/2, 3/2]_{2}^{2}$
3.9.12.21$x^{9} + 3 x^{4} + 6$$9$$1$$12$$C_3^2:C_4$$[3/2, 3/2]_{2}^{2}$
$17$17.6.3.1$x^{6} - 34 x^{4} + 289 x^{2} - 44217$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
17.6.3.1$x^{6} - 34 x^{4} + 289 x^{2} - 44217$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
17.6.3.1$x^{6} - 34 x^{4} + 289 x^{2} - 44217$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$