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Properties

Label 18.0.70657689165...8416.1
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 $28741824$ (GRH)
Class group $[2, 2, 12, 598788]$ (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![17172029521969, 1687026042884, 3528664372527, 201391197924, 323641640959, 7923669502, 17905672065, -27166024, 686399371, -12565692, 19896362, -483714, 451361, -11158, 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 - 11158*x^13 + 451361*x^12 - 483714*x^11 + 19896362*x^10 - 12565692*x^9 + 686399371*x^8 - 27166024*x^7 + 17905672065*x^6 + 7923669502*x^5 + 323641640959*x^4 + 201391197924*x^3 + 3528664372527*x^2 + 1687026042884*x + 17172029521969)
 
gp: K = bnfinit(x^18 - 2*x^17 + 98*x^16 - 230*x^15 + 7698*x^14 - 11158*x^13 + 451361*x^12 - 483714*x^11 + 19896362*x^10 - 12565692*x^9 + 686399371*x^8 - 27166024*x^7 + 17905672065*x^6 + 7923669502*x^5 + 323641640959*x^4 + 201391197924*x^3 + 3528664372527*x^2 + 1687026042884*x + 17172029521969, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} + 98 x^{16} - 230 x^{15} + 7698 x^{14} - 11158 x^{13} + 451361 x^{12} - 483714 x^{11} + 19896362 x^{10} - 12565692 x^{9} + 686399371 x^{8} - 27166024 x^{7} + 17905672065 x^{6} + 7923669502 x^{5} + 323641640959 x^{4} + 201391197924 x^{3} + 3528664372527 x^{2} + 1687026042884 x + 17172029521969 \)

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$$\chi_{1596}(1,·)$, $\chi_{1596}(709,·)$, $\chi_{1596}(1537,·)$, $\chi_{1596}(1487,·)$, $\chi_{1596}(529,·)$, $\chi_{1596}(83,·)$, $\chi_{1596}(215,·)$, $\chi_{1596}(923,·)$, $\chi_{1596}(541,·)$, $\chi_{1596}(289,·)$, $\chi_{1596}(419,·)$, $\chi_{1596}(613,·)$, $\chi_{1596}(815,·)$, $\chi_{1596}(1261,·)$, $\chi_{1596}(47,·)$, $\chi_{1596}(1391,·)$, $\chi_{1596}(505,·)$, $\chi_{1596}(1403,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{7} a^{8} + \frac{3}{7} a^{7} + \frac{1}{7} a^{6} - \frac{3}{7} a^{5} - \frac{2}{7} a^{3}$, $\frac{1}{7} a^{9} - \frac{1}{7} a^{7} + \frac{1}{7} a^{6} + \frac{2}{7} a^{5} - \frac{2}{7} a^{4} - \frac{1}{7} a^{3}$, $\frac{1}{7} a^{10} - \frac{3}{7} a^{7} + \frac{3}{7} a^{6} + \frac{2}{7} a^{5} - \frac{1}{7} a^{4} - \frac{2}{7} a^{3}$, $\frac{1}{7} a^{11} - \frac{2}{7} a^{7} - \frac{2}{7} a^{6} - \frac{3}{7} a^{5} - \frac{2}{7} a^{4} + \frac{1}{7} a^{3}$, $\frac{1}{7} a^{12} - \frac{3}{7} a^{7} - \frac{1}{7} a^{6} - \frac{1}{7} a^{5} + \frac{1}{7} a^{4} + \frac{3}{7} a^{3}$, $\frac{1}{7} a^{13} + \frac{1}{7} a^{7} + \frac{2}{7} a^{6} - \frac{1}{7} a^{5} + \frac{3}{7} a^{4} + \frac{1}{7} a^{3}$, $\frac{1}{7} a^{14} - \frac{1}{7} a^{7} - \frac{2}{7} a^{6} - \frac{1}{7} a^{5} + \frac{1}{7} a^{4} + \frac{2}{7} a^{3}$, $\frac{1}{7} a^{15} + \frac{1}{7} a^{7} - \frac{2}{7} a^{5} + \frac{2}{7} a^{4} - \frac{2}{7} a^{3}$, $\frac{1}{12074606411} a^{16} + \frac{357071511}{12074606411} a^{15} - \frac{251104843}{12074606411} a^{14} - \frac{29964478}{1724943773} a^{13} + \frac{664898581}{12074606411} a^{12} - \frac{414905347}{12074606411} a^{11} + \frac{301416084}{12074606411} a^{10} - \frac{631823875}{12074606411} a^{9} - \frac{154118680}{12074606411} a^{8} + \frac{746833334}{1724943773} a^{7} + \frac{3038062521}{12074606411} a^{6} + \frac{242573486}{1724943773} a^{5} - \frac{667900123}{1724943773} a^{4} + \frac{594603185}{1724943773} a^{3} + \frac{32091242}{246420539} a^{2} + \frac{101125187}{246420539} a - \frac{88519306}{246420539}$, $\frac{1}{72627703902077159563399119885331698585168603707395367348066204346595240784501} a^{17} + \frac{67154419148525078218171878617929615556054463867591911320526843209}{2342829158131521276238681286623603180166729151851463462840845301503072283371} a^{16} - \frac{1366682257672590060247187945044127048734910928457233800479988091234409668529}{72627703902077159563399119885331698585168603707395367348066204346595240784501} a^{15} + \frac{280846265294215623387577796709440682160469162590433049955181351918882803060}{72627703902077159563399119885331698585168603707395367348066204346595240784501} a^{14} - \frac{4677011448132582772478489907830925243512988059915312519671653445291752797061}{72627703902077159563399119885331698585168603707395367348066204346595240784501} a^{13} - \frac{686271160186774189467244688510880423036763342847029019480424064208683889899}{10375386271725308509057017126475956940738371958199338192580886335227891540643} a^{12} + \frac{4510424913110927218155645519098104646227174797428807067668451299067982874}{1482198038817901215579573875210850991534053136885619741797269476461127362949} a^{11} + \frac{4817454896547072657250317544903122645601094719650670364429761968321935577082}{72627703902077159563399119885331698585168603707395367348066204346595240784501} a^{10} + \frac{4426182329238513391355317002762479552036896355617219779714214824905273646636}{72627703902077159563399119885331698585168603707395367348066204346595240784501} a^{9} - \frac{4712274981413939913664597760125684415141286821402562808039257865284829934471}{72627703902077159563399119885331698585168603707395367348066204346595240784501} a^{8} - \frac{1299924811909612308205505847810127768577596193470408594436176168243435531859}{72627703902077159563399119885331698585168603707395367348066204346595240784501} a^{7} - \frac{30036174440475148502812958861791174897401017990571122450077430426751007296487}{72627703902077159563399119885331698585168603707395367348066204346595240784501} a^{6} + \frac{2362494744487996790590197767094110661787515026059680938205432022490465349497}{10375386271725308509057017126475956940738371958199338192580886335227891540643} a^{5} - \frac{3000099554317510972701382239157200276332454353278021446234019963894058777121}{10375386271725308509057017126475956940738371958199338192580886335227891540643} a^{4} - \frac{1715557121184903527629274672376185516636255492157139040866916330815546724012}{10375386271725308509057017126475956940738371958199338192580886335227891540643} a^{3} + \frac{4456666570187128855898660599009272587405724929465377183179750358259945361}{47812839961867781147728189522930677146259778609213540057976434724552495579} a^{2} + \frac{732167909350889955835220408963628031157708977845579998823053009903347581706}{1482198038817901215579573875210850991534053136885619741797269476461127362949} a - \frac{6839051178934258301283469446546444300544842273843770135838034820389678537}{14390272221533021510481299759328650403243234338695337299002616276321624883}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{12}\times C_{598788}$, which has order $28741824$ (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:  \( 4369063.348005476 \) (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, 9.9.1998099208210609.2

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 ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ $18$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ $18$ $18$ $18$ $18$

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$7.6.5.3$x^{6} - 112$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.3$x^{6} - 112$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.3$x^{6} - 112$$6$$1$$5$$C_6$$[\ ]_{6}$
$19$19.9.8.6$x^{9} + 1216$$9$$1$$8$$C_9$$[\ ]_{9}$
19.9.8.6$x^{9} + 1216$$9$$1$$8$$C_9$$[\ ]_{9}$