Normalized defining polynomial
\( x^{27} - 9 x^{26} - 279 x^{25} + 3156 x^{24} + 28773 x^{23} - 460404 x^{22} - 955542 x^{21} + 35974809 x^{20} - 57567654 x^{19} - 1579631155 x^{18} + 7062218181 x^{17} + 34301043441 x^{16} - 306209006808 x^{15} - 6539221512 x^{14} + 6604533011718 x^{13} - 17817297444354 x^{12} - 55843161650181 x^{11} + 386037100051812 x^{10} - 379987896358162 x^{9} - 2701094621543055 x^{8} + 10001937729994218 x^{7} - 7711001730164253 x^{6} - 34488237931683786 x^{5} + 120712409085829428 x^{4} - 187652271757816539 x^{3} + 165535685719269948 x^{2} - 80461586519455374 x + 16831477313082953 \)
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: | \(17860517092005028417206516767820811734461334327640142959315501467081=3^{66}\cdot 97^{18}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $309.61$ | 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: | $3, 97$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(2619=3^{3}\cdot 97\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{2619}(1,·)$, $\chi_{2619}(643,·)$, $\chi_{2619}(583,·)$, $\chi_{2619}(520,·)$, $\chi_{2619}(1225,·)$, $\chi_{2619}(1165,·)$, $\chi_{2619}(1102,·)$, $\chi_{2619}(1807,·)$, $\chi_{2619}(1747,·)$, $\chi_{2619}(1684,·)$, $\chi_{2619}(2389,·)$, $\chi_{2619}(2329,·)$, $\chi_{2619}(2266,·)$, $\chi_{2619}(352,·)$, $\chi_{2619}(292,·)$, $\chi_{2619}(229,·)$, $\chi_{2619}(934,·)$, $\chi_{2619}(874,·)$, $\chi_{2619}(811,·)$, $\chi_{2619}(1516,·)$, $\chi_{2619}(2557,·)$, $\chi_{2619}(1456,·)$, $\chi_{2619}(1393,·)$, $\chi_{2619}(2098,·)$, $\chi_{2619}(2038,·)$, $\chi_{2619}(1975,·)$, $\chi_{2619}(61,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{109} a^{24} + \frac{32}{109} a^{23} - \frac{1}{109} a^{22} + \frac{41}{109} a^{21} + \frac{36}{109} a^{20} + \frac{39}{109} a^{19} + \frac{43}{109} a^{18} + \frac{23}{109} a^{17} + \frac{9}{109} a^{16} - \frac{39}{109} a^{15} - \frac{36}{109} a^{14} - \frac{46}{109} a^{13} + \frac{6}{109} a^{12} - \frac{18}{109} a^{11} - \frac{9}{109} a^{10} - \frac{11}{109} a^{9} + \frac{20}{109} a^{8} + \frac{21}{109} a^{7} - \frac{20}{109} a^{6} - \frac{1}{109} a^{5} + \frac{44}{109} a^{4} + \frac{32}{109} a^{3} + \frac{40}{109} a^{2} - \frac{19}{109} a + \frac{31}{109}$, $\frac{1}{109} a^{25} - \frac{44}{109} a^{23} - \frac{36}{109} a^{22} + \frac{32}{109} a^{21} - \frac{23}{109} a^{20} - \frac{6}{109} a^{19} - \frac{45}{109} a^{18} + \frac{36}{109} a^{17} + \frac{13}{109} a^{15} + \frac{16}{109} a^{14} - \frac{48}{109} a^{13} + \frac{8}{109} a^{12} + \frac{22}{109} a^{11} - \frac{50}{109} a^{10} + \frac{45}{109} a^{9} + \frac{35}{109} a^{8} - \frac{38}{109} a^{7} - \frac{15}{109} a^{6} - \frac{33}{109} a^{5} + \frac{41}{109} a^{4} - \frac{3}{109} a^{3} + \frac{9}{109} a^{2} - \frac{15}{109} a - \frac{11}{109}$, $\frac{1}{92819077513821243378251202072622647438268596219941309896007932080388704368177433639670232195059805792712529929129997149827316460845176537484408539} a^{26} + \frac{236636225941234928650563710041775808263060559799858757535767991507208226132932753426364931533413003034149305957342228491290179677396411934552813}{92819077513821243378251202072622647438268596219941309896007932080388704368177433639670232195059805792712529929129997149827316460845176537484408539} a^{25} - \frac{190187761013252839968181147143797853779107649849341708951481715296652547226080125018309948037760222186927596097977263265561207711965534790889316}{92819077513821243378251202072622647438268596219941309896007932080388704368177433639670232195059805792712529929129997149827316460845176537484408539} a^{24} + \frac{22551121205148772629002154167393477022153779562262491037459592942557329592725487611787422443480101611157870013845014184693121351486763134666996608}{92819077513821243378251202072622647438268596219941309896007932080388704368177433639670232195059805792712529929129997149827316460845176537484408539} a^{23} - \frac{8042537724155965160381341744790230037558973994869053277458551452955364867443123057380200696806014374187631292229113042688945227030266464552445276}{92819077513821243378251202072622647438268596219941309896007932080388704368177433639670232195059805792712529929129997149827316460845176537484408539} a^{22} + \frac{9854003082921353925889426050495295976565280544563063639938526208159241485655400893432122447060029433533472763416699223090942867944742706747451737}{92819077513821243378251202072622647438268596219941309896007932080388704368177433639670232195059805792712529929129997149827316460845176537484408539} a^{21} + \frac{40567248664018858090166662780899699930554027190881508645707029587244235424971327957764273821908335119185294222260022412179830087923755495960210724}{92819077513821243378251202072622647438268596219941309896007932080388704368177433639670232195059805792712529929129997149827316460845176537484408539} a^{20} + \frac{21082241678982203746623863961343418145798857364788661737045366507420197594614374272385743438864250794630497037895001854120106452826921792939779139}{92819077513821243378251202072622647438268596219941309896007932080388704368177433639670232195059805792712529929129997149827316460845176537484408539} a^{19} - \frac{13646322926646823376981253626285583427702082899159350774954166045174256351862180628914472490643510952900324748524420747065153163803979735202866498}{92819077513821243378251202072622647438268596219941309896007932080388704368177433639670232195059805792712529929129997149827316460845176537484408539} a^{18} - \frac{31883800887505901327362517710966087248079735668844113611422573765263206898058763049132596299581814326369857707569280234734125870480854752427608860}{92819077513821243378251202072622647438268596219941309896007932080388704368177433639670232195059805792712529929129997149827316460845176537484408539} a^{17} - \frac{3754707372296092204532550127132256528959454344145405719875593593703761645442818578619187278362496358034772905514743689948742991062726701849683211}{92819077513821243378251202072622647438268596219941309896007932080388704368177433639670232195059805792712529929129997149827316460845176537484408539} a^{16} + \frac{36206667615176655838940888307080553619891877646755754388949597276566991630384265833952133129550386932501773287216202455938987425692679965355763870}{92819077513821243378251202072622647438268596219941309896007932080388704368177433639670232195059805792712529929129997149827316460845176537484408539} a^{15} - \frac{19192814779880917662389471460521210824472477272900502965901212769067589788393784966092608323827073712401695573587171303641182134332491239399198691}{92819077513821243378251202072622647438268596219941309896007932080388704368177433639670232195059805792712529929129997149827316460845176537484408539} a^{14} + \frac{11229363333403172661291480474939708258459939592974516633835074624337547993968652107255108072487770449659806367227527195114879946599896838281888642}{92819077513821243378251202072622647438268596219941309896007932080388704368177433639670232195059805792712529929129997149827316460845176537484408539} a^{13} + \frac{3786592349444190115927651385981393932915539539342653091030744590284001638698924178881808990319804263447320512053677129381900259225589559598392354}{92819077513821243378251202072622647438268596219941309896007932080388704368177433639670232195059805792712529929129997149827316460845176537484408539} a^{12} - \frac{84263883808641211564057056864439574751786463674129630684008398407106410460548390287971413429704776009964134235054002432436984468481636647035015}{851551169851571040167442220849749059066684368990287246752366349361364260258508565501561763257429410942316788340642175686489141842616298509031271} a^{11} - \frac{44200576646422068483555251352781400095507382333281543719642931740450715551296851749573502161978828489990321987878268742041274325146461834542495183}{92819077513821243378251202072622647438268596219941309896007932080388704368177433639670232195059805792712529929129997149827316460845176537484408539} a^{10} + \frac{9325485942438632668602887922617263429168689666929807665760525151780335783068263192978250896722534896766760570569393549732795531901077562867888902}{92819077513821243378251202072622647438268596219941309896007932080388704368177433639670232195059805792712529929129997149827316460845176537484408539} a^{9} + \frac{24415477261751809209761324703708689512924122309077135040738604449061690380611466034316991212071207097111950011341436693207677294600281172894401318}{92819077513821243378251202072622647438268596219941309896007932080388704368177433639670232195059805792712529929129997149827316460845176537484408539} a^{8} - \frac{27311594700092916917690240769379832251475713785372051015507664316339769681329201025362280680388367166725165788269236169447176099162662892960851259}{92819077513821243378251202072622647438268596219941309896007932080388704368177433639670232195059805792712529929129997149827316460845176537484408539} a^{7} - \frac{33879794806268165270555478891947952403240161999545081009691882598687742041440440019829374104981408909882084245128995237686704291432051670514017293}{92819077513821243378251202072622647438268596219941309896007932080388704368177433639670232195059805792712529929129997149827316460845176537484408539} a^{6} - \frac{6350195946081712553212932688517978902720710892288625702836166822327803234955839614729203015967864207797956000948353806387114456734773353082953634}{92819077513821243378251202072622647438268596219941309896007932080388704368177433639670232195059805792712529929129997149827316460845176537484408539} a^{5} - \frac{24739468407361838487963045871416667982260398573745014021457712393978711177910703180902914038085277427629113736817399914648848966852210783333463220}{92819077513821243378251202072622647438268596219941309896007932080388704368177433639670232195059805792712529929129997149827316460845176537484408539} a^{4} - \frac{13212363563325799367508797450193793396101174938564348172776027680821575560931313571955771761001051472628419148793663672095547345733451149707722488}{92819077513821243378251202072622647438268596219941309896007932080388704368177433639670232195059805792712529929129997149827316460845176537484408539} a^{3} - \frac{27440457553917566174493987625882581230244476478745231823397921962422558804123175909486408882299151289768933859323142562749129500821263808084815564}{92819077513821243378251202072622647438268596219941309896007932080388704368177433639670232195059805792712529929129997149827316460845176537484408539} a^{2} - \frac{32238594873722427362121815856060517709836773560739718516911303542760392473082638932901815711857770911445393162706426264833130517230356373219429038}{92819077513821243378251202072622647438268596219941309896007932080388704368177433639670232195059805792712529929129997149827316460845176537484408539} a + \frac{32702802032666696254977628151635689342570255656263035355624890550549849642928951292803228737020120303239163207020046691190451574013402520706417536}{92819077513821243378251202072622647438268596219941309896007932080388704368177433639670232195059805792712529929129997149827316460845176537484408539}$
Class group and class number
Not computed
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: | Not computed | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | Not computed | 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
| 3.3.762129.2, \(\Q(\zeta_{9})^+\), 3.3.9409.1, 3.3.762129.1, 9.9.442675475271472689.1, 9.9.26139544139305190812761.2, 9.9.26139544139305190812761.1, \(\Q(\zeta_{27})^+\) |
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 | ${\href{/LocalNumberField/5.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/7.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/11.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/13.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/17.3.0.1}{3} }^{9}$ | ${\href{/LocalNumberField/19.3.0.1}{3} }^{9}$ | ${\href{/LocalNumberField/23.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/29.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/31.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/37.3.0.1}{3} }^{9}$ | ${\href{/LocalNumberField/41.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/43.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/47.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/53.3.0.1}{3} }^{9}$ | ${\href{/LocalNumberField/59.9.0.1}{9} }^{3}$ |
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 | Data not computed | ||||||
| 97 | Data not computed | ||||||