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Properties

Label 18.12.1280976322...4448.1
Degree $18$
Signature $[12, 3]$
Discriminant $-\,2^{12}\cdot 3^{31}\cdot 7^{9}\cdot 12547$
Root discriminant $47.06$
Ramified primes $2, 3, 7, 12547$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 18T544

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1731, 2799, -9972, -19314, 5715, 47691, 29328, -49365, -23400, 26211, 2898, -6525, 1074, 495, -195, 36, -6, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 6*x^16 + 36*x^15 - 195*x^14 + 495*x^13 + 1074*x^12 - 6525*x^11 + 2898*x^10 + 26211*x^9 - 23400*x^8 - 49365*x^7 + 29328*x^6 + 47691*x^5 + 5715*x^4 - 19314*x^3 - 9972*x^2 + 2799*x + 1731)
 
gp: K = bnfinit(x^18 - 3*x^17 - 6*x^16 + 36*x^15 - 195*x^14 + 495*x^13 + 1074*x^12 - 6525*x^11 + 2898*x^10 + 26211*x^9 - 23400*x^8 - 49365*x^7 + 29328*x^6 + 47691*x^5 + 5715*x^4 - 19314*x^3 - 9972*x^2 + 2799*x + 1731, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 6 x^{16} + 36 x^{15} - 195 x^{14} + 495 x^{13} + 1074 x^{12} - 6525 x^{11} + 2898 x^{10} + 26211 x^{9} - 23400 x^{8} - 49365 x^{7} + 29328 x^{6} + 47691 x^{5} + 5715 x^{4} - 19314 x^{3} - 9972 x^{2} + 2799 x + 1731 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1280976322666113476586551144448=-\,2^{12}\cdot 3^{31}\cdot 7^{9}\cdot 12547\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.06$
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:  $2, 3, 7, 12547$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{18832000520947394721546187288079912040454} a^{17} + \frac{623356414023087062249963018334951959071}{9416000260473697360773093644039956020227} a^{16} - \frac{522583616139062182885583945247236871985}{18832000520947394721546187288079912040454} a^{15} - \frac{2419487632817130511007227487716834449586}{9416000260473697360773093644039956020227} a^{14} + \frac{3353327961090853476412901852080703990974}{9416000260473697360773093644039956020227} a^{13} + \frac{7405217804175325338248334649874323836137}{18832000520947394721546187288079912040454} a^{12} + \frac{2066875545465158051301880265555819937229}{9416000260473697360773093644039956020227} a^{11} + \frac{2057384644318210065947453045338317790515}{18832000520947394721546187288079912040454} a^{10} - \frac{2453003828428942540363344319530777840319}{9416000260473697360773093644039956020227} a^{9} - \frac{4793104807972137879790164894248090524061}{18832000520947394721546187288079912040454} a^{8} - \frac{3297688331245009874319656373948744258529}{9416000260473697360773093644039956020227} a^{7} + \frac{3983043530490690716771537703228506158721}{18832000520947394721546187288079912040454} a^{6} + \frac{2690236876247123807079125161951663937212}{9416000260473697360773093644039956020227} a^{5} - \frac{8799632376594804829108265741961800885129}{18832000520947394721546187288079912040454} a^{4} - \frac{7795185373609415837522202267011118653257}{18832000520947394721546187288079912040454} a^{3} - \frac{1540594895078031999615913646158083156975}{18832000520947394721546187288079912040454} a^{2} + \frac{8454328153095819622827815376831458845197}{18832000520947394721546187288079912040454} a - \frac{1014612457372898929162789797294488540049}{18832000520947394721546187288079912040454}$

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:  $14$
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:  \( 393020282.333 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T544:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 9216
The 96 conjugacy class representatives for t18n544 are not computed
Character table for t18n544 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 3.3.756.1, 9.9.314987206464.1

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

Sibling fields

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 ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.6.0.1}{6} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
3Data not computed
$7$7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
12547Data not computed