Normalized defining polynomial
\( x^{27} - 180 x^{25} + 13581 x^{23} - 560310 x^{21} + 13841163 x^{19} - 110106 x^{18} - 211280076 x^{17} + 8431398 x^{16} + 1997088057 x^{15} - 229078692 x^{14} - 11538409068 x^{13} + 2713319154 x^{12} + 40044205305 x^{11} - 14681285424 x^{10} - 81200156077 x^{9} + 39061121994 x^{8} + 90794275164 x^{7} - 50957966052 x^{6} - 49424314320 x^{5} + 28585642770 x^{4} + 10186413309 x^{3} - 3822753528 x^{2} - 1276399539 x - 67976091 \)
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: | \(5659106580522258442740055894009895600932750567734671570320232266209=3^{66}\cdot 7^{18}\cdot 13^{18}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $296.70$ | 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: | $3, 7, 13$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(2457=3^{3}\cdot 7\cdot 13\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{2457}(256,·)$, $\chi_{2457}(1,·)$, $\chi_{2457}(835,·)$, $\chi_{2457}(1030,·)$, $\chi_{2457}(1738,·)$, $\chi_{2457}(2263,·)$, $\chi_{2457}(1933,·)$, $\chi_{2457}(781,·)$, $\chi_{2457}(16,·)$, $\chi_{2457}(1810,·)$, $\chi_{2457}(211,·)$, $\chi_{2457}(919,·)$, $\chi_{2457}(100,·)$, $\chi_{2457}(1114,·)$, $\chi_{2457}(991,·)$, $\chi_{2457}(1444,·)$, $\chi_{2457}(1894,·)$, $\chi_{2457}(295,·)$, $\chi_{2457}(1639,·)$, $\chi_{2457}(172,·)$, $\chi_{2457}(625,·)$, $\chi_{2457}(1075,·)$, $\chi_{2457}(820,·)$, $\chi_{2457}(1654,·)$, $\chi_{2457}(1600,·)$, $\chi_{2457}(1849,·)$, $\chi_{2457}(2419,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{19} a^{16} - \frac{1}{19} a^{15} - \frac{6}{19} a^{14} + \frac{3}{19} a^{13} - \frac{8}{19} a^{12} + \frac{9}{19} a^{11} - \frac{2}{19} a^{10} + \frac{1}{19} a^{7} - \frac{1}{19} a^{6} - \frac{6}{19} a^{5} + \frac{3}{19} a^{4} - \frac{8}{19} a^{3} + \frac{9}{19} a^{2} - \frac{2}{19} a$, $\frac{1}{19} a^{17} - \frac{7}{19} a^{15} - \frac{3}{19} a^{14} - \frac{5}{19} a^{13} + \frac{1}{19} a^{12} + \frac{7}{19} a^{11} - \frac{2}{19} a^{10} + \frac{1}{19} a^{8} - \frac{7}{19} a^{6} - \frac{3}{19} a^{5} - \frac{5}{19} a^{4} + \frac{1}{19} a^{3} + \frac{7}{19} a^{2} - \frac{2}{19} a$, $\frac{1}{19} a^{18} + \frac{9}{19} a^{15} - \frac{9}{19} a^{14} + \frac{3}{19} a^{13} + \frac{8}{19} a^{12} + \frac{4}{19} a^{11} + \frac{5}{19} a^{10} + \frac{1}{19} a^{9} + \frac{9}{19} a^{6} - \frac{9}{19} a^{5} + \frac{3}{19} a^{4} + \frac{8}{19} a^{3} + \frac{4}{19} a^{2} + \frac{5}{19} a$, $\frac{1}{19} a^{19} - \frac{1}{19} a$, $\frac{1}{57} a^{20} - \frac{1}{57} a^{2}$, $\frac{1}{57} a^{21} - \frac{1}{57} a^{3}$, $\frac{1}{3249} a^{22} - \frac{3}{361} a^{21} + \frac{4}{361} a^{19} - \frac{6}{361} a^{18} - \frac{7}{361} a^{17} + \frac{1}{1083} a^{16} + \frac{77}{361} a^{15} + \frac{111}{361} a^{14} + \frac{37}{361} a^{13} - \frac{45}{361} a^{12} - \frac{146}{361} a^{11} - \frac{118}{361} a^{10} + \frac{32}{361} a^{9} + \frac{145}{361} a^{8} + \frac{165}{361} a^{7} + \frac{58}{361} a^{6} + \frac{16}{361} a^{5} + \frac{332}{3249} a^{4} + \frac{148}{361} a^{3} - \frac{127}{361} a^{2} - \frac{65}{361} a + \frac{6}{19}$, $\frac{1}{11231793} a^{23} + \frac{1688}{11231793} a^{22} + \frac{7623}{1247977} a^{21} - \frac{22294}{3743931} a^{20} + \frac{11129}{1247977} a^{19} + \frac{6233}{1247977} a^{18} - \frac{96548}{3743931} a^{17} - \frac{16864}{3743931} a^{16} - \frac{397516}{1247977} a^{15} - \frac{402075}{1247977} a^{14} + \frac{248869}{1247977} a^{13} - \frac{491616}{1247977} a^{12} + \frac{407804}{1247977} a^{11} + \frac{267494}{1247977} a^{10} + \frac{481290}{1247977} a^{9} - \frac{456155}{1247977} a^{8} - \frac{331522}{1247977} a^{7} + \frac{142711}{1247977} a^{6} + \frac{1572200}{11231793} a^{5} - \frac{3725321}{11231793} a^{4} - \frac{218134}{1247977} a^{3} + \frac{1526659}{3743931} a^{2} + \frac{301490}{1247977} a + \frac{4077}{65683}$, $\frac{1}{572821443} a^{24} + \frac{5}{190940481} a^{23} + \frac{1352}{11231793} a^{22} - \frac{9154}{1116611} a^{21} - \frac{92286}{21215609} a^{20} + \frac{13796}{21215609} a^{19} - \frac{4975409}{190940481} a^{18} - \frac{155446}{63646827} a^{17} - \frac{537259}{21215609} a^{16} - \frac{10094649}{21215609} a^{15} + \frac{1488151}{21215609} a^{14} - \frac{10430081}{21215609} a^{13} + \frac{22796749}{63646827} a^{12} - \frac{1533273}{21215609} a^{11} + \frac{1358224}{21215609} a^{10} - \frac{1046477}{21215609} a^{9} + \frac{5399044}{21215609} a^{8} - \frac{2204528}{21215609} a^{7} + \frac{4614184}{33695379} a^{6} + \frac{25661407}{190940481} a^{5} + \frac{24020249}{190940481} a^{4} + \frac{3294365}{21215609} a^{3} + \frac{7914879}{21215609} a^{2} - \frac{9393080}{21215609} a - \frac{106068}{1116611}$, $\frac{1}{10883607417} a^{25} - \frac{1}{1209289713} a^{24} - \frac{103}{3627869139} a^{23} + \frac{429361}{3627869139} a^{22} - \frac{74743}{23711563} a^{21} - \frac{511837}{1209289713} a^{20} - \frac{77364254}{3627869139} a^{19} - \frac{8226770}{403096571} a^{18} - \frac{28201100}{1209289713} a^{17} - \frac{21898325}{1209289713} a^{16} + \frac{117921120}{403096571} a^{15} + \frac{16468066}{403096571} a^{14} + \frac{563195260}{1209289713} a^{13} - \frac{142818418}{403096571} a^{12} + \frac{290602}{403096571} a^{11} - \frac{86065546}{403096571} a^{10} - \frac{60386140}{403096571} a^{9} + \frac{48460270}{403096571} a^{8} + \frac{1504150127}{10883607417} a^{7} - \frac{375984611}{1209289713} a^{6} + \frac{1236287806}{3627869139} a^{5} + \frac{1359464429}{3627869139} a^{4} + \frac{138423008}{403096571} a^{3} + \frac{544167283}{1209289713} a^{2} + \frac{139456784}{403096571} a - \frac{6228391}{21215609}$, $\frac{1}{88093486350961317059489760743881440022748335958998324899660324847186140898367779777087032159669} a^{26} + \frac{138181918375522156172262048047420040356544474440527160367113725749132861124620715225}{29364495450320439019829920247960480007582778652999441633220108282395380299455926592362344053223} a^{25} - \frac{7136319723161315823459220819561959641881445611889264569299370897493748694947819376849}{9788165150106813006609973415986826669194259550999813877740036094131793433151975530787448017741} a^{24} - \frac{9142287610341104188693991308020399972769029357802086869585786568562527143301539432459}{1087573905567423667401108157331869629910473283444423764193337343792421492572441725643049779749} a^{23} + \frac{661709639690081314631246922882692236500794781662535835589746326667550945384466178960093940}{9788165150106813006609973415986826669194259550999813877740036094131793433151975530787448017741} a^{22} + \frac{8754510812239773722739499275883043146871396938316318865482958919832790987764200755440499122}{3262721716702271002203324471995608889731419850333271292580012031377264477717325176929149339247} a^{21} - \frac{199118432108317582153537239815403603281546350028446774778022148223711892481888431201166192529}{29364495450320439019829920247960480007582778652999441633220108282395380299455926592362344053223} a^{20} + \frac{10909568182644825973265897681948899953317127946714347082634808069796987618116759641032711043}{575774420594518412153527847999225098187897620647047875161178593772458437244233854752202824573} a^{19} + \frac{236566976562635974051804046467880410115529612136808618486386459338246067059313500506882148494}{9788165150106813006609973415986826669194259550999813877740036094131793433151975530787448017741} a^{18} + \frac{3790483315926778348594071297354178096601406937417298797349087563437886814687635705248426043}{3262721716702271002203324471995608889731419850333271292580012031377264477717325176929149339247} a^{17} + \frac{28951767593968732968640691280757333944060061476805674381315646835403938987145533096417135180}{3262721716702271002203324471995608889731419850333271292580012031377264477717325176929149339247} a^{16} + \frac{679377070075339146900505250100180049821572197376487084957866123257488764630303535430893662512}{3262721716702271002203324471995608889731419850333271292580012031377264477717325176929149339247} a^{15} - \frac{2555516918668061286966891340765227320786827666313194506174038782694917847472422235080126611049}{9788165150106813006609973415986826669194259550999813877740036094131793433151975530787448017741} a^{14} - \frac{567555132875259417304168581123977482300260624296852989440293041142796366178801235168458013990}{3262721716702271002203324471995608889731419850333271292580012031377264477717325176929149339247} a^{13} - \frac{1283495697101797504182112097527359885468731222350233287156584825517384449812680085586067147108}{3262721716702271002203324471995608889731419850333271292580012031377264477717325176929149339247} a^{12} - \frac{220691074278726116139072852526562594967767686325378127568827018271471187297658409299452408924}{3262721716702271002203324471995608889731419850333271292580012031377264477717325176929149339247} a^{11} + \frac{523201598793495424654797321305389418919613913735946088344012692420128636823941032293814812845}{1087573905567423667401108157331869629910473283444423764193337343792421492572441725643049779749} a^{10} - \frac{70941126717941059872186126908899572755606274597763996509367501852036374223711355340752440205}{191924806864839470717842615999741699395965873549015958387059531257486145748077951584067608191} a^{9} - \frac{31905752771172972582603733811824004540161086117266295283302152674331315224518526813113185806282}{88093486350961317059489760743881440022748335958998324899660324847186140898367779777087032159669} a^{8} + \frac{5989224048192756313658831000332717058568390779284540514872703970423746140898051844311122732472}{29364495450320439019829920247960480007582778652999441633220108282395380299455926592362344053223} a^{7} + \frac{185552298546429162899222708489169033512928040515017628519817030837893180939520860995984056604}{515166586847727000347893337683517193115487344789463888302107162849041759639577659515128843039} a^{6} - \frac{53009444645903757361400061873395410080758930782805483172253486121491890395002876461881731415}{171722195615909000115964445894505731038495781596487962767369054283013919879859219838376281013} a^{5} - \frac{3508576310575137608487370088273843837411643089342990478874067642959706443978633265740488236663}{9788165150106813006609973415986826669194259550999813877740036094131793433151975530787448017741} a^{4} + \frac{764354136325036636821905933470817236224000904797231292544302425701525803379677910591119357165}{3262721716702271002203324471995608889731419850333271292580012031377264477717325176929149339247} a^{3} + \frac{509680521931246010666746462342253110666452032343292698561760564476558114606437727832063052477}{1087573905567423667401108157331869629910473283444423764193337343792421492572441725643049779749} a^{2} - \frac{331018693843889307498204841749049332925085054909426159576112596456140807668063220735993155479}{1087573905567423667401108157331869629910473283444423764193337343792421492572441725643049779749} a + \frac{10800287278515711040153461674111013643275366012892909483170689756761577586827348095657716999}{57240731871969666705321481964835243679498593865495987589123018094337973293286406612792093671}$
Class group and class number
$C_{3}$, which has order $3$ (assuming GRH)
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: | \( 20448691760823430000000000 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_3\times C_9$ (as 27T2):
| 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
| 3.3.8281.2, \(\Q(\zeta_{9})^+\), 3.3.670761.2, 3.3.670761.4, 9.9.301789003173921081.10, 9.9.3691950281939241.2, 9.9.151470380950257681.2, 9.9.17820338848416865911969.2 |
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 | ${\href{/LocalNumberField/2.9.0.1}{9} }^{3}$ | R | ${\href{/LocalNumberField/5.9.0.1}{9} }^{3}$ | R | ${\href{/LocalNumberField/11.9.0.1}{9} }^{3}$ | R | ${\href{/LocalNumberField/17.3.0.1}{3} }^{9}$ | ${\href{/LocalNumberField/19.1.0.1}{1} }^{27}$ | ${\href{/LocalNumberField/23.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/29.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/31.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/37.3.0.1}{3} }^{9}$ | ${\href{/LocalNumberField/41.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/43.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/47.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/53.3.0.1}{3} }^{9}$ | ${\href{/LocalNumberField/59.9.0.1}{9} }^{3}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| $3$ | 3.9.22.6 | $x^{9} + 9 x^{8} + 21 x^{6} + 18 x^{5} + 9 x^{3} + 24$ | $9$ | $1$ | $22$ | $C_9$ | $[2, 3]$ |
| 3.9.22.6 | $x^{9} + 9 x^{8} + 21 x^{6} + 18 x^{5} + 9 x^{3} + 24$ | $9$ | $1$ | $22$ | $C_9$ | $[2, 3]$ | |
| 3.9.22.6 | $x^{9} + 9 x^{8} + 21 x^{6} + 18 x^{5} + 9 x^{3} + 24$ | $9$ | $1$ | $22$ | $C_9$ | $[2, 3]$ | |
| 7 | Data not computed | ||||||
| 13 | Data not computed | ||||||