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Properties

Label 18.0.70657689165...8416.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{9}\cdot 7^{15}\cdot 19^{16}$
Root discriminant $240.16$
Ramified primes $2, 3, 7, 19$
Class number $67995072$ (GRH)
Class group $[2, 2, 228, 74556]$ (GRH)
Galois group $C_{18}$ (as 18T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![17934199170349, 106026030264, 3384649707670, 51432868270, 301709383024, 4958427054, 17189420781, 163570074, 691813801, -1036720, 20316110, -294056, 450829, -12222, 7698, -230, 98, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 98*x^16 - 230*x^15 + 7698*x^14 - 12222*x^13 + 450829*x^12 - 294056*x^11 + 20316110*x^10 - 1036720*x^9 + 691813801*x^8 + 163570074*x^7 + 17189420781*x^6 + 4958427054*x^5 + 301709383024*x^4 + 51432868270*x^3 + 3384649707670*x^2 + 106026030264*x + 17934199170349)
 
gp: K = bnfinit(x^18 - 2*x^17 + 98*x^16 - 230*x^15 + 7698*x^14 - 12222*x^13 + 450829*x^12 - 294056*x^11 + 20316110*x^10 - 1036720*x^9 + 691813801*x^8 + 163570074*x^7 + 17189420781*x^6 + 4958427054*x^5 + 301709383024*x^4 + 51432868270*x^3 + 3384649707670*x^2 + 106026030264*x + 17934199170349, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} + 98 x^{16} - 230 x^{15} + 7698 x^{14} - 12222 x^{13} + 450829 x^{12} - 294056 x^{11} + 20316110 x^{10} - 1036720 x^{9} + 691813801 x^{8} + 163570074 x^{7} + 17189420781 x^{6} + 4958427054 x^{5} + 301709383024 x^{4} + 51432868270 x^{3} + 3384649707670 x^{2} + 106026030264 x + 17934199170349 \)

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:  \(-7065768916593110856047783889584316216508416=-\,2^{18}\cdot 3^{9}\cdot 7^{15}\cdot 19^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $240.16$
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:  $2, 3, 7, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1596=2^{2}\cdot 3\cdot 7\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{1596}(1,·)$, $\chi_{1596}(899,·)$, $\chi_{1596}(923,·)$, $\chi_{1596}(1261,·)$, $\chi_{1596}(1297,·)$, $\chi_{1596}(83,·)$, $\chi_{1596}(719,·)$, $\chi_{1596}(25,·)$, $\chi_{1596}(731,·)$, $\chi_{1596}(803,·)$, $\chi_{1596}(479,·)$, $\chi_{1596}(419,·)$, $\chi_{1596}(131,·)$, $\chi_{1596}(625,·)$, $\chi_{1596}(1453,·)$, $\chi_{1596}(1201,·)$, $\chi_{1596}(505,·)$, $\chi_{1596}(1213,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{7} a^{9} - \frac{1}{7} a^{8} + \frac{3}{7} a^{7} - \frac{2}{7} a^{5} - \frac{2}{7} a^{4} + \frac{1}{7} a^{3}$, $\frac{1}{77} a^{10} - \frac{3}{77} a^{9} - \frac{2}{77} a^{8} - \frac{27}{77} a^{7} + \frac{26}{77} a^{6} - \frac{12}{77} a^{5} - \frac{30}{77} a^{4} - \frac{9}{77} a^{3} - \frac{4}{11} a^{2} + \frac{1}{11} a$, $\frac{1}{77} a^{11} + \frac{3}{7} a^{8} - \frac{2}{7} a^{7} - \frac{1}{7} a^{6} - \frac{1}{7} a^{5} + \frac{3}{7} a^{4} + \frac{3}{7} a^{3} + \frac{3}{11} a$, $\frac{1}{77} a^{12} + \frac{1}{7} a^{8} - \frac{3}{7} a^{7} - \frac{1}{7} a^{6} + \frac{2}{7} a^{5} + \frac{2}{7} a^{4} - \frac{3}{7} a^{3} + \frac{3}{11} a^{2}$, $\frac{1}{77} a^{13} - \frac{2}{7} a^{8} + \frac{3}{7} a^{7} + \frac{2}{7} a^{6} - \frac{3}{7} a^{5} - \frac{1}{7} a^{4} + \frac{10}{77} a^{3}$, $\frac{1}{77} a^{14} + \frac{1}{7} a^{8} + \frac{1}{7} a^{7} - \frac{3}{7} a^{6} + \frac{2}{7} a^{5} - \frac{34}{77} a^{4} + \frac{2}{7} a^{3}$, $\frac{1}{77} a^{15} + \frac{2}{7} a^{8} + \frac{1}{7} a^{7} + \frac{2}{7} a^{6} - \frac{12}{77} a^{5} - \frac{3}{7} a^{4} - \frac{1}{7} a^{3}$, $\frac{1}{145211747527} a^{16} - \frac{719298430}{145211747527} a^{15} - \frac{49280791}{20744535361} a^{14} + \frac{560965711}{145211747527} a^{13} + \frac{369482632}{145211747527} a^{12} + \frac{355272858}{145211747527} a^{11} - \frac{551907275}{145211747527} a^{10} - \frac{10336064685}{145211747527} a^{9} + \frac{29134572261}{145211747527} a^{8} - \frac{58615526671}{145211747527} a^{7} - \frac{6165988797}{20744535361} a^{6} + \frac{3767822654}{13201067957} a^{5} - \frac{5146997145}{20744535361} a^{4} - \frac{4542520419}{20744535361} a^{3} - \frac{6482095366}{20744535361} a^{2} - \frac{462637952}{1885866851} a + \frac{2916214}{171442441}$, $\frac{1}{3489928427847705588425503704554336164555977539208844551870399932844917407} a^{17} + \frac{430988561989477030860902634300258202655594181136372457087608}{3489928427847705588425503704554336164555977539208844551870399932844917407} a^{16} + \frac{207042086894115353410358953983856651138776179485118294040650593657581}{498561203978243655489357672079190880650853934172692078838628561834988201} a^{15} - \frac{1866054238621418024683945528139290111622899691875437740098557567132114}{498561203978243655489357672079190880650853934172692078838628561834988201} a^{14} + \frac{8209893885671262792763744799036470947077039541246279384749947881895784}{3489928427847705588425503704554336164555977539208844551870399932844917407} a^{13} + \frac{17216184892920625652794088400949101213730055101033346432881749055707682}{3489928427847705588425503704554336164555977539208844551870399932844917407} a^{12} - \frac{5520450973823286636886573554438586096286436779464832886642937032384407}{3489928427847705588425503704554336164555977539208844551870399932844917407} a^{11} + \frac{16977441567943329161118921791707442019916392307907393332950121821539864}{3489928427847705588425503704554336164555977539208844551870399932844917407} a^{10} - \frac{43542117863701968995830890048206405504275295417471381189051639220702256}{3489928427847705588425503704554336164555977539208844551870399932844917407} a^{9} - \frac{515143476781461325292712815900864652828199387390149998933332420587025644}{3489928427847705588425503704554336164555977539208844551870399932844917407} a^{8} + \frac{185009486333222710835156583388130395831581526857839894556943503568732723}{3489928427847705588425503704554336164555977539208844551870399932844917407} a^{7} - \frac{1553150538626058004304181363783717933705971696313178406288898250118983384}{3489928427847705588425503704554336164555977539208844551870399932844917407} a^{6} + \frac{63322771361696851243212236919117559289011535185644544058295485300349570}{3489928427847705588425503704554336164555977539208844551870399932844917407} a^{5} + \frac{277995333880223933194301743474658137232000445835196974491849984165220002}{3489928427847705588425503704554336164555977539208844551870399932844917407} a^{4} + \frac{1716848719745079141216491316029647421164214161479353600333243415805371039}{3489928427847705588425503704554336164555977539208844551870399932844917407} a^{3} - \frac{182105870414243330687000017631616428116588945296567389845438575735631222}{498561203978243655489357672079190880650853934172692078838628561834988201} a^{2} - \frac{3673827615033103169296322540918419527642793337392405741103344130405478}{45323745816203968680850697461744625513713994015699279894420778348635291} a - \frac{231127967825760639239511837473066227262186747711760781162690234117875}{4120340528745815334622790678340420501246726728699934535856434395330481}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{228}\times C_{74556}$, which has order $67995072$ (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:  \( -1 \) (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:  \( 7595459.562747852 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{18}$ (as 18T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 18
The 18 conjugacy class representatives for $C_{18}$
Character table for $C_{18}$

Intermediate fields

\(\Q(\sqrt{-21}) \), 3.3.361.1, 6.0.77241777984.6, 9.9.1998099208210609.1

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

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 ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/11.1.0.1}{1} }^{18}$ $18$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ $18$ $18$ $18$ $18$

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
2Data not computed
3Data not computed
$7$7.6.5.1$x^{6} - 28$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.1$x^{6} - 28$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.1$x^{6} - 28$$6$$1$$5$$C_6$$[\ ]_{6}$
$19$19.9.8.3$x^{9} - 77824$$9$$1$$8$$C_9$$[\ ]_{9}$
19.9.8.3$x^{9} - 77824$$9$$1$$8$$C_9$$[\ ]_{9}$