Normalized defining polynomial
\( x^{27} - 3 x^{26} - 72 x^{25} + 194 x^{24} + 2136 x^{23} - 5166 x^{22} - 34511 x^{21} + 74439 x^{20} + 337788 x^{19} - 642199 x^{18} - 2109834 x^{17} + 3469977 x^{16} + 8634362 x^{15} - 11919870 x^{14} - 23358597 x^{13} + 25816709 x^{12} + 41360031 x^{11} - 34011447 x^{10} - 46156293 x^{9} + 25045044 x^{8} + 29760855 x^{7} - 8403372 x^{6} - 9106128 x^{5} + 587949 x^{4} + 751973 x^{3} + 31935 x^{2} - 14445 x - 1153 \)
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: | \(6504389611606252637488188857147585077648657433962663281=3^{36}\cdot 37^{24}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $107.18$ | 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: | \(333=3^{2}\cdot 37\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{333}(256,·)$, $\chi_{333}(1,·)$, $\chi_{333}(322,·)$, $\chi_{333}(70,·)$, $\chi_{333}(7,·)$, $\chi_{333}(10,·)$, $\chi_{333}(268,·)$, $\chi_{333}(271,·)$, $\chi_{333}(16,·)$, $\chi_{333}(145,·)$, $\chi_{333}(211,·)$, $\chi_{333}(238,·)$, $\chi_{333}(100,·)$, $\chi_{333}(157,·)$, $\chi_{333}(223,·)$, $\chi_{333}(160,·)$, $\chi_{333}(34,·)$, $\chi_{333}(292,·)$, $\chi_{333}(229,·)$, $\chi_{333}(232,·)$, $\chi_{333}(46,·)$, $\chi_{333}(112,·)$, $\chi_{333}(49,·)$, $\chi_{333}(181,·)$, $\chi_{333}(118,·)$, $\chi_{333}(121,·)$, $\chi_{333}(127,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{73} a^{23} - \frac{8}{73} a^{22} + \frac{31}{73} a^{21} + \frac{18}{73} a^{20} - \frac{27}{73} a^{19} + \frac{35}{73} a^{18} - \frac{11}{73} a^{17} + \frac{6}{73} a^{16} - \frac{13}{73} a^{15} + \frac{27}{73} a^{14} + \frac{15}{73} a^{13} - \frac{9}{73} a^{12} - \frac{1}{73} a^{11} - \frac{19}{73} a^{10} - \frac{13}{73} a^{9} + \frac{29}{73} a^{8} + \frac{5}{73} a^{7} - \frac{29}{73} a^{6} + \frac{3}{73} a^{5} + \frac{21}{73} a^{4} + \frac{9}{73} a^{3} + \frac{2}{73} a^{2} + \frac{32}{73} a - \frac{31}{73}$, $\frac{1}{17418311} a^{24} + \frac{111664}{17418311} a^{23} + \frac{5795207}{17418311} a^{22} + \frac{50508}{561881} a^{21} + \frac{8052279}{17418311} a^{20} - \frac{3473330}{17418311} a^{19} + \frac{997050}{17418311} a^{18} + \frac{7189025}{17418311} a^{17} - \frac{7527516}{17418311} a^{16} + \frac{8424242}{17418311} a^{15} + \frac{5409924}{17418311} a^{14} - \frac{146666}{561881} a^{13} - \frac{946576}{17418311} a^{12} + \frac{7798589}{17418311} a^{11} + \frac{7800817}{17418311} a^{10} + \frac{947292}{17418311} a^{9} - \frac{161140}{405077} a^{8} - \frac{2193696}{17418311} a^{7} - \frac{2390}{405077} a^{6} - \frac{6677270}{17418311} a^{5} - \frac{5952205}{17418311} a^{4} + \frac{1661393}{17418311} a^{3} - \frac{4259116}{17418311} a^{2} - \frac{6231376}{17418311} a - \frac{738932}{17418311}$, $\frac{1}{95922638677} a^{25} + \frac{1046}{95922638677} a^{24} - \frac{494166241}{95922638677} a^{23} - \frac{20290284908}{95922638677} a^{22} + \frac{14640047629}{95922638677} a^{21} + \frac{22840613031}{95922638677} a^{20} + \frac{15418779471}{95922638677} a^{19} + \frac{728656568}{3094278667} a^{18} + \frac{29900235617}{95922638677} a^{17} + \frac{3847836117}{95922638677} a^{16} - \frac{43249588302}{95922638677} a^{15} - \frac{40847411230}{95922638677} a^{14} + \frac{26977195760}{95922638677} a^{13} + \frac{199862693}{535880663} a^{12} + \frac{39674030435}{95922638677} a^{11} + \frac{33988115830}{95922638677} a^{10} - \frac{6689298570}{95922638677} a^{9} + \frac{37650161055}{95922638677} a^{8} - \frac{25907435453}{95922638677} a^{7} - \frac{27389196638}{95922638677} a^{6} + \frac{26669744802}{95922638677} a^{5} - \frac{38166840324}{95922638677} a^{4} + \frac{638926737}{1314008749} a^{3} + \frac{22883699575}{95922638677} a^{2} - \frac{759138237}{95922638677} a + \frac{9930641933}{95922638677}$, $\frac{1}{1144797407714753100747175828984851081158380549474139629} a^{26} - \frac{3077663034353784856234337225535405648490151}{1144797407714753100747175828984851081158380549474139629} a^{25} - \frac{16407973833761570074936597381113529835345723879}{1144797407714753100747175828984851081158380549474139629} a^{24} + \frac{3657508249583348424082515739720128644534793372288997}{1144797407714753100747175828984851081158380549474139629} a^{23} + \frac{139045532266746462544340119197960283155461511046759542}{1144797407714753100747175828984851081158380549474139629} a^{22} + \frac{533418967941066445176253369994890906482922327663404307}{1144797407714753100747175828984851081158380549474139629} a^{21} + \frac{450743751187144699327433625643330381376070061389501820}{1144797407714753100747175828984851081158380549474139629} a^{20} - \frac{84796845273873578212473044167904994150735417138122547}{1144797407714753100747175828984851081158380549474139629} a^{19} - \frac{157972278884200211098456226653559534316900536457677867}{1144797407714753100747175828984851081158380549474139629} a^{18} - \frac{532039999460441073337244916418813516179197743870000873}{1144797407714753100747175828984851081158380549474139629} a^{17} + \frac{190529666724389240511623738969593326531997192387886345}{1144797407714753100747175828984851081158380549474139629} a^{16} - \frac{393493874523146658233588444946471703961248711320549721}{1144797407714753100747175828984851081158380549474139629} a^{15} + \frac{552103313778725042555143918513287224386358263954403888}{1144797407714753100747175828984851081158380549474139629} a^{14} - \frac{180550252181574528383427686651050973776091145925693285}{1144797407714753100747175828984851081158380549474139629} a^{13} - \frac{260743816150656419836957317367060894147977698478238294}{1144797407714753100747175828984851081158380549474139629} a^{12} + \frac{175388685706992458471229242413424687488101490165735870}{1144797407714753100747175828984851081158380549474139629} a^{11} + \frac{371487589386260056453450452181983170077854532225788846}{1144797407714753100747175828984851081158380549474139629} a^{10} - \frac{16864024591250197004883970336670187031843849590412019}{1144797407714753100747175828984851081158380549474139629} a^{9} - \frac{197530256236451792126817074478377756188360778021650019}{1144797407714753100747175828984851081158380549474139629} a^{8} + \frac{488278474765786342619134558322012440907193472916585452}{1144797407714753100747175828984851081158380549474139629} a^{7} - \frac{273492066242828655300930925160947428230415250902248915}{1144797407714753100747175828984851081158380549474139629} a^{6} - \frac{2465958554860963692063529661098781532168267043088774}{6395516244216497769537295133993581458985366198179551} a^{5} - \frac{367944965738337269387718412144637854530701309941164238}{1144797407714753100747175828984851081158380549474139629} a^{4} + \frac{57680110012288191737555313728318765674133749751695151}{1144797407714753100747175828984851081158380549474139629} a^{3} + \frac{5865054345969788765524033813249911258284425134270435}{1144797407714753100747175828984851081158380549474139629} a^{2} + \frac{389691300272050397947133668048899803427553834580048460}{1144797407714753100747175828984851081158380549474139629} a + \frac{348787562689315338682953907830204305535403015432368458}{1144797407714753100747175828984851081158380549474139629}$
Class group and class number
Trivial group, which has order $1$ (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: | \( 6779568523744629000 \) (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
| \(\Q(\zeta_{9})^+\), 3.3.1369.1, 3.3.110889.2, 3.3.110889.1, 9.9.1363532208525369.2, 9.9.3512479453921.1, 9.9.1866675593471230161.2, 9.9.1866675593471230161.1 |
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 | ${\href{/LocalNumberField/5.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/7.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/11.3.0.1}{3} }^{9}$ | ${\href{/LocalNumberField/13.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/17.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/19.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/23.3.0.1}{3} }^{9}$ | ${\href{/LocalNumberField/29.3.0.1}{3} }^{9}$ | ${\href{/LocalNumberField/31.3.0.1}{3} }^{9}$ | R | ${\href{/LocalNumberField/41.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/43.3.0.1}{3} }^{9}$ | ${\href{/LocalNumberField/47.3.0.1}{3} }^{9}$ | ${\href{/LocalNumberField/53.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/59.9.0.1}{9} }^{3}$ |
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 | ||||||
| $37$ | 37.9.8.1 | $x^{9} - 37$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ |
| 37.9.8.1 | $x^{9} - 37$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ | |
| 37.9.8.1 | $x^{9} - 37$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ | |