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Properties

Label 27.27.1197336751...1009.2
Degree $27$
Signature $[27, 0]$
Discriminant $3^{36}\cdot 7^{18}\cdot 19^{24}$
Root discriminant $216.89$
Ramified primes $3, 7, 19$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_3\times C_9$ (as 27T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-77502107663, -495953919873, 1384579835685, 6984224959239, 2873413735140, -18015765483522, -30692199651546, -17452227111315, 1612960677906, 6315551962953, 2029302136221, -610037882562, -470794277397, -16231036935, 47517477258, 7915049615, -2617308423, -727641144, 79547261, 35785290, -1072779, -1071027, -5319, 19761, 341, -210, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 3*x^26 - 210*x^25 + 341*x^24 + 19761*x^23 - 5319*x^22 - 1071027*x^21 - 1072779*x^20 + 35785290*x^19 + 79547261*x^18 - 727641144*x^17 - 2617308423*x^16 + 7915049615*x^15 + 47517477258*x^14 - 16231036935*x^13 - 470794277397*x^12 - 610037882562*x^11 + 2029302136221*x^10 + 6315551962953*x^9 + 1612960677906*x^8 - 17452227111315*x^7 - 30692199651546*x^6 - 18015765483522*x^5 + 2873413735140*x^4 + 6984224959239*x^3 + 1384579835685*x^2 - 495953919873*x - 77502107663)
 
gp: K = bnfinit(x^27 - 3*x^26 - 210*x^25 + 341*x^24 + 19761*x^23 - 5319*x^22 - 1071027*x^21 - 1072779*x^20 + 35785290*x^19 + 79547261*x^18 - 727641144*x^17 - 2617308423*x^16 + 7915049615*x^15 + 47517477258*x^14 - 16231036935*x^13 - 470794277397*x^12 - 610037882562*x^11 + 2029302136221*x^10 + 6315551962953*x^9 + 1612960677906*x^8 - 17452227111315*x^7 - 30692199651546*x^6 - 18015765483522*x^5 + 2873413735140*x^4 + 6984224959239*x^3 + 1384579835685*x^2 - 495953919873*x - 77502107663, 1)
 

Normalized defining polynomial

\( x^{27} - 3 x^{26} - 210 x^{25} + 341 x^{24} + 19761 x^{23} - 5319 x^{22} - 1071027 x^{21} - 1072779 x^{20} + 35785290 x^{19} + 79547261 x^{18} - 727641144 x^{17} - 2617308423 x^{16} + 7915049615 x^{15} + 47517477258 x^{14} - 16231036935 x^{13} - 470794277397 x^{12} - 610037882562 x^{11} + 2029302136221 x^{10} + 6315551962953 x^{9} + 1612960677906 x^{8} - 17452227111315 x^{7} - 30692199651546 x^{6} - 18015765483522 x^{5} + 2873413735140 x^{4} + 6984224959239 x^{3} + 1384579835685 x^{2} - 495953919873 x - 77502107663 \)

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

Invariants

Degree:  $27$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[27, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1197336751300349460100016353165918520894496565611371829528951009=3^{36}\cdot 7^{18}\cdot 19^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $216.89$
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:  $3, 7, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1197=3^{2}\cdot 7\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{1197}(64,·)$, $\chi_{1197}(1,·)$, $\chi_{1197}(1033,·)$, $\chi_{1197}(970,·)$, $\chi_{1197}(655,·)$, $\chi_{1197}(529,·)$, $\chi_{1197}(403,·)$, $\chi_{1197}(340,·)$, $\chi_{1197}(277,·)$, $\chi_{1197}(214,·)$, $\chi_{1197}(121,·)$, $\chi_{1197}(25,·)$, $\chi_{1197}(883,·)$, $\chi_{1197}(442,·)$, $\chi_{1197}(940,·)$, $\chi_{1197}(814,·)$, $\chi_{1197}(688,·)$, $\chi_{1197}(625,·)$, $\chi_{1197}(562,·)$, $\chi_{1197}(499,·)$, $\chi_{1197}(757,·)$, $\chi_{1197}(310,·)$, $\chi_{1197}(631,·)$, $\chi_{1197}(568,·)$, $\chi_{1197}(505,·)$, $\chi_{1197}(58,·)$, $\chi_{1197}(253,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{17} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{17} - \frac{1}{2} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{20} - \frac{1}{2} a^{17} - \frac{1}{2} a^{16} - \frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{21} - \frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{22} - \frac{1}{2} a^{15} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{23} - \frac{1}{2} a^{16} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{24} - \frac{1}{2} a^{17} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{22} a^{25} - \frac{5}{22} a^{24} - \frac{3}{22} a^{23} - \frac{3}{22} a^{21} + \frac{1}{11} a^{19} + \frac{3}{22} a^{18} + \frac{5}{22} a^{17} - \frac{1}{2} a^{16} + \frac{7}{22} a^{15} - \frac{7}{22} a^{14} - \frac{2}{11} a^{13} + \frac{1}{11} a^{12} + \frac{4}{11} a^{11} + \frac{5}{22} a^{9} + \frac{9}{22} a^{8} - \frac{5}{11} a^{7} + \frac{3}{11} a^{6} - \frac{9}{22} a^{5} + \frac{4}{11} a^{4} + \frac{3}{11} a^{2} - \frac{1}{22} a + \frac{9}{22}$, $\frac{1}{546928725792031623763625816007427681949795042570377204556420038308465665043527573565459988093230320726346282016266143807021070726353849408426} a^{26} - \frac{943690202068695147532960928523106054650356653288136947803305656846652052821612272425816498326793332030341154724219034362956257760304153129}{546928725792031623763625816007427681949795042570377204556420038308465665043527573565459988093230320726346282016266143807021070726353849408426} a^{25} - \frac{89942792260507129923494982550111284429002493990548198848851028889888495843377123509763735621744812015135794973643543476983251196936354346551}{546928725792031623763625816007427681949795042570377204556420038308465665043527573565459988093230320726346282016266143807021070726353849408426} a^{24} - \frac{4047216328864968143921206930850326100427034209007594242257364519669617380532844221146459219422079587250898205568444227396026618666422337883}{273464362896015811881812908003713840974897521285188602278210019154232832521763786782729994046615160363173141008133071903510535363176924704213} a^{23} + \frac{95040945925840187971326875442392885378493420508182011979792136403384794247109878333578547370652729863394454838309857401249774551028363259697}{546928725792031623763625816007427681949795042570377204556420038308465665043527573565459988093230320726346282016266143807021070726353849408426} a^{22} - \frac{18016065171931277797900368040610463076492158161230628481421500275599037597942745602417413339150206570123517431875128772218147026307903122973}{546928725792031623763625816007427681949795042570377204556420038308465665043527573565459988093230320726346282016266143807021070726353849408426} a^{21} + \frac{120485275348255198006504630246240736419180712116051565191984762064711629962886509919076012721731689211424111397849463499451404184579639229413}{546928725792031623763625816007427681949795042570377204556420038308465665043527573565459988093230320726346282016266143807021070726353849408426} a^{20} + \frac{43289413762031489699560966641055511332461014143270338939284130035658734111747648813906557529092459275554542733083271599345037518867742637732}{273464362896015811881812908003713840974897521285188602278210019154232832521763786782729994046615160363173141008133071903510535363176924704213} a^{19} + \frac{10933305757842776870895665162654122169146420796033932702763353759854365704371310432212175943910408480602928063391051942907429514949853189600}{273464362896015811881812908003713840974897521285188602278210019154232832521763786782729994046615160363173141008133071903510535363176924704213} a^{18} + \frac{56545314930912937505774655676362735633310507702432227373491280437406132666018487797204391089247121290586943654118938480981469347364204411192}{273464362896015811881812908003713840974897521285188602278210019154232832521763786782729994046615160363173141008133071903510535363176924704213} a^{17} + \frac{132308375730134121303934820354828219348000356366329450872859376058503746640832445986901591751539411494606704606475851011070721170893732446199}{273464362896015811881812908003713840974897521285188602278210019154232832521763786782729994046615160363173141008133071903510535363176924704213} a^{16} + \frac{2176317130153256311389889278839564295380050448415398406471486946397424091563349900460327090848315087825985064639369558248845817034865112271}{24860396626910528352892082545792167361354320116835327479837274468566621138342162434793635822419560033015740091648461082137321396652447700383} a^{15} + \frac{9917762544790249228346266100106399175669609524273371693690599315012956831723966984821834023701388694884931752597441265989155281729416318943}{24860396626910528352892082545792167361354320116835327479837274468566621138342162434793635822419560033015740091648461082137321396652447700383} a^{14} + \frac{133571689050025381170408649790038396752814884036765591476035844658506042979506835495649833109247509743748076797287361109552763375625336621150}{273464362896015811881812908003713840974897521285188602278210019154232832521763786782729994046615160363173141008133071903510535363176924704213} a^{13} + \frac{51744551883360224489805274392885892004394145058350923152624402445018326976238282760912084981125384975429566367120136593353329110541592899333}{273464362896015811881812908003713840974897521285188602278210019154232832521763786782729994046615160363173141008133071903510535363176924704213} a^{12} - \frac{50776805007897476581791615903913098373651796235673446240477562228778703001402356763969556814443327187275094215473648881181761723063774966775}{546928725792031623763625816007427681949795042570377204556420038308465665043527573565459988093230320726346282016266143807021070726353849408426} a^{11} - \frac{51642553694073217151702609102483524555079283886564552339304882864296026156900955881983464431907538834993464723629243398787797040969185826319}{546928725792031623763625816007427681949795042570377204556420038308465665043527573565459988093230320726346282016266143807021070726353849408426} a^{10} + \frac{112913918936457173442443776986940144765105787485431285752633569042137064712672836955216513789420985566050201153866822046172592382810210515869}{546928725792031623763625816007427681949795042570377204556420038308465665043527573565459988093230320726346282016266143807021070726353849408426} a^{9} - \frac{85250091022321127166919519208777999106044192653522552538288958952180504445375691548687580338289346372381988360679693718507665061795533993549}{546928725792031623763625816007427681949795042570377204556420038308465665043527573565459988093230320726346282016266143807021070726353849408426} a^{8} - \frac{110325991695542546104066407611408243155060856032038171367420389080435195939196659081317264667482636984538894058553857233162902369555233108135}{546928725792031623763625816007427681949795042570377204556420038308465665043527573565459988093230320726346282016266143807021070726353849408426} a^{7} + \frac{11463261701041482569781369662837220484609971687491263676495464153448124354471018210468790159305599014506177602587392962017072621815465497677}{273464362896015811881812908003713840974897521285188602278210019154232832521763786782729994046615160363173141008133071903510535363176924704213} a^{6} - \frac{106151835608137324709771489464223641999684283212134712987404560422349405453456632979312034818221555102876417804364286796288169690246658081830}{273464362896015811881812908003713840974897521285188602278210019154232832521763786782729994046615160363173141008133071903510535363176924704213} a^{5} - \frac{154447213486110836678137939636603878549135323150054548044823217396294670432994088402794113159987222363981926234470394411190273730693278555341}{546928725792031623763625816007427681949795042570377204556420038308465665043527573565459988093230320726346282016266143807021070726353849408426} a^{4} + \frac{174484172151624970103651539827340440121277985947264535370724001869718488135408363710084367198695478732864120820933561514507257717817638378103}{546928725792031623763625816007427681949795042570377204556420038308465665043527573565459988093230320726346282016266143807021070726353849408426} a^{3} + \frac{246566056969197847967440539998208140045210189088776174776258711911868818798143379604767604480816325680847656240852185376959168235180428130819}{546928725792031623763625816007427681949795042570377204556420038308465665043527573565459988093230320726346282016266143807021070726353849408426} a^{2} + \frac{120206175599059969728758537962988281784635477486880802791307109569885609234654919379011721098097436555637138073298254940412496282962183018987}{273464362896015811881812908003713840974897521285188602278210019154232832521763786782729994046615160363173141008133071903510535363176924704213} a + \frac{38026598068405803561894557923654092725867005523261806621498041872707731527656814548324481781652368705925350187293138878012096255806893758486}{273464362896015811881812908003713840974897521285188602278210019154232832521763786782729994046615160363173141008133071903510535363176924704213}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  $26$
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:  \( 21303579559761460000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_9$ (as 27T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 27
The 27 conjugacy class representatives for $C_3\times C_9$
Character table for $C_3\times C_9$ is not computed

Intermediate fields

3.3.1432809.4, 3.3.361.1, 3.3.3969.1, 3.3.1432809.1, 9.9.2941473244627851129.10, 9.9.1061871841310654257569.1, 9.9.1061871841310654257569.5, \(\Q(\zeta_{19})^+\)

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{3}$ R ${\href{/LocalNumberField/5.9.0.1}{9} }^{3}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{3}$ R ${\href{/LocalNumberField/23.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{3}$

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
3Data not computed
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
19Data not computed