Normalized defining polynomial
\( x^{27} - 513 x^{25} + 116964 x^{23} - 15617943 x^{21} + 1354686795 x^{19} - 644328 x^{18} - 80025043581 x^{17} + 220360176 x^{16} + 3282297025608 x^{15} - 31401325080 x^{14} - 93545465229828 x^{13} + 2413017380592 x^{12} + 1824136571981646 x^{11} - 108068706973656 x^{10} - 23533855837754592 x^{9} + 2843038291153104 x^{8} + 189417763194150150 x^{7} - 42013788080373648 x^{6} - 861008849860765383 x^{5} + 311011158517051680 x^{4} + 1862509228623929799 x^{3} - 886381801773597288 x^{2} - 1419315119960781051 x + 851940306004555639 \)
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: | \(3463254175600113063839837232871102966170246194081956872095753304432278349849=3^{94}\cdot 19^{24}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $627.71$ | 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: | $3, 19$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1539=3^{4}\cdot 19\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1539}(64,·)$, $\chi_{1539}(1,·)$, $\chi_{1539}(514,·)$, $\chi_{1539}(1027,·)$, $\chi_{1539}(577,·)$, $\chi_{1539}(1090,·)$, $\chi_{1539}(85,·)$, $\chi_{1539}(598,·)$, $\chi_{1539}(1111,·)$, $\chi_{1539}(346,·)$, $\chi_{1539}(859,·)$, $\chi_{1539}(1372,·)$, $\chi_{1539}(1531,·)$, $\chi_{1539}(169,·)$, $\chi_{1539}(682,·)$, $\chi_{1539}(43,·)$, $\chi_{1539}(556,·)$, $\chi_{1539}(1069,·)$, $\chi_{1539}(1195,·)$, $\chi_{1539}(310,·)$, $\chi_{1539}(823,·)$, $\chi_{1539}(1336,·)$, $\chi_{1539}(505,·)$, $\chi_{1539}(1018,·)$, $\chi_{1539}(187,·)$, $\chi_{1539}(700,·)$, $\chi_{1539}(1213,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{19} a^{9}$, $\frac{1}{19} a^{10}$, $\frac{1}{19} a^{11}$, $\frac{1}{19} a^{12}$, $\frac{1}{19} a^{13}$, $\frac{1}{25631} a^{14} + \frac{21}{1349} a^{13} - \frac{14}{1349} a^{12} - \frac{4}{1349} a^{11} - \frac{28}{1349} a^{10} + \frac{25}{1349} a^{9} - \frac{14}{71} a^{8} + \frac{11}{71} a^{7} - \frac{11}{71} a^{6} + \frac{592}{1349} a^{5} + \frac{12}{71} a^{4} - \frac{5}{71} a^{3} + \frac{34}{71} a^{2} + \frac{12}{71} a + \frac{15}{71}$, $\frac{1}{25631} a^{15} - \frac{15}{1349} a^{13} - \frac{27}{1349} a^{12} + \frac{6}{1349} a^{11} - \frac{21}{1349} a^{10} - \frac{17}{1349} a^{9} - \frac{12}{71} a^{8} + \frac{2}{71} a^{7} + \frac{345}{1349} a^{6} + \frac{5}{71} a^{5} + \frac{35}{71} a^{4} - \frac{30}{71} a^{3} + \frac{7}{71} a^{2} - \frac{16}{71} a - \frac{21}{71}$, $\frac{1}{25631} a^{16} - \frac{6}{1349} a^{13} - \frac{8}{1349} a^{12} - \frac{25}{1349} a^{11} + \frac{26}{1349} a^{10} + \frac{10}{1349} a^{9} - \frac{12}{71} a^{8} + \frac{554}{1349} a^{7} - \frac{6}{71} a^{6} - \frac{31}{71} a^{5} - \frac{18}{71} a^{4} + \frac{2}{71} a^{3} + \frac{18}{71} a^{2} - \frac{9}{71} a + \frac{15}{71}$, $\frac{1}{25631} a^{17} - \frac{28}{1349} a^{13} + \frac{12}{1349} a^{12} - \frac{4}{1349} a^{11} + \frac{13}{1349} a^{10} - \frac{5}{1349} a^{9} - \frac{92}{1349} a^{8} - \frac{30}{71} a^{7} - \frac{7}{71} a^{6} - \frac{16}{71} a^{5} + \frac{21}{71} a^{4} + \frac{16}{71} a^{3} + \frac{33}{71} a^{2} + \frac{34}{71} a + \frac{6}{71}$, $\frac{1}{5752309428455797} a^{18} - \frac{18}{302753127813463} a^{16} + \frac{135}{15934375148077} a^{14} - \frac{546}{838651323583} a^{12} + \frac{24453}{838651323583} a^{10} + \frac{95774828}{17809007518439} a^{9} - \frac{643302}{838651323583} a^{8} - \frac{861973452}{937316185181} a^{7} + \frac{9506574}{838651323583} a^{6} + \frac{2585920356}{49332430799} a^{5} - \frac{70373340}{838651323583} a^{4} - \frac{5259221161}{49332430799} a^{3} + \frac{200564019}{838651323583} a^{2} + \frac{15177831378}{49332430799} a - \frac{130163161159}{838651323583}$, $\frac{1}{615497108844770279} a^{19} + \frac{35435971401}{32394584676040541} a^{17} - \frac{70871940273}{32394584676040541} a^{15} - \frac{767779391119}{1704978140844239} a^{13} - \frac{30}{1349} a^{12} - \frac{22371909491255}{1704978140844239} a^{11} + \frac{13426160004815}{1905563804472973} a^{10} + \frac{32057729920984}{1704978140844239} a^{9} + \frac{37313215930755}{100292831814367} a^{8} + \frac{21438772215069}{89735691623381} a^{7} + \frac{9581408798475}{100292831814367} a^{6} + \frac{7465107605596}{89735691623381} a^{5} - \frac{354060351599}{5278570095493} a^{4} + \frac{29530176746519}{89735691623381} a^{3} - \frac{2115844214545}{5278570095493} a^{2} + \frac{36593315219398}{89735691623381} a + \frac{783}{7597}$, $\frac{1}{615497108844770279} a^{20} + \frac{48}{615497108844770279} a^{18} - \frac{590599528105}{32394584676040541} a^{16} + \frac{70872746082}{32394584676040541} a^{14} - \frac{460671209700}{1704978140844239} a^{12} + \frac{2125559236999}{1905563804472973} a^{11} + \frac{1559191635804}{89735691623381} a^{10} - \frac{43029561648939}{1905563804472973} a^{9} - \frac{34782673514}{292298669783} a^{8} + \frac{991069579450}{5278570095493} a^{7} - \frac{2264264980296}{89735691623381} a^{6} - \frac{21745208333469}{100292831814367} a^{5} + \frac{21517825584171}{89735691623381} a^{4} - \frac{2619569715372}{5278570095493} a^{3} - \frac{35109930917613}{89735691623381} a^{2} - \frac{1216477654210}{5278570095493} a - \frac{186743791396}{838651323583}$, $\frac{1}{11694445068050635301} a^{21} - \frac{519727581001}{32394584676040541} a^{17} + \frac{23623982830}{1704978140844239} a^{15} - \frac{9792140264396}{1704978140844239} a^{13} - \frac{625057783376789}{36205712284986487} a^{12} - \frac{23742093131106}{1704978140844239} a^{11} - \frac{21834644650342}{1905563804472973} a^{10} + \frac{42771007000023}{1704978140844239} a^{9} - \frac{31339717982114}{100292831814367} a^{8} + \frac{29069475968525}{89735691623381} a^{7} - \frac{52212248154}{5278570095493} a^{6} + \frac{522432980681}{5278570095493} a^{5} - \frac{1966817315052}{5278570095493} a^{4} - \frac{20578997718262}{1704978140844239} a^{3} - \frac{106867199239}{5278570095493} a^{2} + \frac{25639668529675}{89735691623381} a + \frac{145}{7597}$, $\frac{1}{11694445068050635301} a^{22} - \frac{2}{615497108844770279} a^{18} - \frac{354359742996}{32394584676040541} a^{16} + \frac{1243384378}{89735691623381} a^{14} - \frac{7762466434840}{36205712284986487} a^{13} - \frac{5965084701527}{1704978140844239} a^{12} + \frac{21955183324945}{1905563804472973} a^{11} - \frac{125270033034}{13425024731057} a^{10} - \frac{4443289501512}{1905563804472973} a^{9} + \frac{5514058154485}{89735691623381} a^{8} - \frac{29851775593581}{100292831814367} a^{7} + \frac{33283616111462}{89735691623381} a^{6} + \frac{1939595177417}{5278570095493} a^{5} + \frac{51109638976}{13425024731057} a^{4} - \frac{2397482499007}{5278570095493} a^{3} - \frac{18738686832048}{89735691623381} a^{2} - \frac{2395938728101}{5278570095493} a - \frac{167274203267}{838651323583}$, $\frac{1}{15775806396800307021049} a^{23} + \frac{21}{830305599831595106371} a^{22} - \frac{23}{830305599831595106371} a^{21} + \frac{35}{43700294727978689809} a^{20} + \frac{1}{2570605572234040577} a^{19} + \frac{7}{344096808881721967} a^{18} - \frac{7547861909724}{2300015511998878411} a^{17} - \frac{24320889131276}{2300015511998878411} a^{16} + \frac{15095723367297}{2300015511998878411} a^{15} + \frac{5991382553840387}{830305599831595106371} a^{14} + \frac{667443206741717404}{43700294727978689809} a^{13} - \frac{150484527872455043}{43700294727978689809} a^{12} - \frac{14149362186453952}{2300015511998878411} a^{11} - \frac{32601265896732028}{2300015511998878411} a^{10} + \frac{29636454631141615}{2300015511998878411} a^{9} + \frac{25298852033454122}{121053447999940969} a^{8} - \frac{38072486489356772}{121053447999940969} a^{7} - \frac{1127930850661315}{121053447999940969} a^{6} - \frac{752004377163124921}{2300015511998878411} a^{5} - \frac{42012522564274411}{121053447999940969} a^{4} + \frac{11319242776374593}{121053447999940969} a^{3} - \frac{133492776549786}{374778476780003} a^{2} - \frac{2313992564953175}{6371234105260051} a + \frac{979565643527114}{6371234105260051}$, $\frac{1}{15775806396800307021049} a^{24} - \frac{24}{830305599831595106371} a^{22} - \frac{27}{830305599831595106371} a^{21} - \frac{32}{43700294727978689809} a^{20} - \frac{1}{43700294727978689809} a^{19} + \frac{10}{142346236898953387} a^{18} - \frac{20127631760871}{2300015511998878411} a^{17} - \frac{11741117974952}{2300015511998878411} a^{16} + \frac{1450029416817506}{830305599831595106371} a^{15} - \frac{20127748271826}{2300015511998878411} a^{14} + \frac{742398449686952794}{43700294727978689809} a^{13} - \frac{549592377507979722}{43700294727978689809} a^{12} - \frac{13592341835687070}{2300015511998878411} a^{11} + \frac{14945533628200828}{2300015511998878411} a^{10} + \frac{28742032087581550}{2300015511998878411} a^{9} + \frac{27194095827707697}{121053447999940969} a^{8} - \frac{57957946118632005}{121053447999940969} a^{7} - \frac{588202809440983616}{2300015511998878411} a^{6} + \frac{31933552646015127}{121053447999940969} a^{5} + \frac{7573686413460265}{121053447999940969} a^{4} + \frac{45342156624010343}{121053447999940969} a^{3} - \frac{199387281934886}{6371234105260051} a^{2} - \frac{3080412214445355}{6371234105260051} a - \frac{912560619220086}{6371234105260051}$, $\frac{1}{299740321539205833399931} a^{25} + \frac{28}{830305599831595106371} a^{22} - \frac{16}{830305599831595106371} a^{21} - \frac{27}{43700294727978689809} a^{20} + \frac{24}{43700294727978689809} a^{19} + \frac{1323}{43700294727978689809} a^{18} - \frac{19288980450167}{2300015511998878411} a^{17} + \frac{219129523627051033}{15775806396800307021049} a^{16} - \frac{39416610380297}{2300015511998878411} a^{15} - \frac{8747900768940210}{830305599831595106371} a^{14} - \frac{855851181369132676}{43700294727978689809} a^{13} + \frac{45859120541697813}{2300015511998878411} a^{12} - \frac{24634168517602762}{2300015511998878411} a^{11} + \frac{7490117468921213}{2300015511998878411} a^{10} - \frac{16881156705687759}{2300015511998878411} a^{9} + \frac{1424510048844176}{6371234105260051} a^{8} + \frac{16989314292268292882}{43700294727978689809} a^{7} + \frac{31909549017268932}{121053447999940969} a^{6} + \frac{175424434335400775}{2300015511998878411} a^{5} - \frac{16730829004265818}{121053447999940969} a^{4} - \frac{14083028844542663}{121053447999940969} a^{3} + \frac{1435714354368838}{6371234105260051} a^{2} - \frac{915407203602058}{6371234105260051} a + \frac{868208090098496}{6371234105260051}$, $\frac{1}{299740321539205833399931} a^{26} + \frac{30}{830305599831595106371} a^{22} + \frac{8}{830305599831595106371} a^{21} + \frac{6}{43700294727978689809} a^{20} + \frac{18}{43700294727978689809} a^{19} - \frac{165}{43700294727978689809} a^{18} + \frac{370239988180551}{927988611576488648297} a^{17} - \frac{44448522234155}{2300015511998878411} a^{16} - \frac{15408525994595557}{830305599831595106371} a^{15} + \frac{16773495889574}{2300015511998878411} a^{14} - \frac{8091967522657492}{615497108844770279} a^{13} + \frac{642234154897013633}{43700294727978689809} a^{12} - \frac{5974495481780709}{2300015511998878411} a^{11} + \frac{38244909446717838}{2300015511998878411} a^{10} + \frac{28150893891331437}{2300015511998878411} a^{9} + \frac{562745099224064973}{2570605572234040577} a^{8} + \frac{53519347133091388}{121053447999940969} a^{7} - \frac{981503597404758627}{2300015511998878411} a^{6} + \frac{46652573821762958}{121053447999940969} a^{5} + \frac{48364533802352549}{121053447999940969} a^{4} - \frac{49938013858344491}{121053447999940969} a^{3} + \frac{34617893852923}{89735691623381} a^{2} - \frac{1429647918001797}{6371234105260051} a - \frac{1974414079203513}{6371234105260051}$
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: | \( 37392701741628760000000000000 \) (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
| \(\Q(\zeta_{9})^+\), 9.9.1476349596018920529.1 |
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 | $27$ | $27$ | $27$ | $27$ | ${\href{/LocalNumberField/17.3.0.1}{3} }^{9}$ | R | $27$ | $27$ | $27$ | ${\href{/LocalNumberField/37.9.0.1}{9} }^{3}$ | $27$ | $27$ | $27$ | ${\href{/LocalNumberField/53.9.0.1}{9} }^{3}$ | $27$ |
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 | ||||||
| $19$ | 19.9.8.2 | $x^{9} - 304$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ |
| 19.9.8.2 | $x^{9} - 304$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ | |
| 19.9.8.2 | $x^{9} - 304$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ | |