Normalized defining polynomial
\( x^{20} - 220 x^{18} - 1980 x^{17} + 23650 x^{16} + 478764 x^{15} + 2050950 x^{14} - 30753360 x^{13} - 536533360 x^{12} - 3362940900 x^{11} + 6278444370 x^{10} + 346639916700 x^{9} + 3998225366245 x^{8} + 29464182568260 x^{7} + 161681814006900 x^{6} + 678220640882532 x^{5} + 2195040871191410 x^{4} + 5453675221682160 x^{3} + 10566566691050700 x^{2} + 13513912852956120 x + 13421585401813457 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 10]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(4663996685529944093936844800000000000000000000000000000000=2^{55}\cdot 5^{32}\cdot 11^{18}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $764.61$ | 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: | $2, 5, 11$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(4400=2^{4}\cdot 5^{2}\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{4400}(1,·)$, $\chi_{4400}(261,·)$, $\chi_{4400}(4161,·)$, $\chi_{4400}(4041,·)$, $\chi_{4400}(1281,·)$, $\chi_{4400}(1421,·)$, $\chi_{4400}(2201,·)$, $\chi_{4400}(3481,·)$, $\chi_{4400}(3621,·)$, $\chi_{4400}(2141,·)$, $\chi_{4400}(901,·)$, $\chi_{4400}(4321,·)$, $\chi_{4400}(1381,·)$, $\chi_{4400}(1961,·)$, $\chi_{4400}(3101,·)$, $\chi_{4400}(1841,·)$, $\chi_{4400}(4341,·)$, $\chi_{4400}(2121,·)$, $\chi_{4400}(2461,·)$, $\chi_{4400}(3581,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{11} a^{5}$, $\frac{1}{11} a^{6}$, $\frac{1}{11} a^{7}$, $\frac{1}{11} a^{8}$, $\frac{1}{121} a^{9} + \frac{2}{11} a^{4}$, $\frac{1}{121} a^{10}$, $\frac{1}{121} a^{11}$, $\frac{1}{3993} a^{12} + \frac{1}{363} a^{10} - \frac{1}{121} a^{8} + \frac{5}{121} a^{7} + \frac{5}{33} a^{4} - \frac{2}{11} a^{3} + \frac{4}{33} a^{2} + \frac{1}{3}$, $\frac{1}{3993} a^{13} + \frac{1}{363} a^{11} + \frac{5}{121} a^{8} - \frac{1}{33} a^{5} + \frac{4}{33} a^{3} + \frac{1}{3} a$, $\frac{1}{3993} a^{14} + \frac{1}{363} a^{10} - \frac{1}{33} a^{6} - \frac{5}{11} a^{4} + \frac{1}{3}$, $\frac{1}{3993} a^{15} + \frac{1}{363} a^{11} - \frac{1}{33} a^{7} + \frac{1}{3} a$, $\frac{1}{1888689} a^{16} - \frac{14}{171699} a^{15} + \frac{4}{57233} a^{14} + \frac{20}{171699} a^{13} - \frac{3}{57233} a^{12} + \frac{35}{57233} a^{11} + \frac{41}{15609} a^{10} - \frac{8}{5203} a^{9} + \frac{125}{15609} a^{8} + \frac{679}{15609} a^{7} - \frac{150}{5203} a^{6} - \frac{32}{1419} a^{5} + \frac{160}{1419} a^{4} - \frac{523}{1419} a^{3} - \frac{34}{473} a^{2} - \frac{1}{473} a - \frac{58}{129}$, $\frac{1}{1888689} a^{17} + \frac{2}{57233} a^{15} + \frac{19}{171699} a^{14} + \frac{6}{57233} a^{13} + \frac{3}{57233} a^{12} - \frac{37}{15609} a^{11} + \frac{5}{1419} a^{10} + \frac{41}{15609} a^{9} + \frac{21}{5203} a^{8} + \frac{162}{5203} a^{7} + \frac{32}{1419} a^{6} - \frac{38}{1419} a^{5} + \frac{84}{473} a^{4} + \frac{51}{473} a^{3} - \frac{206}{473} a^{2} + \frac{29}{129} a + \frac{55}{129}$, $\frac{1}{101097176903563349146912114983951370881} a^{18} + \frac{8643263708586553317939464431376}{33699058967854449715637371661317123627} a^{17} + \frac{4798189143909456706875196491122}{33699058967854449715637371661317123627} a^{16} + \frac{766197268224214598098773873335956}{9190652445778486286082919543995579171} a^{15} - \frac{158955272639014665325857237748598}{3063550815259495428694306514665193057} a^{14} - \frac{1017582670595822026402191015704783}{9190652445778486286082919543995579171} a^{13} + \frac{199812893608682954707089916550218}{9190652445778486286082919543995579171} a^{12} - \frac{37556012262721540358453726539760213}{9190652445778486286082919543995579171} a^{11} + \frac{797960466330952481547045088723982}{835513858707135116916629049454143561} a^{10} - \frac{414901854354975460001065143148702}{278504619569045038972209683151381187} a^{9} + \frac{9300651697352103004106724745692596}{278504619569045038972209683151381187} a^{8} - \frac{9064418002264701543295404824914550}{835513858707135116916629049454143561} a^{7} + \frac{9479206979782360063318053963142997}{835513858707135116916629049454143561} a^{6} - \frac{371622958718031869002234425157366}{75955805337012283356057186314013051} a^{5} - \frac{9059011690667895474108983877805239}{25318601779004094452019062104671017} a^{4} - \frac{33884338477931548195047376369511573}{75955805337012283356057186314013051} a^{3} + \frac{1852585898079886929683398172306953}{75955805337012283356057186314013051} a^{2} + \frac{6168383739883905809247717162174544}{75955805337012283356057186314013051} a + \frac{463057890998478484512522938359492}{2301691070818554041092642009515547}$, $\frac{1}{6445386120648653640818914921179376472969606222431877767490803664764893249} a^{19} - \frac{21110174606236247034752745216321739}{6445386120648653640818914921179376472969606222431877767490803664764893249} a^{18} + \frac{11151318904122974131162900642643043786907609528300449748387884561}{2148462040216217880272971640393125490989868740810625922496934554921631083} a^{17} + \frac{464595907264505353715529674903970523592726599640337259752794677745}{6445386120648653640818914921179376472969606222431877767490803664764893249} a^{16} + \frac{24399652536573182914385784510280329393734801409604825802941886596627}{195314730928747080024815603672102317362715340073693265681539504992875553} a^{15} - \frac{54176371995858331222978018686034418936665232310263009000722317104192}{585944192786241240074446811016306952088146020221079797044618514978626659} a^{14} - \frac{1148468100605467077614220168844186098146913578050196668379561896421}{585944192786241240074446811016306952088146020221079797044618514978626659} a^{13} - \frac{52860569571749950549171374147247074981956825189165589617937222882418}{585944192786241240074446811016306952088146020221079797044618514978626659} a^{12} + \frac{563882066683144186064499601116548754852703004873546643662805960097597}{585944192786241240074446811016306952088146020221079797044618514978626659} a^{11} + \frac{16584916576816064067357689111719109711398948517840698981044285867634}{17755884629886098184074145788372937942065030915790296880139954999352323} a^{10} + \frac{63343238849372639382412261988956383355779941020605926780640190648476}{17755884629886098184074145788372937942065030915790296880139954999352323} a^{9} - \frac{1810869790104707503773579835040619912456147320842576357231600971740849}{53267653889658294552222437365118813826195092747370890640419864998056969} a^{8} - \frac{2074726871984354940610877717958122012816876091441385648270083087012940}{53267653889658294552222437365118813826195092747370890640419864998056969} a^{7} + \frac{1402407021657430671260143015744599203089981464358931320971956976961658}{53267653889658294552222437365118813826195092747370890640419864998056969} a^{6} + \frac{56726283196258657374696487130730706002773775361096035421747329070877}{1614171329989645289461285980761176176551366446890026989103632272668393} a^{5} - \frac{1418761727676337908360808698331865015611072002068723368976616063057022}{4842513989968935868383857942283528529654099340670080967310896818005179} a^{4} - \frac{854630863292012845270373844750979818985442515731862348419482036634158}{4842513989968935868383857942283528529654099340670080967310896818005179} a^{3} + \frac{410350499342258712939511031216869828531686000561662038236714449224043}{4842513989968935868383857942283528529654099340670080967310896818005179} a^{2} - \frac{248417990472261647116688087761224699686085638505684526724475722255279}{1614171329989645289461285980761176176551366446890026989103632272668393} a - \frac{42173339637057792697768756563871206702678378310217122758396853854058}{440228544542630533489441631116684411786736303697280087937354256182289}$
Class group and class number
$C_{5}\times C_{927422050}$, which has order $4637110250$ (assuming GRH)
Unit group
| Rank: | $9$ | 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: | \( 59692546406.70596 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 20 |
| The 20 conjugacy class representatives for $C_{20}$ |
| Character table for $C_{20}$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), 4.0.247808.2, 5.5.5719140625.2, 10.10.1071794405000000000000000.4 |
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 | R | ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ | R | ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ | R | ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ | $20$ | ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ | $20$ | ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ | $20$ | ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/47.1.0.1}{1} }^{20}$ | $20$ | $20$ |
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 |
|---|---|---|---|---|---|---|---|
| 2 | Data not computed | ||||||
| 5 | Data not computed | ||||||
| 11 | Data not computed | ||||||