Normalized defining polynomial
\( x^{22} - x^{21} + 204 x^{20} + 300 x^{19} + 17203 x^{18} + 68673 x^{17} + 825513 x^{16} + 4874907 x^{15} + 29311275 x^{14} + 165265669 x^{13} + 760942580 x^{12} + 3191438973 x^{11} + 12791899873 x^{10} + 37757317040 x^{9} + 122682543835 x^{8} + 272442102752 x^{7} + 760956148725 x^{6} + 1683377741263 x^{5} + 5010846934880 x^{4} + 10572689320889 x^{3} + 20272483083031 x^{2} + 19526867163395 x + 6857181839381 \)
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: | \(-92168952614220241164959742886255742486065996298779296875=-\,5^{11}\cdot 199^{21}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $349.82$ | 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: | $5, 199$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(995=5\cdot 199\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{995}(1,·)$, $\chi_{995}(259,·)$, $\chi_{995}(261,·)$, $\chi_{995}(711,·)$, $\chi_{995}(74,·)$, $\chi_{995}(461,·)$, $\chi_{995}(579,·)$, $\chi_{995}(534,·)$, $\chi_{995}(409,·)$, $\chi_{995}(921,·)$, $\chi_{995}(284,·)$, $\chi_{995}(736,·)$, $\chi_{995}(734,·)$, $\chi_{995}(416,·)$, $\chi_{995}(994,·)$, $\chi_{995}(934,·)$, $\chi_{995}(874,·)$, $\chi_{995}(494,·)$, $\chi_{995}(61,·)$, $\chi_{995}(501,·)$, $\chi_{995}(121,·)$, $\chi_{995}(586,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{37} a^{13} - \frac{7}{37} a^{12} - \frac{4}{37} a^{11} + \frac{17}{37} a^{10} - \frac{9}{37} a^{9} - \frac{13}{37} a^{8} + \frac{1}{37} a^{7} - \frac{16}{37} a^{6} + \frac{14}{37} a^{5} + \frac{9}{37} a^{4} + \frac{17}{37} a^{3} - \frac{11}{37} a^{2} + \frac{17}{37} a - \frac{16}{37}$, $\frac{1}{37} a^{14} - \frac{16}{37} a^{12} - \frac{11}{37} a^{11} - \frac{1}{37} a^{10} - \frac{2}{37} a^{9} - \frac{16}{37} a^{8} - \frac{9}{37} a^{7} + \frac{13}{37} a^{6} - \frac{4}{37} a^{5} + \frac{6}{37} a^{4} - \frac{3}{37} a^{3} + \frac{14}{37} a^{2} - \frac{8}{37} a - \frac{1}{37}$, $\frac{1}{37} a^{15} - \frac{12}{37} a^{12} + \frac{9}{37} a^{11} + \frac{11}{37} a^{10} - \frac{12}{37} a^{9} + \frac{5}{37} a^{8} - \frac{8}{37} a^{7} - \frac{1}{37} a^{6} + \frac{8}{37} a^{5} - \frac{7}{37} a^{4} - \frac{10}{37} a^{3} + \frac{1}{37} a^{2} + \frac{12}{37} a + \frac{3}{37}$, $\frac{1}{37} a^{16} - \frac{1}{37} a^{12} + \frac{7}{37} a^{10} + \frac{8}{37} a^{9} - \frac{16}{37} a^{8} + \frac{11}{37} a^{7} + \frac{1}{37} a^{6} + \frac{13}{37} a^{5} - \frac{13}{37} a^{4} - \frac{17}{37} a^{3} - \frac{9}{37} a^{2} - \frac{15}{37} a - \frac{7}{37}$, $\frac{1}{37} a^{17} - \frac{7}{37} a^{12} + \frac{3}{37} a^{11} - \frac{12}{37} a^{10} + \frac{12}{37} a^{9} - \frac{2}{37} a^{8} + \frac{2}{37} a^{7} - \frac{3}{37} a^{6} + \frac{1}{37} a^{5} - \frac{8}{37} a^{4} + \frac{8}{37} a^{3} + \frac{11}{37} a^{2} + \frac{10}{37} a - \frac{16}{37}$, $\frac{1}{37} a^{18} - \frac{9}{37} a^{12} - \frac{3}{37} a^{11} - \frac{17}{37} a^{10} + \frac{9}{37} a^{9} - \frac{15}{37} a^{8} + \frac{4}{37} a^{7} + \frac{16}{37} a^{5} - \frac{3}{37} a^{4} - \frac{18}{37} a^{3} + \frac{7}{37} a^{2} - \frac{8}{37} a - \frac{1}{37}$, $\frac{1}{37} a^{19} + \frac{8}{37} a^{12} - \frac{16}{37} a^{11} + \frac{14}{37} a^{10} + \frac{15}{37} a^{9} - \frac{2}{37} a^{8} + \frac{9}{37} a^{7} - \frac{17}{37} a^{6} + \frac{12}{37} a^{5} - \frac{11}{37} a^{4} + \frac{12}{37} a^{3} + \frac{4}{37} a^{2} + \frac{4}{37} a + \frac{4}{37}$, $\frac{1}{703} a^{20} - \frac{5}{703} a^{19} - \frac{6}{703} a^{18} - \frac{8}{703} a^{17} - \frac{5}{703} a^{16} + \frac{5}{703} a^{15} - \frac{9}{703} a^{14} + \frac{8}{703} a^{13} - \frac{153}{703} a^{12} + \frac{47}{703} a^{11} - \frac{272}{703} a^{10} - \frac{198}{703} a^{9} - \frac{107}{703} a^{8} - \frac{10}{37} a^{7} + \frac{68}{703} a^{6} - \frac{90}{703} a^{5} - \frac{97}{703} a^{4} + \frac{198}{703} a^{3} + \frac{3}{19} a^{2} + \frac{307}{703} a - \frac{49}{703}$, $\frac{1}{39472992039777305248577396644956073154460827540501925690453379479248878595850522940309232156399473571389594289790320605767034075979} a^{21} - \frac{4425184487055602205246693836934202464994066111406841244174995914931423751565754989675882950155634500976485014304078979720132345}{39472992039777305248577396644956073154460827540501925690453379479248878595850522940309232156399473571389594289790320605767034075979} a^{20} + \frac{239154529723247706843488925917816209091486534133143112670145617374589228465416848458775120604248845073388354144113889414304766162}{39472992039777305248577396644956073154460827540501925690453379479248878595850522940309232156399473571389594289790320605767034075979} a^{19} - \frac{477737430853878500758192509480070287289143063600108878466041484017814084399884199553103435659769505021205136990890368289484009949}{39472992039777305248577396644956073154460827540501925690453379479248878595850522940309232156399473571389594289790320605767034075979} a^{18} + \frac{62378220269287751369070994148635755320381696433834021665348552172710399754777079602183705764144078660854105092149010636713245767}{39472992039777305248577396644956073154460827540501925690453379479248878595850522940309232156399473571389594289790320605767034075979} a^{17} + \frac{20722266739687190701003246474085228853554200644260692090144835969815310393238258611243882780862871031750058085692747209303636984}{39472992039777305248577396644956073154460827540501925690453379479248878595850522940309232156399473571389594289790320605767034075979} a^{16} + \frac{54243101729780409685598621513869763355391198039454283633879187490264514821138438616812860849862751919120729644425895550050417693}{39472992039777305248577396644956073154460827540501925690453379479248878595850522940309232156399473571389594289790320605767034075979} a^{15} - \frac{525446740731665297433050192691084328271704704653325368859603851564307358595497063050249885074339935090584517309124642771192737644}{39472992039777305248577396644956073154460827540501925690453379479248878595850522940309232156399473571389594289790320605767034075979} a^{14} + \frac{264471087869342767330213331132757372704387813918198617619751043848504367488068718014684937370075037704957689121779672817869623986}{39472992039777305248577396644956073154460827540501925690453379479248878595850522940309232156399473571389594289790320605767034075979} a^{13} + \frac{17245539332879198423050890614783952263517569809439148077973139930945771285925389479064462427803878168206483540458945444177660190100}{39472992039777305248577396644956073154460827540501925690453379479248878595850522940309232156399473571389594289790320605767034075979} a^{12} - \frac{15631578890150973066767696404165472653153739292902866102393669953676030785529238791251075864694514642406889457908739587455206543090}{39472992039777305248577396644956073154460827540501925690453379479248878595850522940309232156399473571389594289790320605767034075979} a^{11} + \frac{7894502190257844023861594648861144446422492999723114425186615555153047873865644784243383104307541019034943629205871853633919508720}{39472992039777305248577396644956073154460827540501925690453379479248878595850522940309232156399473571389594289790320605767034075979} a^{10} - \frac{15368982193295780462963831684426256771576724607705263609059728776176773023731849673643702912716999414957549670132674794346876470507}{39472992039777305248577396644956073154460827540501925690453379479248878595850522940309232156399473571389594289790320605767034075979} a^{9} - \frac{15682895283939995285819061698108891438654612309359855106194322704842143217266437860905404817142222730689629047701738470894290562306}{39472992039777305248577396644956073154460827540501925690453379479248878595850522940309232156399473571389594289790320605767034075979} a^{8} + \frac{19702940307062493305578641434801373005958699601737850910661652902538331746027133900918580981309735502883683744779916954737998365422}{39472992039777305248577396644956073154460827540501925690453379479248878595850522940309232156399473571389594289790320605767034075979} a^{7} - \frac{319602008437043532866288667550175244947292588453753391152522815630560603879484685122257165416532121680756436902348733361640642032}{917976559064588494152962712673397050103740175360509899777985569284857641763965649774633305962778455148595216041635362924814745953} a^{6} - \frac{3172768341985343727790935907728249868932235172721536644895909734140517129188271248472236416561080096825864377218351959510285548206}{39472992039777305248577396644956073154460827540501925690453379479248878595850522940309232156399473571389594289790320605767034075979} a^{5} + \frac{13414606141573686143776457118408776465724517252061868325834380860475925247649185405123430429233675511841895164313351039527963290004}{39472992039777305248577396644956073154460827540501925690453379479248878595850522940309232156399473571389594289790320605767034075979} a^{4} + \frac{9131597083610828698053989812917450012904801021565341864058011420002611271199299482923053594672937695755162620834977633263035159214}{39472992039777305248577396644956073154460827540501925690453379479248878595850522940309232156399473571389594289790320605767034075979} a^{3} - \frac{15733754159196541910624992588433945012615481948989013646646204448372939041097551362185430604109249095842450781514301454112065629327}{39472992039777305248577396644956073154460827540501925690453379479248878595850522940309232156399473571389594289790320605767034075979} a^{2} - \frac{13178807837739850674768857197391416201587846823109604882796978538130673081673452926069472058415550483820392038187046784026898268943}{39472992039777305248577396644956073154460827540501925690453379479248878595850522940309232156399473571389594289790320605767034075979} a + \frac{18060074743930781277518710903071106458188990879751697895320740389384355479308695629370555121019022995352300844011692035224895788683}{39472992039777305248577396644956073154460827540501925690453379479248878595850522940309232156399473571389594289790320605767034075979}$
Class group and class number
$C_{69273608}$, which has order $69273608$ (assuming GRH)
Unit group
| Rank: | $10$ | 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: | \( 117135822355.96071 \) (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
| \(\Q(\sqrt{-995}) \), 11.11.97393677359695041798001.1 |
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 | $22$ | ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ | R | $22$ | $22$ | $22$ | ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{11}$ | $22$ | ${\href{/LocalNumberField/29.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/37.1.0.1}{1} }^{22}$ | $22$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{11}$ | $22$ | $22$ | $22$ |
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 |
|---|---|---|---|---|---|---|---|
| 5 | Data not computed | ||||||
| 199 | Data not computed | ||||||