Normalized defining polynomial
\( x^{27} - 6 x^{26} - 149 x^{25} + 788 x^{24} + 9832 x^{23} - 42520 x^{22} - 381437 x^{21} + 1216319 x^{20} + 9582638 x^{19} - 19409897 x^{18} - 160108066 x^{17} + 155008644 x^{16} + 1756031318 x^{15} - 149163963 x^{14} - 12079333253 x^{13} - 7562465658 x^{12} + 47644690593 x^{11} + 58813589752 x^{10} - 88610606092 x^{9} - 177046464592 x^{8} + 27034763677 x^{7} + 205863517381 x^{6} + 81470197594 x^{5} - 63004124267 x^{4} - 47812852922 x^{3} - 8296111176 x^{2} - 149023054 x + 24923387 \)
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: | \(12882515415852435459493331403759124565080304093640247033847689489=7^{18}\cdot 109^{24}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $236.83$ | 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: | $7, 109$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(763=7\cdot 109\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{763}(256,·)$, $\chi_{763}(1,·)$, $\chi_{763}(323,·)$, $\chi_{763}(452,·)$, $\chi_{763}(620,·)$, $\chi_{763}(263,·)$, $\chi_{763}(393,·)$, $\chi_{763}(590,·)$, $\chi_{763}(655,·)$, $\chi_{763}(16,·)$, $\chi_{763}(611,·)$, $\chi_{763}(214,·)$, $\chi_{763}(729,·)$, $\chi_{763}(281,·)$, $\chi_{763}(463,·)$, $\chi_{763}(284,·)$, $\chi_{763}(541,·)$, $\chi_{763}(354,·)$, $\chi_{763}(219,·)$, $\chi_{763}(681,·)$, $\chi_{763}(583,·)$, $\chi_{763}(172,·)$, $\chi_{763}(365,·)$, $\chi_{763}(561,·)$, $\chi_{763}(499,·)$, $\chi_{763}(372,·)$, $\chi_{763}(184,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{6}$, $\frac{1}{4} a^{14} - \frac{1}{4}$, $\frac{1}{4} a^{15} - \frac{1}{4} a$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{18} - \frac{1}{4} a^{4}$, $\frac{1}{164} a^{19} - \frac{19}{164} a^{18} - \frac{4}{41} a^{17} + \frac{3}{164} a^{16} + \frac{17}{164} a^{15} - \frac{7}{82} a^{14} - \frac{10}{41} a^{13} - \frac{7}{82} a^{12} - \frac{19}{82} a^{11} + \frac{17}{82} a^{10} + \frac{10}{41} a^{9} - \frac{5}{82} a^{8} - \frac{3}{82} a^{7} - \frac{14}{41} a^{6} + \frac{65}{164} a^{5} - \frac{19}{164} a^{4} + \frac{17}{82} a^{3} - \frac{19}{164} a^{2} + \frac{1}{4} a + \frac{12}{41}$, $\frac{1}{164} a^{20} - \frac{2}{41} a^{18} - \frac{7}{82} a^{17} - \frac{2}{41} a^{16} - \frac{19}{164} a^{15} - \frac{19}{164} a^{14} - \frac{9}{41} a^{13} + \frac{6}{41} a^{12} - \frac{8}{41} a^{11} + \frac{15}{82} a^{10} + \frac{3}{41} a^{9} - \frac{8}{41} a^{8} - \frac{3}{82} a^{7} + \frac{67}{164} a^{6} + \frac{17}{41} a^{5} - \frac{10}{41} a^{4} + \frac{3}{41} a^{3} + \frac{2}{41} a^{2} + \frac{7}{164} a - \frac{31}{164}$, $\frac{1}{328} a^{21} - \frac{1}{164} a^{18} + \frac{7}{82} a^{17} - \frac{9}{82} a^{16} - \frac{3}{164} a^{15} - \frac{25}{328} a^{14} + \frac{4}{41} a^{13} + \frac{5}{82} a^{12} - \frac{7}{82} a^{11} - \frac{11}{82} a^{10} - \frac{5}{41} a^{9} - \frac{1}{82} a^{8} + \frac{19}{328} a^{7} + \frac{14}{41} a^{6} - \frac{3}{82} a^{5} - \frac{29}{164} a^{4} - \frac{6}{41} a^{3} + \frac{15}{82} a^{2} + \frac{5}{164} a - \frac{67}{328}$, $\frac{1}{328} a^{22} - \frac{5}{164} a^{18} + \frac{7}{164} a^{17} + \frac{9}{328} a^{15} + \frac{1}{82} a^{14} - \frac{15}{82} a^{13} - \frac{7}{41} a^{12} + \frac{11}{82} a^{11} + \frac{7}{82} a^{10} + \frac{19}{82} a^{9} - \frac{1}{328} a^{8} - \frac{8}{41} a^{7} - \frac{31}{82} a^{6} + \frac{9}{41} a^{5} + \frac{39}{164} a^{4} + \frac{23}{164} a^{3} - \frac{7}{82} a^{2} + \frac{15}{328} a - \frac{17}{82}$, $\frac{1}{328} a^{23} - \frac{3}{82} a^{18} + \frac{1}{82} a^{17} + \frac{39}{328} a^{16} + \frac{5}{164} a^{15} - \frac{9}{82} a^{14} + \frac{9}{82} a^{13} + \frac{17}{82} a^{12} - \frac{3}{41} a^{11} - \frac{19}{82} a^{10} + \frac{71}{328} a^{9} - \frac{5}{82} a^{7} + \frac{1}{82} a^{6} - \frac{23}{82} a^{5} + \frac{5}{82} a^{4} - \frac{2}{41} a^{3} + \frac{153}{328} a^{2} + \frac{7}{164} a + \frac{19}{41}$, $\frac{1}{237063968} a^{24} + \frac{179125}{237063968} a^{23} - \frac{89723}{118531984} a^{22} - \frac{118111}{237063968} a^{21} - \frac{11821}{118531984} a^{20} + \frac{25293}{14816498} a^{19} + \frac{12704081}{118531984} a^{18} - \frac{5530641}{237063968} a^{17} + \frac{10207603}{237063968} a^{16} - \frac{13441797}{118531984} a^{15} - \frac{18621783}{237063968} a^{14} + \frac{1181327}{59265992} a^{13} + \frac{5920535}{59265992} a^{12} + \frac{791651}{7408249} a^{11} + \frac{4306959}{237063968} a^{10} - \frac{33766929}{237063968} a^{9} - \frac{18373161}{118531984} a^{8} - \frac{52866797}{237063968} a^{7} - \frac{46157145}{118531984} a^{6} + \frac{5309667}{59265992} a^{5} + \frac{25861855}{118531984} a^{4} + \frac{62550961}{237063968} a^{3} + \frac{84617545}{237063968} a^{2} + \frac{13882341}{118531984} a + \frac{25766219}{237063968}$, $\frac{1}{4504215392} a^{25} + \frac{1}{563026924} a^{24} - \frac{650475}{4504215392} a^{23} - \frac{1416517}{4504215392} a^{22} - \frac{1572043}{4504215392} a^{21} - \frac{3744459}{2252107696} a^{20} + \frac{1934869}{2252107696} a^{19} + \frac{500725773}{4504215392} a^{18} - \frac{11356634}{140756731} a^{17} + \frac{42404875}{4504215392} a^{16} + \frac{72063159}{4504215392} a^{15} - \frac{554525133}{4504215392} a^{14} + \frac{30431384}{140756731} a^{13} + \frac{253125701}{1126053848} a^{12} + \frac{310111807}{4504215392} a^{11} - \frac{19321}{11149048} a^{10} + \frac{574642463}{4504215392} a^{9} + \frac{228979025}{4504215392} a^{8} - \frac{20281895}{237063968} a^{7} - \frac{871796641}{2252107696} a^{6} - \frac{349512171}{2252107696} a^{5} - \frac{28532797}{4504215392} a^{4} - \frac{504906553}{1126053848} a^{3} + \frac{723473497}{4504215392} a^{2} - \frac{80699425}{237063968} a - \frac{2194302011}{4504215392}$, $\frac{1}{127621751571328455945164737216869288220448516805122187170778931274205078935183410149934552820896} a^{26} - \frac{2177395652620897865231986782063119775153819091606787991218416979909004147502277519}{315895424681506079072189943606112099555565635656243037551432998203477918156394579579045922824} a^{25} - \frac{1049399748898817270030180604173446855786219586399996401798179034605264795602131867883}{778181412020295463080272787907739562319808029299525531529139824842713895946240305792283858664} a^{24} + \frac{86316795822180461952254780029336982968289693078348768651294002305804067763392950482265112237}{63810875785664227972582368608434644110224258402561093585389465637102539467591705074967276410448} a^{23} + \frac{54587372161009286198406157302637231584489356414798352885323630670404428531259760507714094991}{127621751571328455945164737216869288220448516805122187170778931274205078935183410149934552820896} a^{22} + \frac{248180584436205604893512530646219127857602590330667797427442001477426010633009668209404261}{127621751571328455945164737216869288220448516805122187170778931274205078935183410149934552820896} a^{21} - \frac{59746443895174972317628246782018949249554562275666385452545062045826143204185800110156040403}{31905437892832113986291184304217322055112129201280546792694732818551269733795852537483638205224} a^{20} + \frac{217155505437799026419788251708449058982192241032064619370437818757557127886075635091227746237}{127621751571328455945164737216869288220448516805122187170778931274205078935183410149934552820896} a^{19} + \frac{5044441258973722100629335405169523911142283963734625078582233547399863648255720522269567791373}{63810875785664227972582368608434644110224258402561093585389465637102539467591705074967276410448} a^{18} + \frac{2659980737382936548132150322010504324672917233876401277150025370637819283926133393377332492121}{31905437892832113986291184304217322055112129201280546792694732818551269733795852537483638205224} a^{17} + \frac{1644166951703969075006279344264978588235782456017566381624008863612041722341092536753985450243}{15952718946416056993145592152108661027556064600640273396347366409275634866897926268741819102612} a^{16} - \frac{11546232224301118285503882050348192894714950464840069332667459743887023223947527587993952578943}{127621751571328455945164737216869288220448516805122187170778931274205078935183410149934552820896} a^{15} - \frac{13039685349394897402083677532665789659939402695139000974119200073154476642430366147941333999197}{127621751571328455945164737216869288220448516805122187170778931274205078935183410149934552820896} a^{14} + \frac{3212106442281318149642462463312267122779254865679682227328262925394115897185189523134465618697}{15952718946416056993145592152108661027556064600640273396347366409275634866897926268741819102612} a^{13} - \frac{6182137029762776565108499880725362100689213562360922628487951466537318353319682130670721773917}{127621751571328455945164737216869288220448516805122187170778931274205078935183410149934552820896} a^{12} + \frac{1020234201050106645166963539783282627431355135151421722200986186916109484571066132373534562747}{7976359473208028496572796076054330513778032300320136698173683204637817433448963134370909551306} a^{11} - \frac{1410508753660168737175018537006230375373682715740781544895473946006231982058467498592834199615}{7976359473208028496572796076054330513778032300320136698173683204637817433448963134370909551306} a^{10} + \frac{1195477780538491622399757066296659233130445215738579153950536190073089417389715729201735367327}{63810875785664227972582368608434644110224258402561093585389465637102539467591705074967276410448} a^{9} + \frac{20019244668160114971894796582350211368456897132315073695790702085615024365679161686767363625289}{127621751571328455945164737216869288220448516805122187170778931274205078935183410149934552820896} a^{8} - \frac{26929212363991561567665734695524090922577510507142838805316782221723144792503829073385177115729}{127621751571328455945164737216869288220448516805122187170778931274205078935183410149934552820896} a^{7} - \frac{122164110500375271575999711175068240185964944995726510271781218046288378655877011642753514837}{315895424681506079072189943606112099555565635656243037551432998203477918156394579579045922824} a^{6} + \frac{43068487376828678124875098763018749219990612991205519650589303840913504736064350543193638841719}{127621751571328455945164737216869288220448516805122187170778931274205078935183410149934552820896} a^{5} - \frac{4121897465768686880961677257759231454389941545792787185417138067437059055950359097272003014871}{63810875785664227972582368608434644110224258402561093585389465637102539467591705074967276410448} a^{4} + \frac{26732661198170281651321886951923583512862566214297157714617311278225881601852655688171625057}{97272676502536932885034098488467445289976003662440691441142478105339236993280038224035482333} a^{3} + \frac{942108504353551015548258662168246544535824103359966616889697375440092263846148889485460821876}{3988179736604014248286398038027165256889016150160068349086841602318908716724481567185454775653} a^{2} - \frac{60203271685667089110642979254001623758832716396242326275919711900718323322458556902995976937321}{127621751571328455945164737216869288220448516805122187170778931274205078935183410149934552820896} a + \frac{18838331698254941574046185345810721782046383084958081082956600564652312704833479931971918735113}{127621751571328455945164737216869288220448516805122187170778931274205078935183410149934552820896}$
Class group and class number
$C_{2}\times C_{2}$, which has order $4$ (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: | \( 293970223750363250000000 \) (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
| 3.3.582169.2, 3.3.11881.1, 3.3.582169.1, \(\Q(\zeta_{7})^+\), 9.9.197309150940332809.1, 9.9.2344230022322094103729.2, 9.9.19925626416901921.1, 9.9.2344230022322094103729.1 |
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 | ${\href{/LocalNumberField/2.3.0.1}{3} }^{9}$ | ${\href{/LocalNumberField/3.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/5.9.0.1}{9} }^{3}$ | R | ${\href{/LocalNumberField/11.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/13.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/17.3.0.1}{3} }^{9}$ | ${\href{/LocalNumberField/19.3.0.1}{3} }^{9}$ | ${\href{/LocalNumberField/23.3.0.1}{3} }^{9}$ | ${\href{/LocalNumberField/29.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/31.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/37.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/41.1.0.1}{1} }^{27}$ | ${\href{/LocalNumberField/43.3.0.1}{3} }^{9}$ | ${\href{/LocalNumberField/47.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/53.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/59.9.0.1}{9} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
| 7 | Data not computed | ||||||
| 109 | Data not computed | ||||||