Normalized defining polynomial
\( x^{27} - 180 x^{25} + 13581 x^{23} - 568500 x^{21} + 14742063 x^{19} - 174192 x^{18} - 249963084 x^{17} + 11243754 x^{16} + 2843438562 x^{15} - 300038904 x^{14} - 21840737832 x^{13} + 4313717772 x^{12} + 112040890065 x^{11} - 36374296608 x^{10} - 371584562893 x^{9} + 182892909312 x^{8} + 742994668764 x^{7} - 530201597184 x^{6} - 756546767280 x^{5} + 798056616960 x^{4} + 173258396544 x^{3} - 459871520256 x^{2} + 182205736704 x - 21684382208 \)
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$$\chi_{2457}(1,·)$, $\chi_{2457}(898,·)$, $\chi_{2457}(2146,·)$, $\chi_{2457}(1030,·)$, $\chi_{2457}(1537,·)$, $\chi_{2457}(2440,·)$, $\chi_{2457}(1639,·)$, $\chi_{2457}(1933,·)$, $\chi_{2457}(718,·)$, $\chi_{2457}(79,·)$, $\chi_{2457}(211,·)$, $\chi_{2457}(1108,·)$, $\chi_{2457}(1621,·)$, $\chi_{2457}(1849,·)$, $\chi_{2457}(1114,·)$, $\chi_{2457}(2011,·)$, $\chi_{2457}(289,·)$, $\chi_{2457}(802,·)$, $\chi_{2457}(295,·)$, $\chi_{2457}(1192,·)$, $\chi_{2457}(1927,·)$, $\chi_{2457}(1327,·)$, $\chi_{2457}(2356,·)$, $\chi_{2457}(1717,·)$, $\chi_{2457}(820,·)$, $\chi_{2457}(508,·)$, $\chi_{2457}(373,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{148} a^{20} - \frac{13}{74} a^{19} - \frac{5}{74} a^{18} + \frac{11}{74} a^{17} - \frac{13}{148} a^{16} + \frac{3}{74} a^{15} + \frac{3}{37} a^{14} - \frac{3}{74} a^{13} - \frac{3}{148} a^{12} + \frac{8}{37} a^{11} - \frac{3}{74} a^{10} + \frac{5}{74} a^{9} - \frac{11}{74} a^{8} - \frac{6}{37} a^{7} - \frac{3}{74} a^{6} - \frac{3}{37} a^{5} + \frac{43}{148} a^{4} + \frac{10}{37} a^{3} - \frac{33}{148} a^{2} + \frac{8}{37} a$, $\frac{1}{296} a^{21} - \frac{5}{74} a^{19} - \frac{2}{37} a^{18} + \frac{41}{296} a^{17} - \frac{9}{74} a^{16} + \frac{5}{74} a^{15} - \frac{8}{37} a^{14} + \frac{63}{296} a^{13} + \frac{7}{74} a^{12} + \frac{3}{74} a^{11} + \frac{1}{148} a^{10} + \frac{45}{148} a^{9} - \frac{1}{74} a^{8} + \frac{9}{74} a^{7} - \frac{5}{74} a^{6} + \frac{101}{296} a^{5} - \frac{25}{74} a^{4} - \frac{29}{296} a^{3} - \frac{3}{74} a^{2} + \frac{23}{74} a$, $\frac{1}{592} a^{22} + \frac{7}{74} a^{19} - \frac{11}{592} a^{18} - \frac{5}{74} a^{17} + \frac{7}{74} a^{16} + \frac{7}{74} a^{15} + \frac{7}{592} a^{14} + \frac{7}{74} a^{13} - \frac{3}{37} a^{12} + \frac{25}{296} a^{11} - \frac{15}{296} a^{10} - \frac{31}{74} a^{9} - \frac{16}{37} a^{8} - \frac{51}{148} a^{7} + \frac{129}{592} a^{6} + \frac{13}{74} a^{5} + \frac{239}{592} a^{4} - \frac{31}{74} a^{3} - \frac{17}{37} a^{2} + \frac{3}{37} a$, $\frac{1}{4358774048} a^{23} + \frac{388387}{1089693512} a^{22} - \frac{720375}{544846756} a^{21} + \frac{401699}{136211689} a^{20} + \frac{961455469}{4358774048} a^{19} - \frac{137176769}{1089693512} a^{18} + \frac{66260819}{544846756} a^{17} - \frac{31633126}{136211689} a^{16} - \frac{236462561}{4358774048} a^{15} + \frac{246652605}{1089693512} a^{14} + \frac{61677057}{544846756} a^{13} - \frac{383753843}{2179387024} a^{12} + \frac{5224661}{2179387024} a^{11} - \frac{71115761}{544846756} a^{10} - \frac{17376983}{272423378} a^{9} + \frac{167611743}{1089693512} a^{8} - \frac{1875652239}{4358774048} a^{7} - \frac{31760069}{1089693512} a^{6} - \frac{2118065537}{4358774048} a^{5} - \frac{116009427}{1089693512} a^{4} + \frac{49686821}{136211689} a^{3} - \frac{53342501}{272423378} a^{2} + \frac{43710166}{136211689} a - \frac{1575776}{3681397}$, $\frac{1}{165633413824} a^{24} + \frac{9}{82816706912} a^{23} + \frac{12352837}{41408353456} a^{22} - \frac{22588851}{20704176728} a^{21} - \frac{551926163}{165633413824} a^{20} + \frac{15831201093}{82816706912} a^{19} + \frac{7574977949}{41408353456} a^{18} + \frac{142334483}{1089693512} a^{17} - \frac{3202517297}{165633413824} a^{16} - \frac{8087959313}{82816706912} a^{15} - \frac{5485391437}{41408353456} a^{14} - \frac{163114691}{2238289376} a^{13} - \frac{13787146853}{82816706912} a^{12} + \frac{3437523527}{41408353456} a^{11} + \frac{2662944187}{20704176728} a^{10} + \frac{18083076239}{41408353456} a^{9} + \frac{625120969}{165633413824} a^{8} - \frac{28814515335}{82816706912} a^{7} + \frac{57691304571}{165633413824} a^{6} + \frac{65062071}{2238289376} a^{5} + \frac{17411556701}{41408353456} a^{4} - \frac{2831903599}{20704176728} a^{3} + \frac{2168806123}{10352088364} a^{2} + \frac{541244102}{2588022091} a + \frac{16770080}{69946543}$, $\frac{1}{331266827648} a^{25} + \frac{1}{82816706912} a^{23} - \frac{1625933}{2588022091} a^{22} - \frac{122720403}{331266827648} a^{21} - \frac{5624367}{10352088364} a^{20} - \frac{14819549063}{82816706912} a^{19} + \frac{645449472}{2588022091} a^{18} - \frac{33025432881}{331266827648} a^{17} + \frac{2714229687}{20704176728} a^{16} - \frac{13467715537}{82816706912} a^{15} + \frac{3166915797}{165633413824} a^{14} + \frac{6257396705}{165633413824} a^{13} - \frac{5535356589}{41408353456} a^{12} - \frac{9650517503}{41408353456} a^{11} - \frac{13338725741}{82816706912} a^{10} - \frac{40000345263}{331266827648} a^{9} + \frac{2240909721}{5176044182} a^{8} - \frac{134924503317}{331266827648} a^{7} - \frac{1290784004}{2588022091} a^{6} - \frac{8031413471}{82816706912} a^{5} + \frac{2715699489}{10352088364} a^{4} + \frac{4941637235}{20704176728} a^{3} - \frac{1121756139}{10352088364} a^{2} - \frac{748095912}{2588022091} a + \frac{20159948}{69946543}$, $\frac{1}{184299837990628945041246773912208577246377544567116776723829626915858340278857467645385523456} a^{26} - \frac{53606825298026060305121528151503498505181335288822025670507542346603451692293619}{92149918995314472520623386956104288623188772283558388361914813457929170139428733822692761728} a^{25} + \frac{122690937408179326423377686266796521812822333857995597225060545417556725097880521}{46074959497657236260311693478052144311594386141779194180957406728964585069714366911346380864} a^{24} + \frac{108565234924326834764476034949417452934333401853866773224332840452844565254727627}{11518739874414309065077923369513036077898596535444798545239351682241146267428591727836595216} a^{23} - \frac{75634653197815061633434974794999535056466774038107780711731632344337716431828566869096739}{184299837990628945041246773912208577246377544567116776723829626915858340278857467645385523456} a^{22} - \frac{13752296384397400116855993185695034668970894804111012708281165160686598979742036941456615}{92149918995314472520623386956104288623188772283558388361914813457929170139428733822692761728} a^{21} + \frac{36740059783547150901020066239821058199263685095452703037406962925668228368669500456744369}{46074959497657236260311693478052144311594386141779194180957406728964585069714366911346380864} a^{20} + \frac{1766441964480740080868126098854519765912428311160319670186742596850436881695196524162662197}{11518739874414309065077923369513036077898596535444798545239351682241146267428591727836595216} a^{19} + \frac{888888444256477786137377958032755997038156238669903856386016860280927229507135467740798245}{9699991473190997107434040732221504065598818135111409301254190890308333698887235139230817024} a^{18} - \frac{6270720665002235437961630984032288146592884744687272267450035783907651777588526130870889829}{92149918995314472520623386956104288623188772283558388361914813457929170139428733822692761728} a^{17} - \frac{10707032250939674840015984818079680101381993979416455479197661533151964893802470871172645249}{46074959497657236260311693478052144311594386141779194180957406728964585069714366911346380864} a^{16} - \frac{12696518589218383639828755567936233641908627967553194969160732962622492214968217895577872675}{92149918995314472520623386956104288623188772283558388361914813457929170139428733822692761728} a^{15} + \frac{19824347419933677630651253875067339701476619087279130382357946025359654508322446717377117291}{92149918995314472520623386956104288623188772283558388361914813457929170139428733822692761728} a^{14} + \frac{6767805536313435584099830765039840506186212950623491491562128720085903967951175128940958275}{46074959497657236260311693478052144311594386141779194180957406728964585069714366911346380864} a^{13} - \frac{1199169468240731393707404632406492155759294610134484195947059542110856083126220904599993439}{23037479748828618130155846739026072155797193070889597090478703364482292534857183455673190432} a^{12} - \frac{290095403979609556225912429559815553986524300682556715999836352529315588125844726119584605}{46074959497657236260311693478052144311594386141779194180957406728964585069714366911346380864} a^{11} - \frac{37511930531230673865878201250487785006611664403623117522625144006348952663292113482284203927}{184299837990628945041246773912208577246377544567116776723829626915858340278857467645385523456} a^{10} - \frac{34635996042976512893029228095394928954717147076189719904791130079890213656507701621406643587}{92149918995314472520623386956104288623188772283558388361914813457929170139428733822692761728} a^{9} - \frac{55992242560647100971337329644328194655801353692222054201105798945393122550302357132037900053}{184299837990628945041246773912208577246377544567116776723829626915858340278857467645385523456} a^{8} - \frac{18788858701581270563746959925431048575686156856920378521955835429465325956831188419158056461}{92149918995314472520623386956104288623188772283558388361914813457929170139428733822692761728} a^{7} + \frac{11659113764518180096155299445215184989911674891535071220380674642152200136259676705755811557}{46074959497657236260311693478052144311594386141779194180957406728964585069714366911346380864} a^{6} + \frac{262182776236622693785635933839932825136210211425928623286011178560797892092367831275858764}{719921242150894316567370210594564754868662283465299909077459480140071641714286982989787201} a^{5} + \frac{1426534751254979110549883793696981048305327796612064794287220243722509262742440858544537091}{2879684968603577266269480842378259019474649133861199636309837920560286566857147931959148804} a^{4} + \frac{18406658746215501665840885587844348593820084296914085432507229964156530727741838173163802}{719921242150894316567370210594564754868662283465299909077459480140071641714286982989787201} a^{3} - \frac{4812188868409148715181002345098756055511553894456421034853092928318699832951387385784292}{719921242150894316567370210594564754868662283465299909077459480140071641714286982989787201} a^{2} + \frac{136364803144279534401805542771501815016423506586888624434606807883250408759224827153872898}{719921242150894316567370210594564754868662283465299909077459480140071641714286982989787201} a + \frac{2369975758429024288790079225575570401331231306362067161423662775991084001827459287463919}{19457330868943089636955951637690939320774656309872970515607012976758693019305053594318573}$
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: | \( 2399583275882119000000000 \) (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
| 3.3.670761.3, \(\Q(\zeta_{9})^+\), 3.3.670761.1, 3.3.8281.1, 9.9.301789003173921081.12, 9.9.151470380950257681.2, 9.9.3691950281939241.1, 9.9.17820338848416865911969.4 |
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 | ${\href{/LocalNumberField/17.3.0.1}{3} }^{9}$ | ${\href{/LocalNumberField/19.3.0.1}{3} }^{9}$ | ${\href{/LocalNumberField/23.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/29.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/31.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/37.1.0.1}{1} }^{27}$ | ${\href{/LocalNumberField/41.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/43.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/47.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/53.3.0.1}{3} }^{9}$ | ${\href{/LocalNumberField/59.9.0.1}{9} }^{3}$ |
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 | Data not computed | ||||||
| 7 | Data not computed | ||||||
| 13 | Data not computed | ||||||