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Properties

Label 18.2.11756112812...6512.1
Degree $18$
Signature $[2, 8]$
Discriminant $2^{16}\cdot 3^{33}\cdot 19^{9}$
Root discriminant $60.49$
Ramified primes $2, 3, 19$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times D_9:C_3$ (as 18T45)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-786432, -1769472, -1179648, 884736, -626688, -622080, -357504, 165312, -155556, -90315, -40599, 9252, -360, -1404, -54, -18, 24, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 24*x^16 - 18*x^15 - 54*x^14 - 1404*x^13 - 360*x^12 + 9252*x^11 - 40599*x^10 - 90315*x^9 - 155556*x^8 + 165312*x^7 - 357504*x^6 - 622080*x^5 - 626688*x^4 + 884736*x^3 - 1179648*x^2 - 1769472*x - 786432)
 
gp: K = bnfinit(x^18 - 9*x^17 + 24*x^16 - 18*x^15 - 54*x^14 - 1404*x^13 - 360*x^12 + 9252*x^11 - 40599*x^10 - 90315*x^9 - 155556*x^8 + 165312*x^7 - 357504*x^6 - 622080*x^5 - 626688*x^4 + 884736*x^3 - 1179648*x^2 - 1769472*x - 786432, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 24 x^{16} - 18 x^{15} - 54 x^{14} - 1404 x^{13} - 360 x^{12} + 9252 x^{11} - 40599 x^{10} - 90315 x^{9} - 155556 x^{8} + 165312 x^{7} - 357504 x^{6} - 622080 x^{5} - 626688 x^{4} + 884736 x^{3} - 1179648 x^{2} - 1769472 x - 786432 \)

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:  \(117561128126763835119847566016512=2^{16}\cdot 3^{33}\cdot 19^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.49$
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:  $2, 3, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{2} + \frac{1}{4} a$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{10} - \frac{1}{8} a^{8} - \frac{3}{8} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{7}{16} a^{3} - \frac{3}{16} a^{2} + \frac{1}{4} a$, $\frac{1}{64} a^{12} - \frac{1}{64} a^{11} - \frac{1}{32} a^{9} - \frac{3}{32} a^{8} - \frac{3}{16} a^{7} + \frac{3}{8} a^{6} - \frac{7}{16} a^{5} + \frac{9}{64} a^{4} - \frac{3}{64} a^{3} - \frac{3}{16} a^{2} + \frac{1}{4} a$, $\frac{1}{256} a^{13} - \frac{1}{256} a^{12} - \frac{1}{128} a^{10} - \frac{35}{128} a^{9} + \frac{29}{64} a^{8} - \frac{13}{32} a^{7} - \frac{23}{64} a^{6} + \frac{9}{256} a^{5} + \frac{61}{256} a^{4} - \frac{19}{64} a^{3} - \frac{7}{16} a^{2} - \frac{1}{4} a$, $\frac{1}{1024} a^{14} - \frac{1}{1024} a^{13} - \frac{1}{512} a^{11} - \frac{35}{512} a^{10} + \frac{29}{256} a^{9} - \frac{45}{128} a^{8} + \frac{105}{256} a^{7} + \frac{265}{1024} a^{6} + \frac{317}{1024} a^{5} + \frac{109}{256} a^{4} - \frac{7}{64} a^{3} + \frac{3}{16} a^{2} - \frac{1}{4} a$, $\frac{1}{4096} a^{15} - \frac{1}{4096} a^{14} - \frac{1}{2048} a^{12} - \frac{35}{2048} a^{11} + \frac{29}{1024} a^{10} - \frac{45}{512} a^{9} + \frac{105}{1024} a^{8} + \frac{1289}{4096} a^{7} - \frac{707}{4096} a^{6} - \frac{403}{1024} a^{5} - \frac{7}{256} a^{4} - \frac{13}{64} a^{3} + \frac{7}{16} a^{2} + \frac{1}{4} a$, $\frac{1}{1163264} a^{16} - \frac{61}{1163264} a^{15} + \frac{91}{290816} a^{14} - \frac{345}{581632} a^{13} - \frac{2215}{581632} a^{12} - \frac{8609}{290816} a^{11} + \frac{3945}{145408} a^{10} + \frac{12209}{290816} a^{9} - \frac{460007}{1163264} a^{8} + \frac{471521}{1163264} a^{7} - \frac{13437}{145408} a^{6} + \frac{14213}{72704} a^{5} - \frac{2791}{18176} a^{4} + \frac{39}{568} a^{3} - \frac{103}{568} a^{2} + \frac{17}{71} a + \frac{6}{71}$, $\frac{1}{868961856437114977813474036306957606739443712} a^{17} + \frac{200922090457841699357290820472538886863}{868961856437114977813474036306957606739443712} a^{16} - \frac{1283438094856049016752486647121698808429}{27155058013659843056671063634592425210607616} a^{15} - \frac{10565179302785235099374753875168845684683}{39498266201687044446067001650316254851792896} a^{14} - \frac{201878901100229306239153825536836566304979}{434480928218557488906737018153478803369721856} a^{13} + \frac{555462283146942266103439239289382794486989}{217240464109278744453368509076739401684860928} a^{12} - \frac{3162211562118139281850990366809832510339897}{108620232054639372226684254538369700842430464} a^{11} - \frac{13823737256870197593407264338696413470426087}{217240464109278744453368509076739401684860928} a^{10} - \frac{146094863957224238297309559521605972482228791}{868961856437114977813474036306957606739443712} a^{9} - \frac{197021508783402975006298389478500666312606899}{868961856437114977813474036306957606739443712} a^{8} - \frac{7203869532333125779443637616496162010840233}{19749133100843522223033500825158127425896448} a^{7} + \frac{75060979528394849007213393290039275700963}{1697191125853740191041941477162026575662976} a^{6} + \frac{6574764183990465166854051596483286385236257}{13577529006829921528335531817296212605303808} a^{5} + \frac{854147381100225496314159231024117030960815}{3394382251707480382083882954324053151325952} a^{4} + \frac{296909359956680098238651772773978516445731}{848595562926870095520970738581013287831488} a^{3} + \frac{59367520921041851380520459844190293580019}{212148890731717523880242684645253321957872} a^{2} - \frac{10668268751853769414551791009438878970095}{53037222682929380970060671161313330489468} a - \frac{4595443987186763815611104534908238585226}{13259305670732345242515167790328332622367}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $9$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 2717691757.986842 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_9:C_3$ (as 18T45):

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 $C_2\times D_9:C_3$
Character table for $C_2\times D_9:C_3$

Intermediate fields

\(\Q(\sqrt{57}) \), 3.1.108.1, 6.2.240010128.2, 9.1.11019960576.1

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

Sibling fields

Degree 18 sibling: 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ $18$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ $18$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.9.8.1$x^{9} - 2$$9$$1$$8$$(C_9:C_3):C_2$$[\ ]_{9}^{6}$
2.9.8.1$x^{9} - 2$$9$$1$$8$$(C_9:C_3):C_2$$[\ ]_{9}^{6}$
3Data not computed
19Data not computed