Properties

Label 18.18.133...125.1
Degree $18$
Signature $[18, 0]$
Discriminant $1.331\times 10^{25}$
Root discriminant \(24.88\)
Ramified primes $3,5,17$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 24*x^16 - 3*x^15 + 216*x^14 + 42*x^13 - 943*x^12 - 180*x^11 + 2190*x^10 + 272*x^9 - 2772*x^8 - 54*x^7 + 1819*x^6 - 180*x^5 - 522*x^4 + 110*x^3 + 36*x^2 - 6*x - 1)
 
gp: K = bnfinit(y^18 - 24*y^16 - 3*y^15 + 216*y^14 + 42*y^13 - 943*y^12 - 180*y^11 + 2190*y^10 + 272*y^9 - 2772*y^8 - 54*y^7 + 1819*y^6 - 180*y^5 - 522*y^4 + 110*y^3 + 36*y^2 - 6*y - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 24*x^16 - 3*x^15 + 216*x^14 + 42*x^13 - 943*x^12 - 180*x^11 + 2190*x^10 + 272*x^9 - 2772*x^8 - 54*x^7 + 1819*x^6 - 180*x^5 - 522*x^4 + 110*x^3 + 36*x^2 - 6*x - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 24*x^16 - 3*x^15 + 216*x^14 + 42*x^13 - 943*x^12 - 180*x^11 + 2190*x^10 + 272*x^9 - 2772*x^8 - 54*x^7 + 1819*x^6 - 180*x^5 - 522*x^4 + 110*x^3 + 36*x^2 - 6*x - 1)
 

\( x^{18} - 24 x^{16} - 3 x^{15} + 216 x^{14} + 42 x^{13} - 943 x^{12} - 180 x^{11} + 2190 x^{10} + 272 x^{9} - 2772 x^{8} - 54 x^{7} + 1819 x^{6} - 180 x^{5} - 522 x^{4} + 110 x^{3} + \cdots - 1 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $18$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[18, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(13314770360250302126953125\) \(\medspace = 3^{24}\cdot 5^{9}\cdot 17^{6}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(24.88\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $3^{4/3}5^{1/2}17^{1/2}\approx 39.890652095882146$
Ramified primes:   \(3\), \(5\), \(17\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{5}) \)
$\card{ \Aut(K/\Q) }$:  $6$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{19}a^{15}+\frac{5}{19}a^{14}+\frac{1}{19}a^{13}+\frac{7}{19}a^{12}-\frac{9}{19}a^{11}+\frac{2}{19}a^{10}-\frac{5}{19}a^{9}-\frac{3}{19}a^{8}+\frac{6}{19}a^{5}-\frac{5}{19}a^{4}+\frac{8}{19}a^{3}+\frac{4}{19}a^{2}+\frac{5}{19}a+\frac{4}{19}$, $\frac{1}{19}a^{16}-\frac{5}{19}a^{14}+\frac{2}{19}a^{13}-\frac{6}{19}a^{12}+\frac{9}{19}a^{11}+\frac{4}{19}a^{10}+\frac{3}{19}a^{9}-\frac{4}{19}a^{8}+\frac{6}{19}a^{6}+\frac{3}{19}a^{5}-\frac{5}{19}a^{4}+\frac{2}{19}a^{3}+\frac{4}{19}a^{2}-\frac{2}{19}a-\frac{1}{19}$, $\frac{1}{4993181}a^{17}+\frac{60376}{4993181}a^{16}-\frac{23577}{4993181}a^{15}-\frac{20807}{45809}a^{14}-\frac{428917}{4993181}a^{13}+\frac{971906}{4993181}a^{12}-\frac{1381394}{4993181}a^{11}+\frac{2262703}{4993181}a^{10}-\frac{11097}{262799}a^{9}+\frac{1178060}{4993181}a^{8}+\frac{261237}{4993181}a^{7}-\frac{1013721}{4993181}a^{6}+\frac{22033}{262799}a^{5}-\frac{2298243}{4993181}a^{4}+\frac{2305698}{4993181}a^{3}-\frac{1326521}{4993181}a^{2}+\frac{2168174}{4993181}a+\frac{761136}{4993181}$ Copy content Toggle raw display

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $17$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{598824}{262799}a^{17}+\frac{288198}{262799}a^{16}-\frac{14051118}{262799}a^{15}-\frac{78372}{2411}a^{14}+\frac{121170984}{262799}a^{13}+\frac{82447653}{262799}a^{12}-\frac{492479280}{262799}a^{11}-\frac{333001578}{262799}a^{10}+\frac{1033840199}{262799}a^{9}+\frac{614222676}{262799}a^{8}-\frac{1162421424}{262799}a^{7}-\frac{524286813}{262799}a^{6}+\frac{673745184}{262799}a^{5}+\frac{184566600}{262799}a^{4}-\frac{171360343}{262799}a^{3}-\frac{17405100}{262799}a^{2}+\frac{9833424}{262799}a+\frac{1321213}{262799}$, $\frac{11279978}{4993181}a^{17}+\frac{7906384}{4993181}a^{16}-\frac{266136276}{4993181}a^{15}-\frac{2021957}{45809}a^{14}+\frac{2305740430}{4993181}a^{13}+\frac{2093093894}{4993181}a^{12}-\frac{9392562412}{4993181}a^{11}-\frac{8660927344}{4993181}a^{10}+\frac{1034252688}{262799}a^{9}+\frac{17084355297}{4993181}a^{8}-\frac{21748423524}{4993181}a^{7}-\frac{863516500}{262799}a^{6}+\frac{12142672099}{4993181}a^{5}+\frac{7048125497}{4993181}a^{4}-\frac{2901184900}{4993181}a^{3}-\frac{990399906}{4993181}a^{2}+\frac{197591169}{4993181}a+\frac{43356668}{4993181}$, $\frac{2707318}{4993181}a^{17}-\frac{1581241}{4993181}a^{16}-\frac{69617508}{4993181}a^{15}+\frac{241237}{45809}a^{14}+\frac{692722516}{4993181}a^{13}-\frac{131485354}{4993181}a^{12}-\frac{3428195172}{4993181}a^{11}+\frac{130655702}{4993181}a^{10}+\frac{472586436}{262799}a^{9}+\frac{561581150}{4993181}a^{8}-\frac{12215817924}{4993181}a^{7}-\frac{64342444}{262799}a^{6}+\frac{7915227303}{4993181}a^{5}+\frac{567066195}{4993181}a^{4}-\frac{1905409012}{4993181}a^{3}+\frac{110356071}{4993181}a^{2}+\frac{55930940}{4993181}a+\frac{4742745}{4993181}$, $\frac{6677866}{4993181}a^{17}+\frac{5809781}{4993181}a^{16}-\frac{149647217}{4993181}a^{15}-\frac{1392262}{45809}a^{14}+\frac{1185478152}{4993181}a^{13}+\frac{1324988216}{4993181}a^{12}-\frac{4127881072}{4993181}a^{11}-\frac{4772957522}{4993181}a^{10}+\frac{6662825173}{4993181}a^{9}+\frac{7338259984}{4993181}a^{8}-\frac{5079976253}{4993181}a^{7}-\frac{4417598580}{4993181}a^{6}+\frac{1867585216}{4993181}a^{5}+\frac{508309419}{4993181}a^{4}-\frac{426416465}{4993181}a^{3}+\frac{139153262}{4993181}a^{2}+\frac{14852712}{4993181}a-\frac{9956505}{4993181}$, $\frac{26025251}{4993181}a^{17}+\frac{11930627}{4993181}a^{16}-\frac{619989322}{4993181}a^{15}-\frac{175462}{2411}a^{14}+\frac{5476021511}{4993181}a^{13}+\frac{3629000632}{4993181}a^{12}-\frac{1214300092}{262799}a^{11}-\frac{15468976716}{4993181}a^{10}+\frac{50739200739}{4993181}a^{9}+\frac{31054522566}{4993181}a^{8}-\frac{59755089769}{4993181}a^{7}-\frac{29863903950}{4993181}a^{6}+\frac{35771039317}{4993181}a^{5}+\frac{12318758601}{4993181}a^{4}-\frac{9073139270}{4993181}a^{3}-\frac{1366938487}{4993181}a^{2}+\frac{528764928}{4993181}a+\frac{79778901}{4993181}$, $\frac{6628371}{4993181}a^{17}+\frac{5512}{4993181}a^{16}-\frac{163196710}{4993181}a^{15}-\frac{222235}{45809}a^{14}+\frac{1524306959}{4993181}a^{13}+\frac{386876869}{4993181}a^{12}-\frac{6985351600}{4993181}a^{11}-\frac{2124897536}{4993181}a^{10}+\frac{17063809990}{4993181}a^{9}+\frac{5163521086}{4993181}a^{8}-\frac{22378934943}{4993181}a^{7}-\frac{5648801678}{4993181}a^{6}+\frac{14773825097}{4993181}a^{5}+\frac{2265424864}{4993181}a^{4}-\frac{4021563545}{4993181}a^{3}+\frac{3054909}{4993181}a^{2}+\frac{215567869}{4993181}a+\frac{16133725}{4993181}$, $\frac{8392602}{4993181}a^{17}+\frac{7935256}{4993181}a^{16}-\frac{194181157}{4993181}a^{15}-\frac{1926214}{45809}a^{14}+\frac{1623673734}{4993181}a^{13}+\frac{1913931176}{4993181}a^{12}-\frac{6224866928}{4993181}a^{11}-\frac{7611839811}{4993181}a^{10}+\frac{11902634658}{4993181}a^{9}+\frac{14375054486}{4993181}a^{8}-\frac{11866467853}{4993181}a^{7}-\frac{700115108}{262799}a^{6}+\frac{6131080762}{4993181}a^{5}+\frac{5740296067}{4993181}a^{4}-\frac{1576846721}{4993181}a^{3}-\frac{955888238}{4993181}a^{2}+\frac{179016060}{4993181}a+\frac{28628977}{4993181}$, $\frac{97678}{4993181}a^{17}-\frac{2430622}{4993181}a^{16}-\frac{834966}{4993181}a^{15}+\frac{532889}{45809}a^{14}-\frac{3491734}{4993181}a^{13}-\frac{526588487}{4993181}a^{12}+\frac{35456092}{4993181}a^{11}+\frac{2333897362}{4993181}a^{10}-\frac{412489}{262799}a^{9}-\frac{5414124453}{4993181}a^{8}-\frac{337583532}{4993181}a^{7}+\frac{339229687}{262799}a^{6}+\frac{658486397}{4993181}a^{5}-\frac{3541360097}{4993181}a^{4}-\frac{354661617}{4993181}a^{3}+\frac{659703006}{4993181}a^{2}-\frac{10756113}{4993181}a-\frac{13260440}{4993181}$, $\frac{1758953}{4993181}a^{17}-\frac{2471557}{4993181}a^{16}-\frac{43726119}{4993181}a^{15}+\frac{461806}{45809}a^{14}+\frac{420843292}{4993181}a^{13}-\frac{370763459}{4993181}a^{12}-\frac{2015296290}{4993181}a^{11}+\frac{66052347}{262799}a^{10}+\frac{5072093317}{4993181}a^{9}-\frac{2071980804}{4993181}a^{8}-\frac{6490005145}{4993181}a^{7}+\frac{1633763084}{4993181}a^{6}+\frac{3804568187}{4993181}a^{5}-\frac{444662771}{4993181}a^{4}-\frac{818499110}{4993181}a^{3}-\frac{2052472}{262799}a^{2}+\frac{69841869}{4993181}a+\frac{11576561}{4993181}$, $\frac{178156}{262799}a^{17}+\frac{4155717}{4993181}a^{16}-\frac{76970417}{4993181}a^{15}-\frac{994773}{45809}a^{14}+\frac{620943414}{4993181}a^{13}+\frac{990947380}{4993181}a^{12}-\frac{2204376376}{4993181}a^{11}-\frac{4030143986}{4993181}a^{10}+\frac{3536905741}{4993181}a^{9}+\frac{7965125450}{4993181}a^{8}-\frac{116132689}{262799}a^{7}-\frac{7872644249}{4993181}a^{6}-\frac{128620987}{4993181}a^{5}+\frac{3612696844}{4993181}a^{4}+\frac{510252111}{4993181}a^{3}-\frac{572444626}{4993181}a^{2}-\frac{59020627}{4993181}a+\frac{13826010}{4993181}$, $\frac{574391}{262799}a^{17}-\frac{2801411}{4993181}a^{16}-\frac{272725593}{4993181}a^{15}+\frac{185343}{45809}a^{14}+\frac{2609131757}{4993181}a^{13}+\frac{207714271}{4993181}a^{12}-\frac{12336346352}{4993181}a^{11}-\frac{2343745321}{4993181}a^{10}+\frac{31028519066}{4993181}a^{9}+\frac{7801128537}{4993181}a^{8}-\frac{2174401882}{262799}a^{7}-\frac{10642237480}{4993181}a^{6}+\frac{27187891530}{4993181}a^{5}+\frac{5595557994}{4993181}a^{4}-\frac{384176333}{262799}a^{3}-\frac{622975459}{4993181}a^{2}+\frac{471698056}{4993181}a+\frac{51326550}{4993181}$, $\frac{6392283}{4993181}a^{17}+\frac{5955762}{4993181}a^{16}-\frac{151469598}{4993181}a^{15}-\frac{1439080}{45809}a^{14}+\frac{1316878966}{4993181}a^{13}+\frac{75272150}{262799}a^{12}-\frac{5392385601}{4993181}a^{11}-\frac{5712872068}{4993181}a^{10}+\frac{11485856412}{4993181}a^{9}+\frac{10798591242}{4993181}a^{8}-\frac{13440290097}{4993181}a^{7}-\frac{9692233758}{4993181}a^{6}+\frac{8540343963}{4993181}a^{5}+\frac{3655269608}{4993181}a^{4}-\frac{2552489105}{4993181}a^{3}-\frac{383561279}{4993181}a^{2}+\frac{187214465}{4993181}a+\frac{22751158}{4993181}$, $\frac{320330}{4993181}a^{17}+\frac{2442464}{4993181}a^{16}-\frac{5095929}{4993181}a^{15}-\frac{499560}{45809}a^{14}+\frac{1590570}{4993181}a^{13}+\frac{418933658}{4993181}a^{12}+\frac{296767447}{4993181}a^{11}-\frac{1343244453}{4993181}a^{10}-\frac{1545253310}{4993181}a^{9}+\frac{1739888336}{4993181}a^{8}+\frac{2902365992}{4993181}a^{7}-\frac{628118381}{4993181}a^{6}-\frac{2261832612}{4993181}a^{5}-\frac{213389544}{4993181}a^{4}+\frac{32614865}{262799}a^{3}+\frac{73808071}{4993181}a^{2}-\frac{38695610}{4993181}a-\frac{12582511}{4993181}$, $\frac{13348555}{4993181}a^{17}+\frac{3246993}{4993181}a^{16}-\frac{319823183}{4993181}a^{15}-\frac{1119202}{45809}a^{14}+\frac{2858883870}{4993181}a^{13}+\frac{1348700279}{4993181}a^{12}-\frac{12282721557}{4993181}a^{11}-\frac{6125078486}{4993181}a^{10}+\frac{27647956616}{4993181}a^{9}+\frac{12940302582}{4993181}a^{8}-\frac{33121053818}{4993181}a^{7}-\frac{13020911531}{4993181}a^{6}+\frac{19856144477}{4993181}a^{5}+\frac{5614723717}{4993181}a^{4}-\frac{4926810142}{4993181}a^{3}-\frac{649877700}{4993181}a^{2}+\frac{286066838}{4993181}a+\frac{39346506}{4993181}$, $\frac{2792210}{4993181}a^{17}-\frac{4664334}{4993181}a^{16}-\frac{71478601}{4993181}a^{15}+\frac{879128}{45809}a^{14}+\frac{717656304}{4993181}a^{13}-\frac{701249479}{4993181}a^{12}-\frac{3618437728}{4993181}a^{11}+\frac{2288047890}{4993181}a^{10}+\frac{9606536632}{4993181}a^{9}-\frac{3513407275}{4993181}a^{8}-\frac{12947600948}{4993181}a^{7}+\frac{2684672102}{4993181}a^{6}+\frac{7949428486}{4993181}a^{5}-\frac{1213601387}{4993181}a^{4}-\frac{1580335223}{4993181}a^{3}+\frac{22499054}{262799}a^{2}-\frac{24073162}{4993181}a-\frac{7641626}{4993181}$, $\frac{869621}{4993181}a^{17}+\frac{413683}{4993181}a^{16}-\frac{17083870}{4993181}a^{15}-\frac{6913}{2411}a^{14}+\frac{5428307}{262799}a^{13}+\frac{149696764}{4993181}a^{12}-\frac{7361025}{262799}a^{11}-\frac{637088964}{4993181}a^{10}-\frac{569437631}{4993181}a^{9}+\frac{1268320720}{4993181}a^{8}+\frac{1924799905}{4993181}a^{7}-\frac{1328317563}{4993181}a^{6}-\frac{2031125437}{4993181}a^{5}+\frac{742091820}{4993181}a^{4}+\frac{786120389}{4993181}a^{3}-\frac{181973803}{4993181}a^{2}-\frac{59544674}{4993181}a+\frac{96910}{4993181}$, $\frac{11288610}{4993181}a^{17}+\frac{210506}{262799}a^{16}-\frac{271239695}{4993181}a^{15}-\frac{1210440}{45809}a^{14}+\frac{2430207924}{4993181}a^{13}+\frac{1385536302}{4993181}a^{12}-\frac{10456849544}{4993181}a^{11}-\frac{6176783379}{4993181}a^{10}+\frac{23562187687}{4993181}a^{9}+\frac{13049871664}{4993181}a^{8}-\frac{28251465214}{4993181}a^{7}-\frac{13328144584}{4993181}a^{6}+\frac{16924401918}{4993181}a^{5}+\frac{5934646269}{4993181}a^{4}-\frac{4171589596}{4993181}a^{3}-\frac{750315663}{4993181}a^{2}+\frac{253751740}{4993181}a+\frac{47975203}{4993181}$ Copy content Toggle raw display (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 3646318.70374 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{18}\cdot(2\pi)^{0}\cdot 3646318.70374 \cdot 1}{2\cdot\sqrt{13314770360250302126953125}}\cr\approx \mathstrut & 0.130977805954 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 - 24*x^16 - 3*x^15 + 216*x^14 + 42*x^13 - 943*x^12 - 180*x^11 + 2190*x^10 + 272*x^9 - 2772*x^8 - 54*x^7 + 1819*x^6 - 180*x^5 - 522*x^4 + 110*x^3 + 36*x^2 - 6*x - 1)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^18 - 24*x^16 - 3*x^15 + 216*x^14 + 42*x^13 - 943*x^12 - 180*x^11 + 2190*x^10 + 272*x^9 - 2772*x^8 - 54*x^7 + 1819*x^6 - 180*x^5 - 522*x^4 + 110*x^3 + 36*x^2 - 6*x - 1, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^18 - 24*x^16 - 3*x^15 + 216*x^14 + 42*x^13 - 943*x^12 - 180*x^11 + 2190*x^10 + 272*x^9 - 2772*x^8 - 54*x^7 + 1819*x^6 - 180*x^5 - 522*x^4 + 110*x^3 + 36*x^2 - 6*x - 1);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 24*x^16 - 3*x^15 + 216*x^14 + 42*x^13 - 943*x^12 - 180*x^11 + 2190*x^10 + 272*x^9 - 2772*x^8 - 54*x^7 + 1819*x^6 - 180*x^5 - 522*x^4 + 110*x^3 + 36*x^2 - 6*x - 1);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$C_6\times S_3$ (as 18T6):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{9})^+\), 3.3.6885.1, 6.6.820125.1, 6.6.237016125.1, 9.9.326371204125.1

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

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Sibling fields

Galois closure: data not computed
Degree 12 sibling: 12.12.2474477972015625.1
Degree 18 sibling: deg 18
Minimal sibling: 12.12.2474477972015625.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.6.0.1}{6} }^{3}$ R R ${\href{/padicField/7.6.0.1}{6} }^{3}$ ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ ${\href{/padicField/13.6.0.1}{6} }^{3}$ R ${\href{/padicField/19.3.0.1}{3} }^{6}$ ${\href{/padicField/23.6.0.1}{6} }^{3}$ ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ ${\href{/padicField/37.6.0.1}{6} }^{3}$ ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}$ ${\href{/padicField/43.6.0.1}{6} }^{3}$ ${\href{/padicField/47.6.0.1}{6} }^{3}$ ${\href{/padicField/53.2.0.1}{2} }^{9}$ ${\href{/padicField/59.3.0.1}{3} }^{6}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(3\) Copy content Toggle raw display Deg $18$$3$$6$$24$
\(5\) Copy content Toggle raw display 5.6.3.1$x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.12.6.1$x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
\(17\) Copy content Toggle raw display 17.2.0.1$x^{2} + 16 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} + 16 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} + 16 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.4.2.1$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$