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Properties

Label 18.0.56239497739...4368.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{33}\cdot 89^{12}$
Root discriminant $237.14$
Ramified primes $2, 3, 89$
Class number $36$ (GRH)
Class group $[6, 6]$ (GRH)
Galois group $C_3\times C_3:S_3$ (as 18T23)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![47870076, 9854892, -23111064, 11847282, 2088558, -1645560, 1661599, -1223703, 747021, -272778, 73689, -23931, 4408, -531, 177, -90, 39, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 39*x^16 - 90*x^15 + 177*x^14 - 531*x^13 + 4408*x^12 - 23931*x^11 + 73689*x^10 - 272778*x^9 + 747021*x^8 - 1223703*x^7 + 1661599*x^6 - 1645560*x^5 + 2088558*x^4 + 11847282*x^3 - 23111064*x^2 + 9854892*x + 47870076)
 
gp: K = bnfinit(x^18 - 9*x^17 + 39*x^16 - 90*x^15 + 177*x^14 - 531*x^13 + 4408*x^12 - 23931*x^11 + 73689*x^10 - 272778*x^9 + 747021*x^8 - 1223703*x^7 + 1661599*x^6 - 1645560*x^5 + 2088558*x^4 + 11847282*x^3 - 23111064*x^2 + 9854892*x + 47870076, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 39 x^{16} - 90 x^{15} + 177 x^{14} - 531 x^{13} + 4408 x^{12} - 23931 x^{11} + 73689 x^{10} - 272778 x^{9} + 747021 x^{8} - 1223703 x^{7} + 1661599 x^{6} - 1645560 x^{5} + 2088558 x^{4} + 11847282 x^{3} - 23111064 x^{2} + 9854892 x + 47870076 \)

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:  \(-5623949773935752894766481334356864599994368=-\,2^{12}\cdot 3^{33}\cdot 89^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $237.14$
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:  $2, 3, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5}$, $\frac{1}{13286} a^{15} + \frac{183}{6643} a^{14} - \frac{1294}{6643} a^{13} + \frac{89}{949} a^{12} + \frac{166}{511} a^{11} - \frac{250}{6643} a^{10} - \frac{1770}{6643} a^{9} + \frac{122}{949} a^{8} - \frac{1451}{6643} a^{7} - \frac{2658}{6643} a^{6} + \frac{2633}{6643} a^{5} + \frac{355}{6643} a^{4} + \frac{4595}{13286} a^{3} + \frac{2389}{6643} a^{2} + \frac{2573}{6643} a + \frac{2322}{6643}$, $\frac{1}{788685350582086594} a^{16} + \frac{13689117006925}{788685350582086594} a^{15} + \frac{9062320661794501}{60668103890929738} a^{14} - \frac{119821290468061061}{788685350582086594} a^{13} - \frac{88092495070901536}{394342675291043297} a^{12} + \frac{218565984259176457}{788685350582086594} a^{11} + \frac{270720439856946449}{788685350582086594} a^{10} - \frac{181371900136993012}{394342675291043297} a^{9} + \frac{65843501624437855}{788685350582086594} a^{8} + \frac{6786864029904997}{112669335797440942} a^{7} + \frac{15454152007580212}{394342675291043297} a^{6} + \frac{282136179783470521}{788685350582086594} a^{5} - \frac{156966532567573334}{394342675291043297} a^{4} + \frac{10068353005853539}{112669335797440942} a^{3} + \frac{140571737297170612}{394342675291043297} a^{2} + \frac{15594399558312017}{394342675291043297} a + \frac{74359962569451823}{394342675291043297}$, $\frac{1}{29961497018436380822182084079306608798562} a^{17} + \frac{10340963972209633000163}{29961497018436380822182084079306608798562} a^{16} + \frac{330088041609676959454298613952285758}{14980748509218190411091042039653304399281} a^{15} - \frac{4255178300334258623126226012253526537641}{29961497018436380822182084079306608798562} a^{14} - \frac{1484569183617022445142310566098191809089}{14980748509218190411091042039653304399281} a^{13} + \frac{3682975646349070237028988080720141038239}{29961497018436380822182084079306608798562} a^{12} + \frac{9496959866278208882661537014370935161799}{29961497018436380822182084079306608798562} a^{11} + \frac{3624243378125360023509781644035160422282}{14980748509218190411091042039653304399281} a^{10} - \frac{249895217719105632749608181603885522709}{2304730539879721601706314159946662215274} a^{9} - \frac{2870261580974897822775891363566010527593}{29961497018436380822182084079306608798562} a^{8} - \frac{3281038574788051298457572307573691272}{8102081400334337702050320194512333369} a^{7} - \frac{775112889316640672342508300122586473121}{4280213859776625831740297725615229828366} a^{6} + \frac{5735891247421557958995724578613535614466}{14980748509218190411091042039653304399281} a^{5} + \frac{67312269138439804576636931832032142467}{4280213859776625831740297725615229828366} a^{4} - \frac{1915387418315973426525736364228056325927}{29961497018436380822182084079306608798562} a^{3} - \frac{1068895222792240477475717882214978197573}{2140106929888312915870148862807614914183} a^{2} - \frac{5118216834883057827025920827877744259419}{14980748509218190411091042039653304399281} a + \frac{3815502814315132126008593721170704458784}{14980748509218190411091042039653304399281}$

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

Class group and class number

$C_{6}\times C_{6}$, which has order $36$ (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:  \( -\frac{402635200717080130065}{23257411928326387709138518363} a^{17} + \frac{9865926329952181670265}{46514823856652775418277036726} a^{16} - \frac{26659815560594206224063}{23257411928326387709138518363} a^{15} + \frac{164168103982423017431271}{46514823856652775418277036726} a^{14} - \frac{377039273019426674052477}{46514823856652775418277036726} a^{13} + \frac{1056019487057474890761081}{46514823856652775418277036726} a^{12} - \frac{5512979377047464860318713}{46514823856652775418277036726} a^{11} + \frac{31200796843421342289590931}{46514823856652775418277036726} a^{10} - \frac{118460973920898489841570745}{46514823856652775418277036726} a^{9} + \frac{396132869139410064995535849}{46514823856652775418277036726} a^{8} - \frac{1383729126093556884976577163}{46514823856652775418277036726} a^{7} + \frac{3137668414025033469159473265}{46514823856652775418277036726} a^{6} - \frac{5279321924139662239016826777}{46514823856652775418277036726} a^{5} + \frac{5204100648297506369799583332}{23257411928326387709138518363} a^{4} - \frac{12958491733631693219123936797}{46514823856652775418277036726} a^{3} + \frac{3202552556181132489227757141}{23257411928326387709138518363} a^{2} + \frac{17925364288553274640489080348}{23257411928326387709138518363} a - \frac{4689168988362552552859955677}{23257411928326387709138518363} \) (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:  \( 40553438788528.734 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_3:S_3$ (as 18T23):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 18 conjugacy class representatives for $C_3\times C_3:S_3$
Character table for $C_3\times C_3:S_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.23763.1 x3, Deg 6, Deg 6, 6.0.2834352.3, 6.0.1694040507.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{9}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
$3$3.6.11.18$x^{6} + 21 x^{3} + 12$$6$$1$$11$$S_3\times C_3$$[2, 5/2]_{2}$
3.6.11.18$x^{6} + 21 x^{3} + 12$$6$$1$$11$$S_3\times C_3$$[2, 5/2]_{2}$
3.6.11.18$x^{6} + 21 x^{3} + 12$$6$$1$$11$$S_3\times C_3$$[2, 5/2]_{2}$
$89$89.6.4.1$x^{6} + 1513 x^{3} + 1710936$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
89.6.4.1$x^{6} + 1513 x^{3} + 1710936$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
89.6.4.1$x^{6} + 1513 x^{3} + 1710936$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$