Normalized defining polynomial
\( x^{22} + 20 x^{20} + 257 x^{18} - 4 x^{17} + 1934 x^{16} - 106 x^{15} + 10498 x^{14} - 1280 x^{13} + 38977 x^{12} - 9695 x^{11} + 107932 x^{10} - 38992 x^{9} + 209993 x^{8} - 108043 x^{7} + 304613 x^{6} - 168141 x^{5} + 295910 x^{4} - 171702 x^{3} + 186586 x^{2} - 69384 x + 31329 \)
Invariants
| Degree: | $22$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 11]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-174398723209634833558490883294218000698177947=-\,3^{11}\cdot 74843^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $102.56$ | 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: | $3, 74843$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{13} - \frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{14} - \frac{1}{3} a^{13} - \frac{1}{3} a^{12} - \frac{1}{3} a^{10} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{9} a^{17} - \frac{1}{9} a^{16} - \frac{1}{9} a^{15} - \frac{1}{3} a^{14} + \frac{1}{3} a^{13} - \frac{2}{9} a^{12} - \frac{1}{9} a^{11} - \frac{2}{9} a^{10} - \frac{1}{9} a^{9} - \frac{2}{9} a^{8} + \frac{2}{9} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{2}{9} a^{4} + \frac{1}{9} a^{3} + \frac{2}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{27} a^{18} + \frac{1}{27} a^{17} + \frac{4}{27} a^{15} + \frac{4}{9} a^{14} - \frac{8}{27} a^{13} + \frac{10}{27} a^{12} + \frac{5}{27} a^{11} - \frac{8}{27} a^{10} - \frac{4}{27} a^{9} + \frac{13}{27} a^{8} + \frac{1}{27} a^{7} + \frac{2}{9} a^{6} - \frac{11}{27} a^{5} - \frac{2}{9} a^{4} - \frac{5}{27} a^{3} + \frac{1}{27} a^{2} - \frac{4}{9} a + \frac{1}{3}$, $\frac{1}{27} a^{19} - \frac{1}{27} a^{17} + \frac{4}{27} a^{16} - \frac{1}{27} a^{15} + \frac{7}{27} a^{14} + \frac{4}{27} a^{12} - \frac{4}{27} a^{11} + \frac{4}{27} a^{10} - \frac{1}{27} a^{9} - \frac{4}{9} a^{8} - \frac{13}{27} a^{7} + \frac{10}{27} a^{6} - \frac{13}{27} a^{5} + \frac{10}{27} a^{4} - \frac{4}{9} a^{3} - \frac{13}{27} a^{2} + \frac{4}{9} a - \frac{1}{3}$, $\frac{1}{81} a^{20} - \frac{1}{81} a^{19} - \frac{1}{81} a^{18} - \frac{4}{81} a^{17} + \frac{4}{81} a^{16} - \frac{10}{81} a^{15} - \frac{7}{81} a^{14} - \frac{23}{81} a^{13} - \frac{17}{81} a^{12} + \frac{17}{81} a^{11} - \frac{14}{81} a^{10} + \frac{25}{81} a^{9} - \frac{10}{81} a^{8} + \frac{32}{81} a^{7} - \frac{23}{81} a^{6} - \frac{4}{81} a^{5} - \frac{31}{81} a^{4} - \frac{10}{81} a^{3} + \frac{7}{81} a^{2} + \frac{11}{27} a + \frac{4}{9}$, $\frac{1}{471808023562399806951227033399334309527739} a^{21} - \frac{6151805265667535752407984387143517349}{7996746162074572999173339549141259483521} a^{20} + \frac{303137647204878964783132449894686735804}{157269341187466602317075677799778103175913} a^{19} + \frac{25414532733490379509691730374254796}{888527351341619222130371061015695498169} a^{18} + \frac{12263509029388391339781579237356544504521}{471808023562399806951227033399334309527739} a^{17} + \frac{17620149356244351978352914230350596401962}{471808023562399806951227033399334309527739} a^{16} + \frac{8211416891672748170276463061916057931268}{157269341187466602317075677799778103175913} a^{15} - \frac{163898917002460679031883072230999234561055}{471808023562399806951227033399334309527739} a^{14} - \frac{23640871518643492850644059219026927069903}{157269341187466602317075677799778103175913} a^{13} + \frac{7026326966668249837536583822935241255918}{471808023562399806951227033399334309527739} a^{12} + \frac{187970693481581418351053294829045984749498}{471808023562399806951227033399334309527739} a^{11} - \frac{54412847489917067145856590633260673783953}{157269341187466602317075677799778103175913} a^{10} + \frac{161581961983645972057307751546284882566471}{471808023562399806951227033399334309527739} a^{9} - \frac{20006564153779071729945902692963194349714}{52423113729155534105691892599926034391971} a^{8} + \frac{64960739187047758084250992827535375004828}{471808023562399806951227033399334309527739} a^{7} + \frac{25180413528213883015205473766104548825961}{471808023562399806951227033399334309527739} a^{6} - \frac{3214193371912374462897319340069146598933}{52423113729155534105691892599926034391971} a^{5} - \frac{10924025574575169351555043714956924296795}{52423113729155534105691892599926034391971} a^{4} + \frac{214183039554181687580377430658487072584968}{471808023562399806951227033399334309527739} a^{3} - \frac{104976482323184395889017681350082868668783}{471808023562399806951227033399334309527739} a^{2} - \frac{76660932460861883840463590445903874327418}{157269341187466602317075677799778103175913} a + \frac{204996728260261065526762903819883370367}{888527351341619222130371061015695498169}$
Class group and class number
$C_{2}\times C_{2}\times C_{2}\times C_{3580}$, which has order $28640$ (assuming GRH)
Unit group
| Rank: | $10$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{780005801753936508236165602688951723}{52423113729155534105691892599926034391971} a^{21} + \frac{6337205511265447496471023452401072}{888527351341619222130371061015695498169} a^{20} - \frac{579150767410917296966095259549324686}{1941596804783538300210810837034297570073} a^{19} + \frac{38791800808653086907082054788860948}{296175783780539740710123687005231832723} a^{18} - \frac{201003393884562823128175217983624500442}{52423113729155534105691892599926034391971} a^{17} + \frac{87683167843314358486370788595419227421}{52423113729155534105691892599926034391971} a^{16} - \frac{505348365938842051441353078753720209605}{17474371243051844701897297533308678130657} a^{15} + \frac{661567441580962534478140395559403700497}{52423113729155534105691892599926034391971} a^{14} - \frac{305735674019136297331803945423854310474}{1941596804783538300210810837034297570073} a^{13} + \frac{3902168073043082439950450579515045432666}{52423113729155534105691892599926034391971} a^{12} - \frac{30928744070292807923050535040752721376741}{52423113729155534105691892599926034391971} a^{11} + \frac{623099160909762991546498494824359824722}{1941596804783538300210810837034297570073} a^{10} - \frac{87170703265341430107894588869060393016407}{52423113729155534105691892599926034391971} a^{9} + \frac{17525115536459106503230473553349989884763}{17474371243051844701897297533308678130657} a^{8} - \frac{173089446315676651155989703666458775375391}{52423113729155534105691892599926034391971} a^{7} + \frac{114470001652877009473252300532124887057836}{52423113729155534105691892599926034391971} a^{6} - \frac{85860824872185235359320214468871146620024}{17474371243051844701897297533308678130657} a^{5} + \frac{53661791506390092531761056676391254168401}{17474371243051844701897297533308678130657} a^{4} - \frac{244846484586629283522955448580948911433871}{52423113729155534105691892599926034391971} a^{3} + \frac{113620174147161104568687308718760886178032}{52423113729155534105691892599926034391971} a^{2} - \frac{49436359883918823306859676862328154986415}{17474371243051844701897297533308678130657} a + \frac{104421662751167631560332390247769921247}{98725261260179913570041229001743944241} \) (order $6$) | 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: | \( 3049992559.69 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 1320 |
| The 16 conjugacy class representatives for t22n13 |
| Character table for t22n13 |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 11.11.31376518243389673201.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 24 sibling: | data not computed |
| Arithmetically equvalently siblings: | data not computed |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $22$ | R | ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ | $22$ | ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ | $22$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ | $22$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ | ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ | $22$ | ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ | $22$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{11}$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.6.3.2 | $x^{6} - 9 x^{2} + 27$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 3.6.3.2 | $x^{6} - 9 x^{2} + 27$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 3.6.3.2 | $x^{6} - 9 x^{2} + 27$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 74843 | Data not computed | ||||||