Normalized defining polynomial
\( x^{27} - 999 x^{25} + 443556 x^{23} - 115336881 x^{21} + 19481903595 x^{19} - 16541442 x^{18} - 2241127346283 x^{17} + 11016600372 x^{16} + 179005600103112 x^{15} - 3057106603230 x^{14} - 9934810805722716 x^{13} + 457479019247796 x^{12} + 377261368227838926 x^{11} - 39898705892968494 x^{10} - 9477657404647124790 x^{9} + 2044041394209001308 x^{8} + 148376945318831526852 x^{7} - 58822969011125704308 x^{6} - 1294713215954782474563 x^{5} + 847967475355188724440 x^{4} + 4704159643094157231339 x^{3} - 4706219488221297420642 x^{2} + 1052417297711836222461 x + 46181559728398496293 \)
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: | \(30636512167696967158640831949436414926745966008852301931537630975865004722034443609=3^{94}\cdot 37^{24}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $1135.13$ | 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: | $3, 37$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(2997=3^{4}\cdot 37\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{2997}(1,·)$, $\chi_{2997}(2821,·)$, $\chi_{2997}(7,·)$, $\chi_{2997}(2569,·)$, $\chi_{2997}(2764,·)$, $\chi_{2997}(1999,·)$, $\chi_{2997}(403,·)$, $\chi_{2997}(2005,·)$, $\chi_{2997}(1366,·)$, $\chi_{2997}(343,·)$, $\chi_{2997}(1048,·)$, $\chi_{2997}(1822,·)$, $\chi_{2997}(1765,·)$, $\chi_{2997}(2401,·)$, $\chi_{2997}(1570,·)$, $\chi_{2997}(2341,·)$, $\chi_{2997}(1000,·)$, $\chi_{2997}(1006,·)$, $\chi_{2997}(367,·)$, $\chi_{2997}(49,·)$, $\chi_{2997}(766,·)$, $\chi_{2997}(823,·)$, $\chi_{2997}(1402,·)$, $\chi_{2997}(571,·)$, $\chi_{2997}(2365,·)$, $\chi_{2997}(1342,·)$, $\chi_{2997}(2047,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{629} a^{9} + \frac{8}{17} a^{7} - \frac{4}{17} a^{5} + \frac{2}{17} a^{3} + \frac{5}{17} a$, $\frac{1}{629} a^{10} + \frac{8}{17} a^{8} - \frac{4}{17} a^{6} + \frac{2}{17} a^{4} + \frac{5}{17} a^{2}$, $\frac{1}{629} a^{11} + \frac{8}{17} a^{7} - \frac{4}{17} a^{5} + \frac{8}{17} a^{3} - \frac{1}{17} a$, $\frac{1}{629} a^{12} + \frac{8}{17} a^{8} - \frac{4}{17} a^{6} + \frac{8}{17} a^{4} - \frac{1}{17} a^{2}$, $\frac{1}{629} a^{13} + \frac{8}{17} a^{7} + \frac{2}{17} a^{5} + \frac{2}{17} a^{3} - \frac{1}{17} a$, $\frac{1}{23273} a^{14} - \frac{3}{17} a^{8} - \frac{5}{17} a^{6} + \frac{9}{37} a^{5} - \frac{5}{17} a^{4} - \frac{6}{17} a^{2}$, $\frac{1}{23273} a^{15} - \frac{1}{17} a^{7} + \frac{9}{37} a^{6} - \frac{7}{17} a^{5} - \frac{5}{17} a^{3} - \frac{6}{17} a$, $\frac{1}{23273} a^{16} - \frac{1}{17} a^{8} + \frac{9}{37} a^{7} - \frac{7}{17} a^{6} - \frac{5}{17} a^{4} - \frac{6}{17} a^{2}$, $\frac{1}{23273} a^{17} + \frac{9}{37} a^{8} - \frac{2}{17} a$, $\frac{1}{21030632982403633} a^{18} - \frac{18}{568395486010909} a^{16} + \frac{135}{15362040162457} a^{14} - \frac{546}{415190274661} a^{12} + \frac{47619}{415190274661} a^{10} - \frac{536993081}{1759738346783} a^{9} - \frac{2439558}{415190274661} a^{8} + \frac{4832937729}{47560495859} a^{7} + \frac{70205058}{415190274661} a^{6} - \frac{359206310}{1285418807} a^{5} - \frac{1012046940}{415190274661} a^{4} - \frac{372006538}{1285418807} a^{3} + \frac{5616860517}{415190274661} a^{2} - \frac{241151372}{1285418807} a - \frac{623580851}{1436644549}$, $\frac{1}{21030632982403633} a^{19} - \frac{18}{568395486010909} a^{17} + \frac{135}{15362040162457} a^{15} - \frac{546}{415190274661} a^{13} + \frac{47619}{415190274661} a^{11} - \frac{536993081}{1759738346783} a^{10} - \frac{2439558}{415190274661} a^{9} + \frac{4832937729}{47560495859} a^{8} + \frac{70205058}{415190274661} a^{7} - \frac{359206310}{1285418807} a^{6} - \frac{1012046940}{415190274661} a^{5} - \frac{372006538}{1285418807} a^{4} + \frac{5616860517}{415190274661} a^{3} - \frac{241151372}{1285418807} a^{2} - \frac{623580851}{1436644549} a$, $\frac{1}{21030632982403633} a^{20} - \frac{189}{15362040162457} a^{16} + \frac{1884}{415190274661} a^{14} - \frac{316017}{415190274661} a^{12} - \frac{536993081}{1759738346783} a^{11} + \frac{29274696}{415190274661} a^{10} + \frac{6286015}{47560495859} a^{9} - \frac{1554540570}{415190274661} a^{8} - \frac{623580322}{1285418807} a^{7} + \frac{45744521688}{415190274661} a^{6} - \frac{591123767}{1285418807} a^{5} + \frac{9527891047}{24422957333} a^{4} - \frac{516578897}{1285418807} a^{3} - \frac{176098233566}{415190274661} a^{2} - \frac{156301490}{1285418807} a - \frac{114572105}{1436644549}$, $\frac{1}{778133420348934421} a^{21} - \frac{189}{568395486010909} a^{17} + \frac{1884}{15362040162457} a^{15} - \frac{8541}{415190274661} a^{13} - \frac{48097488940}{65110318830971} a^{12} + \frac{791208}{415190274661} a^{11} + \frac{44847925}{103514020399} a^{10} - \frac{42014610}{415190274661} a^{9} + \frac{22135893849}{47560495859} a^{8} + \frac{1236338424}{415190274661} a^{7} + \frac{41244233}{1285418807} a^{6} - \frac{1062649287}{24422957333} a^{5} - \frac{112053965}{1285418807} a^{4} + \frac{239092041095}{15362040162457} a^{3} + \frac{271660435}{1285418807} a^{2} + \frac{268701074}{1436644549} a$, $\frac{1}{778133420348934421} a^{22} - \frac{1518}{15362040162457} a^{16} + \frac{16974}{415190274661} a^{14} - \frac{48097488940}{65110318830971} a^{13} - \frac{3026970}{415190274661} a^{12} + \frac{44847925}{103514020399} a^{11} + \frac{15315003}{21852119719} a^{10} + \frac{37716090}{47560495859} a^{9} - \frac{15823490670}{415190274661} a^{8} - \frac{315071021}{1285418807} a^{7} + \frac{57688658054}{415190274661} a^{6} - \frac{409043788}{1285418807} a^{5} - \frac{24434868404}{808528429603} a^{4} - \frac{87297399}{1285418807} a^{3} - \frac{86715887028}{415190274661} a^{2} - \frac{128908225}{1285418807} a - \frac{484684828}{1436644549}$, $\frac{1}{1777740569733243080408939559649} a^{23} + \frac{26543476679}{48047042425222785956998366477} a^{22} - \frac{23}{48047042425222785956998366477} a^{21} - \frac{28237672505}{1298568714195210431270226121} a^{20} + \frac{230}{1298568714195210431270226121} a^{19} + \frac{15223028119}{1298568714195210431270226121} a^{18} - \frac{69}{1847181670263457227980407} a^{17} - \frac{1293648881831391}{35096451735005687331627733} a^{16} + \frac{276}{55797220564396959191777} a^{15} - \frac{37316756985926667842128}{48047042425222785956998366477} a^{14} + \frac{616236253949691411388340}{1298568714195210431270226121} a^{13} + \frac{529243608147730330853819}{1298568714195210431270226121} a^{12} - \frac{16863061649685818242074}{35096451735005687331627733} a^{11} - \frac{19841324371848394486143}{35096451735005687331627733} a^{10} + \frac{5114740303262712872064}{35096451735005687331627733} a^{9} - \frac{111587502981761356603109}{948552749594748306260209} a^{8} + \frac{247759972427983137227753}{948552749594748306260209} a^{7} - \frac{7328313829508419099231}{25636560799858062331357} a^{6} - \frac{15840393868520914387225144}{35096451735005687331627733} a^{5} - \frac{285996449267729576996598}{948552749594748306260209} a^{4} + \frac{448331642751667009326337}{948552749594748306260209} a^{3} + \frac{6988199626922196382578}{25636560799858062331357} a^{2} - \frac{427946471323982375122}{1508032988226944843021} a - \frac{11068032118128676676}{88707822836879108413}$, $\frac{1}{1777740569733243080408939559649} a^{24} - \frac{24}{48047042425222785956998366477} a^{22} - \frac{5835921869}{48047042425222785956998366477} a^{21} + \frac{252}{1298568714195210431270226121} a^{20} - \frac{938710625}{1298568714195210431270226121} a^{19} - \frac{80}{1847181670263457227980407} a^{18} + \frac{1119886024491}{35096451735005687331627733} a^{17} + \frac{18}{2936695819178787325883} a^{16} - \frac{38062120796449743262522}{48047042425222785956998366477} a^{15} - \frac{864}{1508032988226944843021} a^{14} + \frac{570925912549045373619372}{1298568714195210431270226121} a^{13} + \frac{987994257572914215351838}{1298568714195210431270226121} a^{12} - \frac{21916098877319002630276}{35096451735005687331627733} a^{11} - \frac{5500610746476645733422}{35096451735005687331627733} a^{10} - \frac{319514550217837463456}{948552749594748306260209} a^{9} - \frac{183230434911094580242501}{948552749594748306260209} a^{8} + \frac{6562676899997053372346}{25636560799858062331357} a^{7} - \frac{14176993779234747683745305}{35096451735005687331627733} a^{6} + \frac{6976379285958951228314}{25636560799858062331357} a^{5} + \frac{450540452879550517204858}{948552749594748306260209} a^{4} - \frac{230627654043987415491063}{948552749594748306260209} a^{3} + \frac{2684109814036574751480}{25636560799858062331357} a^{2} - \frac{304268930095502631552}{1508032988226944843021} a - \frac{10732504433005233815}{88707822836879108413}$, $\frac{1}{65776401080129993975130763707013} a^{25} - \frac{15624967332}{48047042425222785956998366477} a^{22} - \frac{300}{48047042425222785956998366477} a^{21} - \frac{186451639}{1298568714195210431270226121} a^{20} + \frac{4000}{1298568714195210431270226121} a^{19} + \frac{3321860859}{1298568714195210431270226121} a^{18} - \frac{1350}{1847181670263457227980407} a^{17} + \frac{32993937532347886346317906}{1777740569733243080408939559649} a^{16} + \frac{5760}{55797220564396959191777} a^{15} - \frac{343051714313937145823070}{48047042425222785956998366477} a^{14} - \frac{46350949188070281124463}{68345721799747917435275059} a^{13} - \frac{956944078196310237874114}{1298568714195210431270226121} a^{12} + \frac{21183002356273631163722}{35096451735005687331627733} a^{11} - \frac{6643804497939078679329}{35096451735005687331627733} a^{10} - \frac{8955261449214572427996}{35096451735005687331627733} a^{9} + \frac{122203928331981472089453}{948552749594748306260209} a^{8} + \frac{60495926703033438938662356}{1298568714195210431270226121} a^{7} + \frac{8546738565301371492726}{25636560799858062331357} a^{6} - \frac{11082613534163143177067497}{35096451735005687331627733} a^{5} - \frac{160473141276771782927418}{948552749594748306260209} a^{4} + \frac{10002172688677533896043}{25636560799858062331357} a^{3} - \frac{10262860446668280729964}{25636560799858062331357} a^{2} - \frac{145141427725471295960}{1508032988226944843021} a + \frac{43583578412790176538}{88707822836879108413}$, $\frac{1}{65776401080129993975130763707013} a^{26} - \frac{325}{48047042425222785956998366477} a^{22} - \frac{28240898720}{48047042425222785956998366477} a^{21} + \frac{4550}{1298568714195210431270226121} a^{20} - \frac{30242398119}{1298568714195210431270226121} a^{19} - \frac{1625}{1847181670263457227980407} a^{18} + \frac{32993893314604620819064291}{1777740569733243080408939559649} a^{17} + \frac{390}{2936695819178787325883} a^{16} - \frac{342556918923444870969816}{48047042425222785956998366477} a^{15} - \frac{19500}{1508032988226944843021} a^{14} - \frac{7790680221775816193574}{1298568714195210431270226121} a^{13} - \frac{1004640453361939497670352}{1298568714195210431270226121} a^{12} + \frac{16532022347575568834084}{35096451735005687331627733} a^{11} - \frac{17365752890670399026522}{35096451735005687331627733} a^{10} - \frac{23389042893094723797}{49923828926039384540011} a^{9} - \frac{216855768407000306368590204}{1298568714195210431270226121} a^{8} - \frac{2052622057693139922525}{25636560799858062331357} a^{7} + \frac{6447907439683022711485673}{35096451735005687331627733} a^{6} + \frac{6667809811944686953071}{25636560799858062331357} a^{5} - \frac{317981912049979049702323}{948552749594748306260209} a^{4} - \frac{170451672829038376866978}{948552749594748306260209} a^{3} + \frac{9073475495390872243641}{25636560799858062331357} a^{2} - \frac{118328769310409437529}{1508032988226944843021} a - \frac{2189142314493520136}{88707822836879108413}$
Class group and class number
$C_{9}$, which has order $9$ (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: | \( 299574682868233200000000000000000 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 27 |
| The 27 conjugacy class representatives for $C_{27}$ |
| Character table for $C_{27}$ is not computed |
Intermediate fields
| \(\Q(\zeta_{9})^+\), 9.9.80515213381214514081.3 |
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 | $27$ | R | $27$ | $27$ | $27$ | $27$ | ${\href{/LocalNumberField/17.1.0.1}{1} }^{27}$ | ${\href{/LocalNumberField/19.3.0.1}{3} }^{9}$ | $27$ | $27$ | $27$ | R | $27$ | $27$ | $27$ | ${\href{/LocalNumberField/53.9.0.1}{9} }^{3}$ | $27$ |
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 | ||||||
| $37$ | 37.9.8.6 | $x^{9} + 1184$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ |
| 37.9.8.6 | $x^{9} + 1184$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ | |
| 37.9.8.6 | $x^{9} + 1184$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ | |