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Properties

Label 27.27.2904636282...3089.5
Degree $27$
Signature $[27, 0]$
Discriminant $3^{66}\cdot 109^{26}$
Root discriminant $1343.58$
Ramified primes $3, 109$
Class number $9$ (GRH)
Class group $[9]$ (GRH)
Galois group $C_{27}$ (as 27T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1938077279322112, 23341809755455488, -97738973934280704, -173303709343893504, 349310299238051328, 586087104825974016, -4158047665852416, -270720680823631872, -60314381965065024, 47366059597665312, 16784501025746016, -3442346576482824, -1920679778347338, 50084440827489, 109156644703548, 6206464972035, -3187424294262, -334654866279, 47816065064, 6852111363, -370235286, -70073157, 1393020, 373761, -1962, -981, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 981*x^25 - 1962*x^24 + 373761*x^23 + 1393020*x^22 - 70073157*x^21 - 370235286*x^20 + 6852111363*x^19 + 47816065064*x^18 - 334654866279*x^17 - 3187424294262*x^16 + 6206464972035*x^15 + 109156644703548*x^14 + 50084440827489*x^13 - 1920679778347338*x^12 - 3442346576482824*x^11 + 16784501025746016*x^10 + 47366059597665312*x^9 - 60314381965065024*x^8 - 270720680823631872*x^7 - 4158047665852416*x^6 + 586087104825974016*x^5 + 349310299238051328*x^4 - 173303709343893504*x^3 - 97738973934280704*x^2 + 23341809755455488*x + 1938077279322112)
 
gp: K = bnfinit(x^27 - 981*x^25 - 1962*x^24 + 373761*x^23 + 1393020*x^22 - 70073157*x^21 - 370235286*x^20 + 6852111363*x^19 + 47816065064*x^18 - 334654866279*x^17 - 3187424294262*x^16 + 6206464972035*x^15 + 109156644703548*x^14 + 50084440827489*x^13 - 1920679778347338*x^12 - 3442346576482824*x^11 + 16784501025746016*x^10 + 47366059597665312*x^9 - 60314381965065024*x^8 - 270720680823631872*x^7 - 4158047665852416*x^6 + 586087104825974016*x^5 + 349310299238051328*x^4 - 173303709343893504*x^3 - 97738973934280704*x^2 + 23341809755455488*x + 1938077279322112, 1)
 

Normalized defining polynomial

\( x^{27} - 981 x^{25} - 1962 x^{24} + 373761 x^{23} + 1393020 x^{22} - 70073157 x^{21} - 370235286 x^{20} + 6852111363 x^{19} + 47816065064 x^{18} - 334654866279 x^{17} - 3187424294262 x^{16} + 6206464972035 x^{15} + 109156644703548 x^{14} + 50084440827489 x^{13} - 1920679778347338 x^{12} - 3442346576482824 x^{11} + 16784501025746016 x^{10} + 47366059597665312 x^{9} - 60314381965065024 x^{8} - 270720680823631872 x^{7} - 4158047665852416 x^{6} + 586087104825974016 x^{5} + 349310299238051328 x^{4} - 173303709343893504 x^{3} - 97738973934280704 x^{2} + 23341809755455488 x + 1938077279322112 \)

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:  \(2904636282635430627257935704503025621208529312293717639293800815805291320812817603089=3^{66}\cdot 109^{26}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1343.58$
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:  $3, 109$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2943=3^{3}\cdot 109\)
Dirichlet character group:    $\lbrace$$\chi_{2943}(1,·)$, $\chi_{2943}(961,·)$, $\chi_{2943}(7,·)$, $\chi_{2943}(2092,·)$, $\chi_{2943}(2059,·)$, $\chi_{2943}(2446,·)$, $\chi_{2943}(2641,·)$, $\chi_{2943}(2134,·)$, $\chi_{2943}(343,·)$, $\chi_{2943}(1561,·)$, $\chi_{2943}(1819,·)$, $\chi_{2943}(223,·)$, $\chi_{2943}(1825,·)$, $\chi_{2943}(2914,·)$, $\chi_{2943}(2407,·)$, $\chi_{2943}(1003,·)$, $\chi_{2943}(2860,·)$, $\chi_{2943}(1522,·)$, $\chi_{2943}(2401,·)$, $\chi_{2943}(49,·)$, $\chi_{2943}(2098,·)$, $\chi_{2943}(2740,·)$, $\chi_{2943}(841,·)$, $\chi_{2943}(2872,·)$, $\chi_{2943}(2362,·)$, $\chi_{2943}(829,·)$, $\chi_{2943}(1135,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{16} a^{6} - \frac{1}{16} a^{5} - \frac{1}{16} a^{4} - \frac{3}{16} a^{3} + \frac{1}{4} a$, $\frac{1}{16} a^{7} + \frac{3}{16} a^{3} - \frac{1}{4} a$, $\frac{1}{128} a^{8} - \frac{1}{32} a^{7} + \frac{1}{64} a^{6} - \frac{1}{16} a^{5} + \frac{1}{128} a^{4} - \frac{1}{32} a^{3} - \frac{1}{32} a^{2} + \frac{1}{8} a$, $\frac{1}{128} a^{9} + \frac{1}{64} a^{7} + \frac{1}{128} a^{5} - \frac{1}{32} a^{3}$, $\frac{1}{256} a^{10} - \frac{1}{256} a^{9} + \frac{3}{128} a^{7} - \frac{3}{256} a^{6} + \frac{15}{256} a^{5} - \frac{3}{128} a^{4} + \frac{3}{64} a^{3} + \frac{1}{32} a^{2} - \frac{1}{8} a$, $\frac{1}{256} a^{11} - \frac{1}{256} a^{9} - \frac{5}{256} a^{7} - \frac{7}{256} a^{5} + \frac{3}{64} a^{3}$, $\frac{1}{1024} a^{12} - \frac{1}{512} a^{11} + \frac{1}{1024} a^{10} - \frac{1}{512} a^{9} - \frac{1}{1024} a^{8} - \frac{15}{512} a^{7} - \frac{5}{1024} a^{6} - \frac{27}{512} a^{5} + \frac{1}{256} a^{4} - \frac{5}{128} a^{3} + \frac{1}{8} a$, $\frac{1}{2048} a^{13} - \frac{1}{2048} a^{12} + \frac{3}{2048} a^{11} - \frac{1}{2048} a^{10} + \frac{1}{2048} a^{9} + \frac{1}{2048} a^{8} + \frac{25}{2048} a^{7} + \frac{5}{2048} a^{6} - \frac{35}{1024} a^{5} + \frac{31}{512} a^{4} - \frac{11}{256} a^{3} - \frac{1}{16} a^{2} + \frac{1}{16} a$, $\frac{1}{2048} a^{14} - \frac{1}{1024} a^{11} - \frac{1}{1024} a^{10} - \frac{1}{1024} a^{9} - \frac{1}{512} a^{8} - \frac{15}{1024} a^{7} + \frac{9}{2048} a^{6} + \frac{37}{1024} a^{5} + \frac{31}{512} a^{4} - \frac{53}{256} a^{3} - \frac{1}{16} a^{2} + \frac{3}{16} a$, $\frac{1}{8192} a^{15} - \frac{1}{2048} a^{12} - \frac{3}{4096} a^{11} - \frac{1}{2048} a^{10} - \frac{1}{1024} a^{9} + \frac{1}{2048} a^{8} - \frac{51}{8192} a^{7} + \frac{5}{2048} a^{6} + \frac{25}{512} a^{5} + \frac{31}{512} a^{4} + \frac{27}{512} a^{3} - \frac{1}{16} a^{2} + \frac{13}{32} a - \frac{1}{2}$, $\frac{1}{32768} a^{16} + \frac{1}{8192} a^{14} + \frac{3}{16384} a^{12} + \frac{1}{1024} a^{11} + \frac{15}{8192} a^{10} + \frac{1}{1024} a^{9} - \frac{79}{32768} a^{8} - \frac{17}{1024} a^{7} - \frac{41}{4096} a^{6} + \frac{27}{1024} a^{5} + \frac{101}{2048} a^{4} - \frac{51}{256} a^{3} - \frac{5}{128} a^{2} + \frac{3}{16} a$, $\frac{1}{32768} a^{17} + \frac{3}{16384} a^{13} - \frac{1}{2048} a^{12} - \frac{11}{8192} a^{11} - \frac{1}{2048} a^{10} + \frac{81}{32768} a^{9} + \frac{1}{2048} a^{8} + \frac{129}{8192} a^{7} + \frac{5}{2048} a^{6} - \frac{71}{2048} a^{5} + \frac{31}{512} a^{4} + \frac{89}{512} a^{3} - \frac{1}{16} a^{2} + \frac{11}{32} a - \frac{1}{2}$, $\frac{1}{3014656} a^{18} - \frac{7}{3014656} a^{17} + \frac{11}{1507328} a^{16} - \frac{45}{753664} a^{15} - \frac{313}{1507328} a^{14} - \frac{245}{1507328} a^{13} + \frac{73}{376832} a^{12} + \frac{555}{753664} a^{11} + \frac{1801}{3014656} a^{10} - \frac{2263}{3014656} a^{9} - \frac{4795}{1507328} a^{8} - \frac{25}{23552} a^{7} - \frac{1005}{94208} a^{6} + \frac{45}{188416} a^{5} - \frac{9}{4096} a^{4} - \frac{2615}{23552} a^{3} - \frac{1381}{5888} a^{2} + \frac{165}{1472} a + \frac{1}{4}$, $\frac{1}{6029312} a^{19} - \frac{27}{6029312} a^{17} - \frac{13}{3014656} a^{16} + \frac{7}{131072} a^{15} + \frac{127}{753664} a^{14} + \frac{49}{3014656} a^{13} + \frac{105}{1507328} a^{12} - \frac{10627}{6029312} a^{11} - \frac{915}{753664} a^{10} - \frac{16599}{6029312} a^{9} + \frac{3107}{3014656} a^{8} - \frac{1283}{47104} a^{7} + \frac{2903}{376832} a^{6} - \frac{18131}{376832} a^{5} - \frac{10437}{188416} a^{4} + \frac{7359}{47104} a^{3} + \frac{2033}{11776} a^{2} - \frac{225}{2944} a - \frac{1}{8}$, $\frac{1}{12058624} a^{20} - \frac{1}{12058624} a^{19} + \frac{1}{12058624} a^{18} - \frac{11}{12058624} a^{17} + \frac{11}{3014656} a^{16} - \frac{333}{6029312} a^{15} - \frac{33}{262144} a^{14} - \frac{259}{6029312} a^{13} - \frac{215}{12058624} a^{12} + \frac{11739}{12058624} a^{11} - \frac{8163}{12058624} a^{10} + \frac{39121}{12058624} a^{9} - \frac{12387}{6029312} a^{8} + \frac{269}{32768} a^{7} - \frac{1351}{376832} a^{6} + \frac{29613}{753664} a^{5} - \frac{22319}{376832} a^{4} - \frac{5839}{94208} a^{3} - \frac{5813}{23552} a^{2} + \frac{61}{5888} a + \frac{5}{16}$, $\frac{1}{96468992} a^{21} + \frac{3}{96468992} a^{20} + \frac{3}{96468992} a^{19} - \frac{11}{96468992} a^{18} - \frac{619}{48234496} a^{17} - \frac{551}{48234496} a^{16} + \frac{1443}{48234496} a^{15} + \frac{6157}{48234496} a^{14} + \frac{23077}{96468992} a^{13} - \frac{895}{4194304} a^{12} - \frac{52985}{96468992} a^{11} - \frac{119727}{96468992} a^{10} - \frac{1435}{753664} a^{9} - \frac{48919}{24117248} a^{8} + \frac{59383}{3014656} a^{7} + \frac{79407}{6029312} a^{6} + \frac{11609}{188416} a^{5} + \frac{61367}{1507328} a^{4} + \frac{84411}{376832} a^{3} + \frac{11433}{94208} a^{2} + \frac{1693}{23552} a + \frac{29}{64}$, $\frac{1}{192937984} a^{22} - \frac{3}{96468992} a^{20} + \frac{3}{48234496} a^{19} + \frac{11}{192937984} a^{18} + \frac{517}{48234496} a^{17} - \frac{17}{12058624} a^{16} + \frac{881}{24117248} a^{15} - \frac{24233}{192937984} a^{14} - \frac{2147}{24117248} a^{13} - \frac{13599}{96468992} a^{12} - \frac{86281}{48234496} a^{11} + \frac{188237}{192937984} a^{10} + \frac{141729}{48234496} a^{9} - \frac{49811}{48234496} a^{8} + \frac{12035}{524288} a^{7} + \frac{182835}{12058624} a^{6} + \frac{59863}{3014656} a^{5} - \frac{79689}{3014656} a^{4} + \frac{178723}{753664} a^{3} + \frac{543}{8192} a^{2} + \frac{22089}{47104} a + \frac{25}{128}$, $\frac{1}{192937984} a^{23} - \frac{1}{96468992} a^{20} - \frac{3}{192937984} a^{19} - \frac{7}{96468992} a^{18} + \frac{595}{48234496} a^{17} - \frac{435}{48234496} a^{16} - \frac{2373}{192937984} a^{15} + \frac{8497}{48234496} a^{14} + \frac{691}{6029312} a^{13} + \frac{6563}{96468992} a^{12} - \frac{60969}{192937984} a^{11} - \frac{25467}{96468992} a^{10} - \frac{146715}{48234496} a^{9} - \frac{57187}{24117248} a^{8} - \frac{85913}{12058624} a^{7} + \frac{107451}{6029312} a^{6} - \frac{2407}{131072} a^{5} + \frac{79331}{1507328} a^{4} - \frac{61989}{376832} a^{3} - \frac{16715}{94208} a^{2} + \frac{1607}{23552} a + \frac{15}{64}$, $\frac{1}{12348030976} a^{24} - \frac{19}{12348030976} a^{23} - \frac{5}{3087007744} a^{22} - \frac{9}{6174015488} a^{21} - \frac{181}{12348030976} a^{20} - \frac{21}{12348030976} a^{19} + \frac{325}{6174015488} a^{18} - \frac{433}{48234496} a^{17} + \frac{17919}{12348030976} a^{16} - \frac{650077}{12348030976} a^{15} - \frac{259771}{1543503872} a^{14} + \frac{1075323}{6174015488} a^{13} - \frac{2752531}{12348030976} a^{12} + \frac{6137045}{12348030976} a^{11} - \frac{9042479}{6174015488} a^{10} + \frac{11878959}{3087007744} a^{9} + \frac{3983077}{1543503872} a^{8} + \frac{22462573}{771751936} a^{7} - \frac{271577}{385875968} a^{6} + \frac{1895093}{192937984} a^{5} - \frac{2537635}{96468992} a^{4} - \frac{3756635}{24117248} a^{3} + \frac{898883}{6029312} a^{2} + \frac{558001}{1507328} a - \frac{535}{4096}$, $\frac{1}{1136018849792} a^{25} + \frac{35}{1136018849792} a^{24} - \frac{907}{568009424896} a^{23} - \frac{293}{568009424896} a^{22} + \frac{1663}{1136018849792} a^{21} - \frac{1461}{49392123904} a^{20} - \frac{18425}{284004712448} a^{19} + \frac{4551}{284004712448} a^{18} - \frac{4206081}{1136018849792} a^{17} + \frac{11178605}{1136018849792} a^{16} - \frac{16235835}{568009424896} a^{15} + \frac{40010419}{568009424896} a^{14} + \frac{30390993}{1136018849792} a^{13} + \frac{11485093}{49392123904} a^{12} + \frac{88128265}{71001178112} a^{11} + \frac{144779133}{142002356224} a^{10} + \frac{109122221}{71001178112} a^{9} - \frac{27842411}{17750294528} a^{8} - \frac{451535597}{17750294528} a^{7} + \frac{155500041}{8875147264} a^{6} - \frac{82215491}{2218786816} a^{5} - \frac{111544231}{4437573632} a^{4} + \frac{28185133}{1109393408} a^{3} + \frac{7967027}{277348352} a^{2} + \frac{26901}{3014656} a - \frac{3547}{8192}$, $\frac{1}{408773268317516834748601115370969777643764628915869361624780707334989011872788684024496663332886113937542745384672886794250976205469646848} a^{26} - \frac{74346162687687335052295158505094430588061266677375351640805375731565431337295127758149077438718282054893413365241884855507823}{204386634158758417374300557685484888821882314457934680812390353667494505936394342012248331666443056968771372692336443397125488102734823424} a^{25} - \frac{13577309976340520214376911077486688442589922661663428985355452012372912056146130590196381642042390565136497762581287064821682897}{408773268317516834748601115370969777643764628915869361624780707334989011872788684024496663332886113937542745384672886794250976205469646848} a^{24} - \frac{140757564572749208571792343349360795356384028196265496558796447654151176860833814956403868147267862045171323238075513116817866939}{102193317079379208687150278842742444410941157228967340406195176833747252968197171006124165833221528484385686346168221698562744051367411712} a^{23} - \frac{870752022660178521065464805727250125900457194794850968907190708782081171753847076992320801278591646914561014049045681266071540311}{408773268317516834748601115370969777643764628915869361624780707334989011872788684024496663332886113937542745384672886794250976205469646848} a^{22} - \frac{109125456213374957117362672356940923164490979057799508289253420564365496014557873781196099430791531129534903126083779981048299785}{204386634158758417374300557685484888821882314457934680812390353667494505936394342012248331666443056968771372692336443397125488102734823424} a^{21} + \frac{13628171289089572947047682872086137690827234674988861992156919001447106729397384348964371290357829572785873340335258264401028164567}{408773268317516834748601115370969777643764628915869361624780707334989011872788684024496663332886113937542745384672886794250976205469646848} a^{20} - \frac{2630052067401844917003398457927690249358998990538860779138980946631117697279781974766836826509696327331185634682620405963166543473}{51096658539689604343575139421371222205470578614483670203097588416873626484098585503062082916610764242192843173084110849281372025683705856} a^{19} + \frac{57506870405104615872675715393912363270705208944969322513051911456367685369939873417726299766583399084308351201944427674887236482547}{408773268317516834748601115370969777643764628915869361624780707334989011872788684024496663332886113937542745384672886794250976205469646848} a^{18} + \frac{2025595954865389761644310490779329005565199444446404479951029169117786451171025501157784097047733785515972295906946956229960695141815}{204386634158758417374300557685484888821882314457934680812390353667494505936394342012248331666443056968771372692336443397125488102734823424} a^{17} - \frac{41511612738510144159148407142219673207694873674636665155100583288387903299435293570284069139032258420428305062558151574640009521483}{408773268317516834748601115370969777643764628915869361624780707334989011872788684024496663332886113937542745384672886794250976205469646848} a^{16} + \frac{4068923030206275333496818233497877229921214950542250971093326408286406632381804036287228814900859062780965141372659315499753141867909}{102193317079379208687150278842742444410941157228967340406195176833747252968197171006124165833221528484385686346168221698562744051367411712} a^{15} - \frac{36823743064521878122432190124608702033301292899322926155657392254547260483738143004538705268240390976053904536212946829773406457780053}{408773268317516834748601115370969777643764628915869361624780707334989011872788684024496663332886113937542745384672886794250976205469646848} a^{14} + \frac{33401612341760590994346989342547493221906416534414092138259355734871940543300827808919935919436188610597910194376144792400570454818041}{204386634158758417374300557685484888821882314457934680812390353667494505936394342012248331666443056968771372692336443397125488102734823424} a^{13} + \frac{29804475831529069620540709022446688997195989456000629745236320527566951508852883450758512225148114932299946587051029187103519074632453}{408773268317516834748601115370969777643764628915869361624780707334989011872788684024496663332886113937542745384672886794250976205469646848} a^{12} - \frac{10266752953125774775739113483637498995776835430026673803249872595956705134517192993235030709470673626900930840645630834173174740563329}{12774164634922401085893784855342805551367644653620917550774397104218406621024646375765520729152691060548210793271027712320343006420926464} a^{11} + \frac{2272599718364564309362156693432788275405896197027744091659067580251540532417227882207920344558946536285227499874866755460298411880119}{51096658539689604343575139421371222205470578614483670203097588416873626484098585503062082916610764242192843173084110849281372025683705856} a^{10} + \frac{5509811345686414854793365042262607830088263749365592610276193284957868629514709685441195146052241457923928995657887722326604367707693}{25548329269844802171787569710685611102735289307241835101548794208436813242049292751531041458305382121096421586542055424640686012841852928} a^{9} - \frac{8186412290509535365973375302613980502495859837837570479572344198327133300859995139331031149215409925845286799098389541951102040577937}{6387082317461200542946892427671402775683822326810458775387198552109203310512323187882760364576345530274105396635513856160171503210463232} a^{8} - \frac{100937119974086382792432977589402005780655324819836452416163061479615927358678940061223536599122778153836010970121361891456827469815599}{6387082317461200542946892427671402775683822326810458775387198552109203310512323187882760364576345530274105396635513856160171503210463232} a^{7} - \frac{76244422241617353580555603169537413746083191402822715303982821694157858928829377893967459185321571200337209283166248244321623330971839}{3193541158730600271473446213835701387841911163405229387693599276054601655256161593941380182288172765137052698317756928080085751605231616} a^{6} - \frac{17938517347838234505551689717380558382208821780371293334002409408849933662618253203771434519065228778393512603515512986794814613344639}{1596770579365300135736723106917850693920955581702614693846799638027300827628080796970690091144086382568526349158878464040042875802615808} a^{5} - \frac{41275783424943408930917528700154038270484143720058755219939471186239132182481874013148790761085292855120790268858801285440259817878785}{1596770579365300135736723106917850693920955581702614693846799638027300827628080796970690091144086382568526349158878464040042875802615808} a^{4} - \frac{38615794292878044934954498432415080636679742544392534263740192111298369480585324813166627694318401347240714689874195281608574614111517}{399192644841325033934180776729462673480238895425653673461699909506825206907020199242672522786021595642131587289719616010010718950653952} a^{3} - \frac{24533069582178087614344805487816946101331679970283954863549026959190593966976386078077512216245477143490362731621957028790584949362635}{99798161210331258483545194182365668370059723856413418365424977376706301726755049810668130696505398910532896822429904002502679737663488} a^{2} + \frac{251648276508440517440891780275562807312034269161077522321363526716349263510629632945591670109629663907987217194996761490745110761479}{1084762621851426722647230371547452917065866563656667590928532362790285888334294019681175333657667379462314095895977217418507388452864} a + \frac{140889430466146226608446354842497334557530127650697251775349147330641020675035769651656679406739135762357441704530298479907674471}{2947724515900616094150082531378948144200724357762683671001446638017081218299712010003193841461052661582375260586894612550291816448}$

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

Class group and class number

$C_{9}$, which has order $9$ (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:  \( 115406326134549590000000000000000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{27}$ (as 27T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 27
The 27 conjugacy class representatives for $C_{27}$
Character table for $C_{27}$ is not computed

Intermediate fields

3.3.11881.1, 9.9.10589294828624773798161.3

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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{27}$ R $27$ $27$ $27$ $27$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{27}$ $27$ $27$ $27$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{9}$ $27$ $27$ $27$

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
109Data not computed