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Properties

Label 18.18.6604314254...3581.1
Degree $18$
Signature $[18, 0]$
Discriminant $3^{9}\cdot 7^{15}\cdot 643^{6}$
Root discriminant $75.66$
Ramified primes $3, 7, 643$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_2\times He_3:C_2$ (as 18T42)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1456771, 5873617, -11477171, -9805981, 21633906, 1512276, -14914626, 3604790, 3991836, -1645149, -426316, 274551, 8209, -20462, 1390, 649, -75, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^17 - 75*x^16 + 649*x^15 + 1390*x^14 - 20462*x^13 + 8209*x^12 + 274551*x^11 - 426316*x^10 - 1645149*x^9 + 3991836*x^8 + 3604790*x^7 - 14914626*x^6 + 1512276*x^5 + 21633906*x^4 - 9805981*x^3 - 11477171*x^2 + 5873617*x + 1456771)
 
gp: K = bnfinit(x^18 - 7*x^17 - 75*x^16 + 649*x^15 + 1390*x^14 - 20462*x^13 + 8209*x^12 + 274551*x^11 - 426316*x^10 - 1645149*x^9 + 3991836*x^8 + 3604790*x^7 - 14914626*x^6 + 1512276*x^5 + 21633906*x^4 - 9805981*x^3 - 11477171*x^2 + 5873617*x + 1456771, 1)
 

Normalized defining polynomial

\( x^{18} - 7 x^{17} - 75 x^{16} + 649 x^{15} + 1390 x^{14} - 20462 x^{13} + 8209 x^{12} + 274551 x^{11} - 426316 x^{10} - 1645149 x^{9} + 3991836 x^{8} + 3604790 x^{7} - 14914626 x^{6} + 1512276 x^{5} + 21633906 x^{4} - 9805981 x^{3} - 11477171 x^{2} + 5873617 x + 1456771 \)

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:  \(6604314254900677539286564516483581=3^{9}\cdot 7^{15}\cdot 643^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $75.66$
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:  $3, 7, 643$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{11} - \frac{1}{3} a^{8} - \frac{1}{3} a^{5} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{65087467979008427621824334271677526837750691833021} a^{17} + \frac{2164102321306850653524005296734497232464666449840}{21695822659669475873941444757225842279250230611007} a^{16} - \frac{10513645301767611600247131494187297983506287919253}{65087467979008427621824334271677526837750691833021} a^{15} + \frac{2362055565533329988138839778600296284446559662734}{21695822659669475873941444757225842279250230611007} a^{14} - \frac{3489899952728652170869025944074745148311959878581}{21695822659669475873941444757225842279250230611007} a^{13} - \frac{4427774659412439962827685645750185252788478952984}{65087467979008427621824334271677526837750691833021} a^{12} + \frac{13256443148603357491169657592507753805412288113402}{65087467979008427621824334271677526837750691833021} a^{11} - \frac{11891116127616103460289084977739682509145579092377}{65087467979008427621824334271677526837750691833021} a^{10} - \frac{12797040516055772755215466789572502488158101053800}{65087467979008427621824334271677526837750691833021} a^{9} + \frac{13874951371497926485193436935698443517385332875295}{65087467979008427621824334271677526837750691833021} a^{8} - \frac{1720790661473021860276481344237250847350250659363}{21695822659669475873941444757225842279250230611007} a^{7} - \frac{14939729323382338547206495570148734998539150941805}{65087467979008427621824334271677526837750691833021} a^{6} - \frac{7564727828137679723146322117143810096234435248080}{21695822659669475873941444757225842279250230611007} a^{5} - \frac{3599174466914938829188782619681330260960408388123}{65087467979008427621824334271677526837750691833021} a^{4} - \frac{17589359566591446997071707863383350301479209414738}{65087467979008427621824334271677526837750691833021} a^{3} - \frac{17199758224539178119220982226213559736360877800027}{65087467979008427621824334271677526837750691833021} a^{2} - \frac{30420607987538017130855095376713528601533449617237}{65087467979008427621824334271677526837750691833021} a + \frac{22772006561685273626255934718376848576349354789183}{65087467979008427621824334271677526837750691833021}$

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:  $17$
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:  \( 41970999915.0 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times He_3:C_2$ (as 18T42):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 20 conjugacy class representatives for $C_2\times He_3:C_2$
Character table for $C_2\times He_3:C_2$

Intermediate fields

\(\Q(\sqrt{21}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{21})^+\), 9.9.844471355782761.3

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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$3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7Data not computed
643Data not computed