Normalized defining polynomial
\( x^{27} - 234 x^{25} - 363 x^{24} + 21600 x^{23} + 58746 x^{22} - 1010232 x^{21} - 3754845 x^{20} + 25508145 x^{19} + 122869178 x^{18} - 334737288 x^{17} - 2243559090 x^{16} + 1661065767 x^{15} + 23417551341 x^{14} + 9070899120 x^{13} - 135972690394 x^{12} - 155269579194 x^{11} + 394444022466 x^{10} + 726143959982 x^{9} - 394126983513 x^{8} - 1382959296534 x^{7} - 230053380286 x^{6} + 1005649338285 x^{5} + 492843065832 x^{4} - 174788627574 x^{3} - 142021806498 x^{2} - 17497701342 x + 1193490397 \)
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: | \(1197336751300349460100016353165918520894496565611371829528951009=3^{36}\cdot 7^{18}\cdot 19^{24}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $216.89$ | 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: | $3, 7, 19$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1197=3^{2}\cdot 7\cdot 19\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1197}(64,·)$, $\chi_{1197}(1,·)$, $\chi_{1197}(1156,·)$, $\chi_{1197}(961,·)$, $\chi_{1197}(520,·)$, $\chi_{1197}(457,·)$, $\chi_{1197}(226,·)$, $\chi_{1197}(529,·)$, $\chi_{1197}(340,·)$, $\chi_{1197}(214,·)$, $\chi_{1197}(100,·)$, $\chi_{1197}(1051,·)$, $\chi_{1197}(1054,·)$, $\chi_{1197}(415,·)$, $\chi_{1197}(802,·)$, $\chi_{1197}(484,·)$, $\chi_{1197}(424,·)$, $\chi_{1197}(967,·)$, $\chi_{1197}(940,·)$, $\chi_{1197}(688,·)$, $\chi_{1197}(232,·)$, $\chi_{1197}(1075,·)$, $\chi_{1197}(310,·)$, $\chi_{1197}(841,·)$, $\chi_{1197}(505,·)$, $\chi_{1197}(634,·)$, $\chi_{1197}(571,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{11} a^{10} + \frac{1}{11} a^{9} + \frac{1}{11} a^{8} + \frac{1}{11} a^{7} + \frac{1}{11} a^{6} + \frac{1}{11} a^{5} + \frac{1}{11} a^{4} + \frac{1}{11} a^{3} + \frac{1}{11} a^{2} + \frac{1}{11} a$, $\frac{1}{11} a^{11} - \frac{1}{11} a$, $\frac{1}{11} a^{12} - \frac{1}{11} a^{2}$, $\frac{1}{11} a^{13} - \frac{1}{11} a^{3}$, $\frac{1}{11} a^{14} - \frac{1}{11} a^{4}$, $\frac{1}{11} a^{15} - \frac{1}{11} a^{5}$, $\frac{1}{11} a^{16} - \frac{1}{11} a^{6}$, $\frac{1}{11} a^{17} - \frac{1}{11} a^{7}$, $\frac{1}{605} a^{18} - \frac{12}{605} a^{17} - \frac{17}{605} a^{16} + \frac{23}{605} a^{15} - \frac{14}{605} a^{14} - \frac{8}{605} a^{13} + \frac{1}{605} a^{12} + \frac{2}{605} a^{11} + \frac{24}{605} a^{10} + \frac{46}{605} a^{9} - \frac{241}{605} a^{8} + \frac{124}{605} a^{7} - \frac{36}{605} a^{6} + \frac{188}{605} a^{5} + \frac{203}{605} a^{4} + \frac{241}{605} a^{3} + \frac{243}{605} a^{2} + \frac{4}{55} a + \frac{2}{5}$, $\frac{1}{605} a^{19} + \frac{4}{605} a^{17} - \frac{16}{605} a^{16} - \frac{13}{605} a^{15} - \frac{1}{55} a^{14} + \frac{3}{121} a^{13} + \frac{14}{605} a^{12} - \frac{7}{605} a^{11} + \frac{4}{605} a^{10} - \frac{19}{605} a^{9} - \frac{73}{605} a^{8} - \frac{23}{55} a^{7} - \frac{134}{605} a^{6} - \frac{16}{605} a^{5} - \frac{238}{605} a^{4} + \frac{5}{11} a^{3} + \frac{42}{121} a^{2} - \frac{2}{11} a - \frac{1}{5}$, $\frac{1}{6655} a^{20} - \frac{1}{1331} a^{18} - \frac{183}{6655} a^{17} - \frac{5}{1331} a^{16} - \frac{53}{6655} a^{15} - \frac{189}{6655} a^{14} + \frac{251}{6655} a^{13} - \frac{71}{6655} a^{12} - \frac{124}{6655} a^{11} - \frac{47}{1331} a^{10} + \frac{1933}{6655} a^{9} + \frac{101}{6655} a^{8} + \frac{652}{1331} a^{7} - \frac{232}{605} a^{6} - \frac{661}{1331} a^{5} + \frac{1198}{6655} a^{4} + \frac{3321}{6655} a^{3} - \frac{427}{6655} a^{2} + \frac{293}{605} a + \frac{7}{55}$, $\frac{1}{6655} a^{21} - \frac{1}{1331} a^{19} + \frac{4}{6655} a^{18} + \frac{151}{6655} a^{17} - \frac{207}{6655} a^{16} - \frac{123}{6655} a^{15} + \frac{53}{6655} a^{14} + \frac{248}{6655} a^{13} + \frac{63}{6655} a^{12} + \frac{139}{6655} a^{11} - \frac{234}{6655} a^{10} + \frac{2048}{6655} a^{9} - \frac{1877}{6655} a^{8} - \frac{159}{605} a^{7} + \frac{248}{6655} a^{6} + \frac{659}{6655} a^{5} - \frac{1068}{6655} a^{4} + \frac{579}{1331} a^{3} + \frac{189}{605} a^{2} + \frac{4}{11} a - \frac{1}{5}$, $\frac{1}{6655} a^{22} + \frac{4}{6655} a^{19} + \frac{1}{1331} a^{18} - \frac{5}{121} a^{17} - \frac{6}{6655} a^{16} + \frac{6}{1331} a^{15} - \frac{213}{6655} a^{14} - \frac{134}{6655} a^{13} + \frac{268}{6655} a^{12} + \frac{114}{6655} a^{11} - \frac{216}{6655} a^{10} - \frac{238}{605} a^{9} + \frac{3112}{6655} a^{8} - \frac{2691}{6655} a^{7} + \frac{508}{1331} a^{6} - \frac{1621}{6655} a^{5} + \frac{657}{6655} a^{4} + \frac{413}{6655} a^{3} - \frac{1288}{6655} a^{2} - \frac{59}{121} a + \frac{13}{55}$, $\frac{1}{73205} a^{23} + \frac{1}{73205} a^{21} + \frac{4}{73205} a^{20} - \frac{1}{6655} a^{19} - \frac{7}{73205} a^{18} + \frac{1773}{73205} a^{17} - \frac{254}{73205} a^{16} + \frac{1644}{73205} a^{15} + \frac{2999}{73205} a^{14} - \frac{2971}{73205} a^{13} - \frac{2738}{73205} a^{12} + \frac{48}{6655} a^{11} - \frac{401}{14641} a^{10} + \frac{5413}{73205} a^{9} - \frac{19594}{73205} a^{8} + \frac{2543}{14641} a^{7} - \frac{25738}{73205} a^{6} + \frac{3329}{73205} a^{5} - \frac{1957}{14641} a^{4} + \frac{4731}{73205} a^{3} - \frac{3174}{6655} a^{2} - \frac{71}{605} a + \frac{8}{55}$, $\frac{1}{3262304472185} a^{24} + \frac{15452317}{3262304472185} a^{23} - \frac{240538066}{3262304472185} a^{22} - \frac{136739642}{3262304472185} a^{21} - \frac{142527766}{3262304472185} a^{20} + \frac{2610359052}{3262304472185} a^{19} + \frac{1388458759}{3262304472185} a^{18} - \frac{4050419309}{652460894437} a^{17} + \frac{33948611646}{3262304472185} a^{16} + \frac{432621674}{3262304472185} a^{15} + \frac{142558191584}{3262304472185} a^{14} - \frac{492106628}{59314626767} a^{13} + \frac{141514201973}{3262304472185} a^{12} + \frac{69604951204}{3262304472185} a^{11} - \frac{17018860949}{3262304472185} a^{10} + \frac{1375224678826}{3262304472185} a^{9} + \frac{350023081547}{3262304472185} a^{8} - \frac{1050896128523}{3262304472185} a^{7} + \frac{10468336052}{3262304472185} a^{6} + \frac{38490194268}{652460894437} a^{5} - \frac{372412611664}{3262304472185} a^{4} + \frac{1064291766396}{3262304472185} a^{3} - \frac{38497097728}{296573133835} a^{2} + \frac{2492258216}{5392238797} a - \frac{484153793}{2451017635}$, $\frac{1}{35885349194035} a^{25} + \frac{4}{35885349194035} a^{24} - \frac{18459846}{35885349194035} a^{23} + \frac{2085956078}{35885349194035} a^{22} - \frac{1201248097}{35885349194035} a^{21} - \frac{456271489}{35885349194035} a^{20} - \frac{18864224823}{35885349194035} a^{19} + \frac{1357483913}{3262304472185} a^{18} + \frac{215969443882}{7177069838807} a^{17} + \frac{178206718056}{35885349194035} a^{16} - \frac{193999801675}{7177069838807} a^{15} + \frac{1480092855366}{35885349194035} a^{14} + \frac{6460967012}{7177069838807} a^{13} - \frac{1181098490321}{35885349194035} a^{12} - \frac{94454866871}{3262304472185} a^{11} + \frac{1106205576672}{35885349194035} a^{10} - \frac{14889974562376}{35885349194035} a^{9} - \frac{15623810260921}{35885349194035} a^{8} - \frac{13677108293384}{35885349194035} a^{7} - \frac{1186199215254}{35885349194035} a^{6} + \frac{6351179374106}{35885349194035} a^{5} - \frac{10657972156042}{35885349194035} a^{4} + \frac{3966524012232}{35885349194035} a^{3} + \frac{6897521094}{21047125627} a^{2} - \frac{90354216237}{296573133835} a + \frac{5846473966}{26961193985}$, $\frac{1}{563451109649880321367460912036114234331202689243565021666718437030586047667974494479820290213977535} a^{26} + \frac{1942788645589711650650865366133159871968382757962625969641653664263441013403063068042}{563451109649880321367460912036114234331202689243565021666718437030586047667974494479820290213977535} a^{25} + \frac{49754559591406002292987002201716983701981315286533717047573893370629098364285859402648}{563451109649880321367460912036114234331202689243565021666718437030586047667974494479820290213977535} a^{24} + \frac{3292504675075836613742412747808937104277168270721421561400568525121528950968813390112167999361}{563451109649880321367460912036114234331202689243565021666718437030586047667974494479820290213977535} a^{23} + \frac{18829585281011048719004724418510570759302073069832606373270821441024576687050352563997301965919}{563451109649880321367460912036114234331202689243565021666718437030586047667974494479820290213977535} a^{22} - \frac{5310827041573926265814343357612736531797874570674340059523121670840413384834780350305161189329}{112690221929976064273492182407222846866240537848713004333343687406117209533594898895964058042795507} a^{21} + \frac{13314489994636687654593634134322954400176537796695924946610330801196906644779022177100460560082}{563451109649880321367460912036114234331202689243565021666718437030586047667974494479820290213977535} a^{20} - \frac{76831549637601598390783644023460673634635481641897384503033870416164123725736050287693112264672}{563451109649880321367460912036114234331202689243565021666718437030586047667974494479820290213977535} a^{19} + \frac{39953626527205825632698473179680721839309709528966127132312239514733809903286619488841264130151}{563451109649880321367460912036114234331202689243565021666718437030586047667974494479820290213977535} a^{18} - \frac{3666737589838569302227912959095902646247247419999686806969976222937366385664146768225000497958090}{112690221929976064273492182407222846866240537848713004333343687406117209533594898895964058042795507} a^{17} - \frac{958233103889053535592885516392829333681923511281873011640626123537118979551372448380108754918517}{51222828149989120124314628366919475848291153567596820151519857911871458878906772225438208201270685} a^{16} + \frac{7217474155197122711165578583985223489787315760192640622827571913295760803731540920666004135993}{406823905884390123731018709051346017567655371294992795427233528541939384597815519480014649973991} a^{15} - \frac{1605638009259732034893840304119172241585856021765112931175298210223995937654620388071495732188872}{563451109649880321367460912036114234331202689243565021666718437030586047667974494479820290213977535} a^{14} - \frac{530027905678221915766116666963573508016598452684089747249920524835950659333250445165002013463947}{563451109649880321367460912036114234331202689243565021666718437030586047667974494479820290213977535} a^{13} - \frac{7537127851978587709360466801693476610610104148844497585097027491987780203351521592822194593937743}{563451109649880321367460912036114234331202689243565021666718437030586047667974494479820290213977535} a^{12} - \frac{2476064278695889193102889762967567814790072438775212709651049128312314134763000093059785173464124}{563451109649880321367460912036114234331202689243565021666718437030586047667974494479820290213977535} a^{11} - \frac{24811769928698940091218819035246791289289240408893094436662778401438352622192956331255690310114851}{563451109649880321367460912036114234331202689243565021666718437030586047667974494479820290213977535} a^{10} - \frac{107058205581683137744713666560293976966134978052635441057289388924419494173665685439904506904232558}{563451109649880321367460912036114234331202689243565021666718437030586047667974494479820290213977535} a^{9} + \frac{49589935767391901256643658132006435962203847718368090924947542487654589631527470245615692735808356}{563451109649880321367460912036114234331202689243565021666718437030586047667974494479820290213977535} a^{8} - \frac{13867231946500208392025881297082916475277086166904632101221184601409852925577826913757169901968962}{51222828149989120124314628366919475848291153567596820151519857911871458878906772225438208201270685} a^{7} + \frac{236749408650629725230427548923759096638553226425500294492656595203655703319351752562483988605082412}{563451109649880321367460912036114234331202689243565021666718437030586047667974494479820290213977535} a^{6} + \frac{194012642590511011620867728160187260552075456342717541255278075244904922216895815298963627043072422}{563451109649880321367460912036114234331202689243565021666718437030586047667974494479820290213977535} a^{5} + \frac{40458461241146246331556607964231923506989107398954599147233614237295442858305903536419683780877990}{112690221929976064273492182407222846866240537848713004333343687406117209533594898895964058042795507} a^{4} - \frac{106950324704484570867272575886953283626820678773162248085959036956075346170548571794900727301865452}{563451109649880321367460912036114234331202689243565021666718437030586047667974494479820290213977535} a^{3} + \frac{7454695313175156154602290593856304255388526557920585104920494222737929374443679807483825191245133}{51222828149989120124314628366919475848291153567596820151519857911871458878906772225438208201270685} a^{2} - \frac{1429969350856609568694348986849405063430556001174113855213192501288906141255278922857589175166044}{4656620740908101829483148033356315986208286687963347286501805264715587170809706565948928018297335} a + \frac{99760022323543278884129899400804518948995176637091226367881664337946869022297482038592765740054}{423329158264372893589377093941483271473480607996667935136527751337780651891791505995357092572485}$
Class group and class number
$C_{2}\times C_{6}$, which has order $12$ (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: | \( 67537284695182510000000 \) (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
| 3.3.1432809.2, 3.3.361.1, 3.3.3969.2, 3.3.1432809.3, 9.9.2941473244627851129.9, 9.9.1061871841310654257569.1, 9.9.1998099208210609.1, 9.9.9025761726072081.1 |
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 | ${\href{/LocalNumberField/2.9.0.1}{9} }^{3}$ | R | ${\href{/LocalNumberField/5.9.0.1}{9} }^{3}$ | R | ${\href{/LocalNumberField/11.1.0.1}{1} }^{27}$ | ${\href{/LocalNumberField/13.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/17.9.0.1}{9} }^{3}$ | R | ${\href{/LocalNumberField/23.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/29.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/31.3.0.1}{3} }^{9}$ | ${\href{/LocalNumberField/37.3.0.1}{3} }^{9}$ | ${\href{/LocalNumberField/41.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/43.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/47.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/53.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/59.9.0.1}{9} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
| 3 | Data not computed | ||||||
| $7$ | 7.9.6.1 | $x^{9} + 42 x^{6} + 539 x^{3} + 2744$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ |
| 7.9.6.1 | $x^{9} + 42 x^{6} + 539 x^{3} + 2744$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ | |
| 7.9.6.1 | $x^{9} + 42 x^{6} + 539 x^{3} + 2744$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ | |
| 19 | Data not computed | ||||||