Normalized defining polynomial
\( x^{22} - x^{21} + 41 x^{20} + 2 x^{19} + 1137 x^{18} + 120 x^{17} + 15187 x^{16} + 4521 x^{15} + 143572 x^{14} + 41908 x^{13} + 810728 x^{12} + 310631 x^{11} + 3278352 x^{10} + 1130011 x^{9} + 7018057 x^{8} + 3087160 x^{7} + 10139143 x^{6} + 2471155 x^{5} + 430399 x^{4} + 38213 x^{3} + 2468 x^{2} + 57 x + 1 \)
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: | \(-172239967478675757728235268038638014589675547=-\,3^{11}\cdot 89^{20}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $102.50$ | 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: | $3, 89$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(267=3\cdot 89\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{267}(128,·)$, $\chi_{267}(1,·)$, $\chi_{267}(2,·)$, $\chi_{267}(67,·)$, $\chi_{267}(4,·)$, $\chi_{267}(134,·)$, $\chi_{267}(8,·)$, $\chi_{267}(194,·)$, $\chi_{267}(256,·)$, $\chi_{267}(16,·)$, $\chi_{267}(217,·)$, $\chi_{267}(91,·)$, $\chi_{267}(223,·)$, $\chi_{267}(32,·)$, $\chi_{267}(97,·)$, $\chi_{267}(167,·)$, $\chi_{267}(64,·)$, $\chi_{267}(242,·)$, $\chi_{267}(179,·)$, $\chi_{267}(245,·)$, $\chi_{267}(182,·)$, $\chi_{267}(121,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{37} a^{16} - \frac{16}{37} a^{14} - \frac{13}{37} a^{13} + \frac{5}{37} a^{12} - \frac{5}{37} a^{11} + \frac{2}{37} a^{10} + \frac{15}{37} a^{9} + \frac{9}{37} a^{8} + \frac{3}{37} a^{7} - \frac{18}{37} a^{6} + \frac{15}{37} a^{5} - \frac{11}{37} a^{4} - \frac{3}{37} a^{2} + \frac{18}{37} a - \frac{2}{37}$, $\frac{1}{37} a^{17} - \frac{16}{37} a^{15} - \frac{13}{37} a^{14} + \frac{5}{37} a^{13} - \frac{5}{37} a^{12} + \frac{2}{37} a^{11} + \frac{15}{37} a^{10} + \frac{9}{37} a^{9} + \frac{3}{37} a^{8} - \frac{18}{37} a^{7} + \frac{15}{37} a^{6} - \frac{11}{37} a^{5} - \frac{3}{37} a^{3} + \frac{18}{37} a^{2} - \frac{2}{37} a$, $\frac{1}{37} a^{18} - \frac{13}{37} a^{15} + \frac{8}{37} a^{14} + \frac{9}{37} a^{13} + \frac{8}{37} a^{12} + \frac{9}{37} a^{11} + \frac{4}{37} a^{10} - \frac{16}{37} a^{9} + \frac{15}{37} a^{8} - \frac{11}{37} a^{7} - \frac{3}{37} a^{6} + \frac{18}{37} a^{5} + \frac{6}{37} a^{4} + \frac{18}{37} a^{3} - \frac{13}{37} a^{2} - \frac{8}{37} a + \frac{5}{37}$, $\frac{1}{37} a^{19} + \frac{8}{37} a^{15} - \frac{14}{37} a^{14} - \frac{13}{37} a^{13} + \frac{13}{37} a^{11} + \frac{10}{37} a^{10} - \frac{12}{37} a^{9} - \frac{5}{37} a^{8} - \frac{1}{37} a^{7} + \frac{6}{37} a^{6} + \frac{16}{37} a^{5} - \frac{14}{37} a^{4} - \frac{13}{37} a^{3} - \frac{10}{37} a^{2} + \frac{17}{37} a + \frac{11}{37}$, $\frac{1}{677618999} a^{20} + \frac{8725065}{677618999} a^{19} - \frac{8725025}{677618999} a^{18} + \frac{1484012}{677618999} a^{17} + \frac{5782310}{677618999} a^{16} - \frac{251964725}{677618999} a^{15} - \frac{291121836}{677618999} a^{14} - \frac{269975762}{677618999} a^{13} + \frac{108978411}{677618999} a^{12} + \frac{257147335}{677618999} a^{11} - \frac{334431280}{677618999} a^{10} + \frac{119144966}{677618999} a^{9} + \frac{123475180}{677618999} a^{8} + \frac{301003860}{677618999} a^{7} - \frac{312011}{18314027} a^{6} - \frac{305751806}{677618999} a^{5} - \frac{174722583}{677618999} a^{4} + \frac{245688669}{677618999} a^{3} + \frac{176777734}{677618999} a^{2} + \frac{325792058}{677618999} a + \frac{17852831}{677618999}$, $\frac{1}{1431402635546355341010628913986304713812742282995233291} a^{21} - \frac{14455656087866041995712211891222207814617540}{38686557717469063270557538215846073346290331972844143} a^{20} - \frac{2739654781010152897839188660175155725602264757822158}{1431402635546355341010628913986304713812742282995233291} a^{19} + \frac{4328700293765315464302064376660377026559618806186674}{1431402635546355341010628913986304713812742282995233291} a^{18} - \frac{253212604315421609420602710893150963358346540655722}{1431402635546355341010628913986304713812742282995233291} a^{17} + \frac{2964341033172080296941058871531909928561400063920778}{1431402635546355341010628913986304713812742282995233291} a^{16} + \frac{338512118305658304683016601997016727667196503442385916}{1431402635546355341010628913986304713812742282995233291} a^{15} - \frac{6644273343983167747902861740225293891983941899884443}{14172303322241141990204246673131729839730121613814191} a^{14} - \frac{30141186410116912992417794585256352899853802503776999}{1431402635546355341010628913986304713812742282995233291} a^{13} - \frac{221547065009198839829313983058166304815897725391254091}{1431402635546355341010628913986304713812742282995233291} a^{12} - \frac{642141801718510365563669938588642383254255806470896152}{1431402635546355341010628913986304713812742282995233291} a^{11} + \frac{262217159348075379110682605302065921728842994660826881}{1431402635546355341010628913986304713812742282995233291} a^{10} - \frac{484301032747408903570645947277215897456242569358169332}{1431402635546355341010628913986304713812742282995233291} a^{9} + \frac{377857679079550695343596619237991070468313894485283481}{1431402635546355341010628913986304713812742282995233291} a^{8} + \frac{420270372638783765312091163024571981072252633775281771}{1431402635546355341010628913986304713812742282995233291} a^{7} - \frac{355556274175537259176499511853388971630309990960307956}{1431402635546355341010628913986304713812742282995233291} a^{6} + \frac{706032880040586976284449475694073101840932177007760704}{1431402635546355341010628913986304713812742282995233291} a^{5} + \frac{714076173370085184918544221864118068054128629828620234}{1431402635546355341010628913986304713812742282995233291} a^{4} + \frac{124729345550843324111251992756875090314724091671322544}{1431402635546355341010628913986304713812742282995233291} a^{3} + \frac{178419308321328370498069130610763448847072021020383273}{1431402635546355341010628913986304713812742282995233291} a^{2} + \frac{699699776346572599770176276616673452069346265395830522}{1431402635546355341010628913986304713812742282995233291} a - \frac{455458692192623861463183658503677333790873950816899083}{1431402635546355341010628913986304713812742282995233291}$
Class group and class number
$C_{26687}$, which has order $26687$ (assuming GRH)
Unit group
| Rank: | $10$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{2082286119811400686673182217458871706156402892}{78158814309182537571372419292944403424366595233} a^{21} + \frac{57683604138671407861421121966735188086948354}{2112400386734663177604659980890389281739637709} a^{20} - \frac{85429537839634429379349219956531887651266580946}{78158814309182537571372419292944403424366595233} a^{19} - \frac{2028194954856803300455009812801688974075048804}{78158814309182537571372419292944403424366595233} a^{18} - \frac{2367611424467478235238356863615047781719681470579}{78158814309182537571372419292944403424366595233} a^{17} - \frac{190738136300728599339667211923660556652865472668}{78158814309182537571372419292944403424366595233} a^{16} - \frac{31621760266270877738471642153126218504333875463876}{78158814309182537571372419292944403424366595233} a^{15} - \frac{8624318256260148820472237415037929449045032238551}{78158814309182537571372419292944403424366595233} a^{14} - \frac{298780589818760219781950368196905963443180817146984}{78158814309182537571372419292944403424366595233} a^{13} - \frac{79810581303879851110042193764949783607789712774030}{78158814309182537571372419292944403424366595233} a^{12} - \frac{1686533196985529575547644367301890856167643103335958}{78158814309182537571372419292944403424366595233} a^{11} - \frac{604778018193833757099335514472262560053218777992570}{78158814309182537571372419292944403424366595233} a^{10} - \frac{6813388077595533375888158968318948151927367080936306}{78158814309182537571372419292944403424366595233} a^{9} - \frac{2183462727494610907131055824125188381450033964044075}{78158814309182537571372419292944403424366595233} a^{8} - \frac{14567245929183247435450871160057877114044113193688586}{78158814309182537571372419292944403424366595233} a^{7} - \frac{6066747523884033041352860517018363332321892775783376}{78158814309182537571372419292944403424366595233} a^{6} - \frac{20978563123909311728581505228572755740867044722610942}{78158814309182537571372419292944403424366595233} a^{5} - \frac{4628150493436084432997131632962821204882910448326376}{78158814309182537571372419292944403424366595233} a^{4} - \frac{805721582240251007790751065441636388607304977423770}{78158814309182537571372419292944403424366595233} a^{3} - \frac{63836035181700145034628814133912692933108544122946}{78158814309182537571372419292944403424366595233} a^{2} - \frac{4619361989575845689866566106333599312178590149410}{78158814309182537571372419292944403424366595233} a - \frac{28527008476144232141858770825719404474429016585}{78158814309182537571372419292944403424366595233} \) (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: | \( 866679281.3791491 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 22 |
| The 22 conjugacy class representatives for $C_{22}$ |
| Character table for $C_{22}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 11.11.31181719929966183601.1 |
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 | $22$ | R | $22$ | ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ | $22$ | ${\href{/LocalNumberField/13.11.0.1}{11} }^{2}$ | $22$ | ${\href{/LocalNumberField/19.11.0.1}{11} }^{2}$ | $22$ | $22$ | ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/37.1.0.1}{1} }^{22}$ | $22$ | ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ | $22$ | $22$ | $22$ |
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 | ||||||
| 89 | Data not computed | ||||||