Normalized defining polynomial
\( x^{27} - 180 x^{25} + 13581 x^{23} - 568500 x^{21} + 14742063 x^{19} - 173190 x^{18} - 249963084 x^{17} + 11056356 x^{16} + 2843438562 x^{15} - 289163682 x^{14} - 21840737832 x^{13} + 4026509760 x^{12} + 112040890065 x^{11} - 32469442902 x^{10} - 371871177991 x^{9} + 154371096648 x^{8} + 749302072020 x^{7} - 421068854304 x^{6} - 802318146192 x^{5} + 605250420480 x^{4} + 297429400704 x^{3} - 354965068800 x^{2} + 75168039168 x + 1993642496 \)
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$$\chi_{2457}(1600,·)$, $\chi_{2457}(1,·)$, $\chi_{2457}(646,·)$, $\chi_{2457}(1927,·)$, $\chi_{2457}(2440,·)$, $\chi_{2457}(841,·)$, $\chi_{2457}(2122,·)$, $\chi_{2457}(2263,·)$, $\chi_{2457}(781,·)$, $\chi_{2457}(1108,·)$, $\chi_{2457}(1621,·)$, $\chi_{2457}(22,·)$, $\chi_{2457}(1303,·)$, $\chi_{2457}(484,·)$, $\chi_{2457}(289,·)$, $\chi_{2457}(802,·)$, $\chi_{2457}(2083,·)$, $\chi_{2457}(1444,·)$, $\chi_{2457}(1639,·)$, $\chi_{2457}(2284,·)$, $\chi_{2457}(1264,·)$, $\chi_{2457}(625,·)$, $\chi_{2457}(2419,·)$, $\chi_{2457}(820,·)$, $\chi_{2457}(1465,·)$, $\chi_{2457}(1660,·)$, $\chi_{2457}(445,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{34} a^{13} - \frac{4}{17} a^{12} + \frac{5}{34} a^{11} + \frac{7}{34} a^{10} + \frac{1}{34} a^{9} - \frac{7}{34} a^{8} + \frac{3}{17} a^{7} + \frac{7}{34} a^{6} - \frac{7}{34} a^{5} + \frac{1}{34} a^{4} - \frac{8}{17} a^{3} - \frac{1}{2} a^{2} - \frac{7}{34} a$, $\frac{1}{34} a^{14} - \frac{4}{17} a^{12} - \frac{2}{17} a^{11} + \frac{3}{17} a^{10} - \frac{8}{17} a^{9} + \frac{1}{34} a^{8} + \frac{2}{17} a^{7} + \frac{15}{34} a^{6} + \frac{13}{34} a^{5} + \frac{9}{34} a^{4} - \frac{9}{34} a^{3} + \frac{5}{17} a^{2} + \frac{6}{17} a$, $\frac{1}{34} a^{15} - \frac{5}{34} a^{11} + \frac{3}{17} a^{10} - \frac{4}{17} a^{9} + \frac{8}{17} a^{8} + \frac{6}{17} a^{7} + \frac{1}{34} a^{6} - \frac{13}{34} a^{5} - \frac{1}{34} a^{4} + \frac{1}{34} a^{3} - \frac{5}{34} a^{2} + \frac{6}{17} a$, $\frac{1}{34} a^{16} - \frac{5}{34} a^{12} + \frac{3}{17} a^{11} - \frac{4}{17} a^{10} + \frac{8}{17} a^{9} + \frac{6}{17} a^{8} + \frac{1}{34} a^{7} - \frac{13}{34} a^{6} - \frac{1}{34} a^{5} + \frac{1}{34} a^{4} - \frac{5}{34} a^{3} + \frac{6}{17} a^{2}$, $\frac{1}{34} a^{17} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{8}{17} a$, $\frac{1}{34} a^{18} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{8}{17} a^{2}$, $\frac{1}{34} a^{19} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{8}{17} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{68} a^{20} - \frac{1}{68} a^{16} + \frac{5}{68} a^{12} - \frac{3}{34} a^{11} + \frac{2}{17} a^{10} - \frac{4}{17} a^{9} + \frac{11}{34} a^{8} + \frac{4}{17} a^{7} + \frac{15}{34} a^{6} + \frac{9}{34} a^{5} + \frac{15}{68} a^{4} - \frac{3}{17} a^{3} + \frac{5}{68} a^{2}$, $\frac{1}{2312} a^{21} + \frac{1}{289} a^{20} + \frac{7}{578} a^{18} + \frac{1}{136} a^{17} - \frac{5}{578} a^{16} - \frac{3}{578} a^{15} + \frac{2}{289} a^{14} + \frac{11}{2312} a^{13} - \frac{41}{1156} a^{12} - \frac{81}{578} a^{11} - \frac{143}{578} a^{10} - \frac{73}{1156} a^{9} - \frac{23}{68} a^{8} - \frac{5}{34} a^{7} - \frac{237}{578} a^{6} - \frac{27}{136} a^{5} + \frac{129}{1156} a^{4} - \frac{759}{2312} a^{3} - \frac{131}{578} a^{2} - \frac{27}{289} a + \frac{8}{17}$, $\frac{1}{411536} a^{22} - \frac{1}{205768} a^{21} + \frac{201}{102884} a^{20} - \frac{753}{51442} a^{19} + \frac{1845}{411536} a^{18} - \frac{2985}{205768} a^{17} - \frac{905}{102884} a^{16} - \frac{37}{3026} a^{15} + \frac{1415}{411536} a^{14} + \frac{785}{102884} a^{13} + \frac{1085}{25721} a^{12} - \frac{19393}{102884} a^{11} - \frac{12343}{205768} a^{10} - \frac{86191}{205768} a^{9} - \frac{959}{6052} a^{8} + \frac{6775}{51442} a^{7} + \frac{41185}{411536} a^{6} + \frac{3975}{25721} a^{5} - \frac{176195}{411536} a^{4} - \frac{6905}{205768} a^{3} + \frac{25685}{102884} a^{2} - \frac{8943}{25721} a - \frac{227}{1513}$, $\frac{1}{4208367136} a^{23} - \frac{1701}{2104183568} a^{22} + \frac{7953}{1052091784} a^{21} - \frac{471050}{131511473} a^{20} - \frac{5213347}{4208367136} a^{19} + \frac{28336967}{2104183568} a^{18} - \frac{503323}{1052091784} a^{17} - \frac{183899}{15471938} a^{16} + \frac{33051183}{4208367136} a^{15} + \frac{392737}{1052091784} a^{14} - \frac{63267}{5910628} a^{13} + \frac{4327745}{1052091784} a^{12} + \frac{240378469}{2104183568} a^{11} - \frac{257300611}{2104183568} a^{10} + \frac{13384091}{61887752} a^{9} - \frac{58070865}{263022946} a^{8} - \frac{606256879}{4208367136} a^{7} + \frac{15747090}{131511473} a^{6} - \frac{287103555}{4208367136} a^{5} - \frac{873098821}{2104183568} a^{4} + \frac{197708859}{1052091784} a^{3} - \frac{104738463}{263022946} a^{2} + \frac{2930599}{15471938} a - \frac{137055}{455057}$, $\frac{1}{8416734272} a^{24} - \frac{429}{526045892} a^{22} + \frac{57019}{526045892} a^{21} + \frac{13905629}{8416734272} a^{20} + \frac{1502992}{131511473} a^{19} + \frac{1882678}{131511473} a^{18} - \frac{4627461}{526045892} a^{17} + \frac{32281151}{8416734272} a^{16} + \frac{12043501}{4208367136} a^{15} + \frac{5724083}{526045892} a^{14} - \frac{15052977}{2104183568} a^{13} - \frac{103782327}{4208367136} a^{12} + \frac{526519031}{4208367136} a^{11} - \frac{40561029}{263022946} a^{10} - \frac{27511365}{1052091784} a^{9} - \frac{3419582927}{8416734272} a^{8} - \frac{620301243}{4208367136} a^{7} + \frac{1061404029}{8416734272} a^{6} + \frac{31549559}{131511473} a^{5} + \frac{379378111}{1052091784} a^{4} + \frac{45852359}{526045892} a^{3} + \frac{64392465}{263022946} a^{2} - \frac{35276925}{263022946} a + \frac{1001951}{7735969}$, $\frac{1}{16833468544} a^{25} + \frac{127}{123775504} a^{22} - \frac{1953107}{16833468544} a^{21} + \frac{483103}{131511473} a^{20} + \frac{7318563}{1052091784} a^{19} - \frac{28178245}{2104183568} a^{18} - \frac{63090481}{16833468544} a^{17} + \frac{7089565}{8416734272} a^{16} + \frac{976920}{131511473} a^{15} + \frac{27241085}{4208367136} a^{14} - \frac{53231559}{8416734272} a^{13} + \frac{92786231}{495102016} a^{12} - \frac{659669}{526045892} a^{11} + \frac{324754017}{2104183568} a^{10} + \frac{8123394081}{16833468544} a^{9} + \frac{2189624805}{8416734272} a^{8} - \frac{515738131}{16833468544} a^{7} - \frac{238760095}{2104183568} a^{6} + \frac{462945065}{2104183568} a^{5} - \frac{388457337}{2104183568} a^{4} - \frac{598305}{1052091784} a^{3} + \frac{45302033}{263022946} a^{2} + \frac{57715501}{131511473} a - \frac{3340656}{7735969}$, $\frac{1}{2626296182008588005159732465610656206243138929567351483785392544808741972436287377664} a^{26} - \frac{3334415287317497437983554325473992723472302707813073287687005098618779777}{656574045502147001289933116402664051560784732391837870946348136202185493109071844416} a^{25} - \frac{9430111092666610851825984533838111581432227990763384886204028331368456715}{656574045502147001289933116402664051560784732391837870946348136202185493109071844416} a^{24} + \frac{31805525763980184077949232736832470517267526082576530669496922278274021953}{328287022751073500644966558201332025780392366195918935473174068101092746554535922208} a^{23} - \frac{705802979049108022316907523979230411160357789144726047144976145974923801927283}{2626296182008588005159732465610656206243138929567351483785392544808741972436287377664} a^{22} - \frac{85123355792193002286342080957430840220011361628410646096656590380406527902428477}{656574045502147001289933116402664051560784732391837870946348136202185493109071844416} a^{21} - \frac{2425587165113609042858467902192724955528778286983675995386145381515345199174957251}{656574045502147001289933116402664051560784732391837870946348136202185493109071844416} a^{20} + \frac{2995001586070097047119562864822141784007286006425474048198848477055322168575478165}{328287022751073500644966558201332025780392366195918935473174068101092746554535922208} a^{19} - \frac{13487269027380133713054469191616039605950279937545313227157665857626605500765118673}{2626296182008588005159732465610656206243138929567351483785392544808741972436287377664} a^{18} - \frac{6885423054671459200243541625909120455636609433428123253976054569542459771614222337}{1313148091004294002579866232805328103121569464783675741892696272404370986218143688832} a^{17} + \frac{6352100896658200511164386762555765194002235718259295935369160913920131695316465761}{656574045502147001289933116402664051560784732391837870946348136202185493109071844416} a^{16} - \frac{8523035571270246694552280869115501810772995315836741566092506339250186939589774037}{656574045502147001289933116402664051560784732391837870946348136202185493109071844416} a^{15} - \frac{5490392715687566693075384982762490961565104867367095462675763628689764345000698751}{1313148091004294002579866232805328103121569464783675741892696272404370986218143688832} a^{14} - \frac{174198698326140154543056914286696795094282986052001522246995437792826430406356997}{1313148091004294002579866232805328103121569464783675741892696272404370986218143688832} a^{13} + \frac{6242029155845874956980822327795295458338229908664600179148688122358446710293520981}{164143511375536750322483279100666012890196183097959467736587034050546373277267961104} a^{12} + \frac{4494044362268484812280078119740732473954416034566895496977072621680797805945325255}{41035877843884187580620819775166503222549045774489866934146758512636593319316990276} a^{11} - \frac{318743418628233041630850165705916237186833364253706561819439699158346599923598454303}{2626296182008588005159732465610656206243138929567351483785392544808741972436287377664} a^{10} - \frac{489239953898153072062873728731725751887272842552737600912826307992260839861626058605}{1313148091004294002579866232805328103121569464783675741892696272404370986218143688832} a^{9} - \frac{46738893681106343361223086559850509825880063943464358158859210103882298794193703927}{154488010706387529715278380330038600367243466445138322575611326165220116025663963392} a^{8} - \frac{3107301890366894102868879636387751934453716812535382017662445248344972273025093589}{38622002676596882428819595082509650091810866611284580643902831541305029006415990848} a^{7} + \frac{216537432727840476364028629696888530070865286627862538411262385937216851401689548371}{656574045502147001289933116402664051560784732391837870946348136202185493109071844416} a^{6} + \frac{17305853557557396384447309214161334068090507385377557996693294846593288517093288165}{328287022751073500644966558201332025780392366195918935473174068101092746554535922208} a^{5} + \frac{64092067765952704327712573094901392066697206660951628973829139982934073275364012711}{164143511375536750322483279100666012890196183097959467736587034050546373277267961104} a^{4} - \frac{10490673220739314038515720877878894171464071658068102436272981899351072360011000509}{41035877843884187580620819775166503222549045774489866934146758512636593319316990276} a^{3} - \frac{18640271945595389410337763259261359997075289473050115541307375392717431302609291513}{41035877843884187580620819775166503222549045774489866934146758512636593319316990276} a^{2} - \frac{1226939685501709066150231232710730328722528429887516311001155641710990103554876615}{10258969460971046895155204943791625805637261443622466733536689628159148329829247569} a + \frac{6400265063766440540677048273544470210603998418422359275911800183086352087383635}{603468791821826287950306173164213282684544790801321572560981742832891078225249857}$
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: | \( 8588321760631236000000000 \) (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, 9.9.151470380950257681.1, 9.9.17820338848416865911969.3, 9.9.3691950281939241.2 |
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 | ${\href{/LocalNumberField/17.1.0.1}{1} }^{27}$ | ${\href{/LocalNumberField/19.3.0.1}{3} }^{9}$ | ${\href{/LocalNumberField/23.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/29.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/31.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/37.3.0.1}{3} }^{9}$ | ${\href{/LocalNumberField/41.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/43.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/47.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/53.3.0.1}{3} }^{9}$ | ${\href{/LocalNumberField/59.9.0.1}{9} }^{3}$ |
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 | ||||||
| 7 | Data not computed | ||||||
| 13 | Data not computed | ||||||