Normalized defining polynomial
\( x^{21} - x^{20} - 314 x^{19} + 641 x^{18} + 39405 x^{17} - 119809 x^{16} - 2475109 x^{15} + 10025575 x^{14} + 78717981 x^{13} - 415588104 x^{12} - 1066077495 x^{11} + 8308415958 x^{10} + 1043228987 x^{9} - 70546887095 x^{8} + 59456573110 x^{7} + 267134979239 x^{6} - 357627209452 x^{5} - 425427767421 x^{4} + 739147737632 x^{3} + 189916353207 x^{2} - 511171099198 x + 73391024141 \)
Invariants
| Degree: | $21$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[21, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(808066270618405716993861719647864148675120272481133649=7^{14}\cdot 127^{20}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $369.00$ | 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: | $7, 127$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(889=7\cdot 127\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{889}(512,·)$, $\chi_{889}(1,·)$, $\chi_{889}(835,·)$, $\chi_{889}(8,·)$, $\chi_{889}(457,·)$, $\chi_{889}(778,·)$, $\chi_{889}(823,·)$, $\chi_{889}(527,·)$, $\chi_{889}(660,·)$, $\chi_{889}(214,·)$, $\chi_{889}(540,·)$, $\chi_{889}(221,·)$, $\chi_{889}(361,·)$, $\chi_{889}(800,·)$, $\chi_{889}(100,·)$, $\chi_{889}(64,·)$, $\chi_{889}(879,·)$, $\chi_{889}(177,·)$, $\chi_{889}(809,·)$, $\chi_{889}(249,·)$, $\chi_{889}(764,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{37} a^{14} + \frac{5}{37} a^{13} - \frac{13}{37} a^{12} + \frac{9}{37} a^{11} - \frac{11}{37} a^{10} - \frac{2}{37} a^{9} + \frac{6}{37} a^{8} - \frac{15}{37} a^{7} - \frac{6}{37} a^{6} + \frac{9}{37} a^{5} + \frac{18}{37} a^{4} - \frac{17}{37} a^{3} + \frac{11}{37} a^{2} + \frac{17}{37} a$, $\frac{1}{37} a^{15} - \frac{1}{37} a^{13} + \frac{18}{37} a^{11} + \frac{16}{37} a^{10} + \frac{16}{37} a^{9} - \frac{8}{37} a^{8} - \frac{5}{37} a^{7} + \frac{2}{37} a^{6} + \frac{10}{37} a^{5} + \frac{4}{37} a^{4} - \frac{15}{37} a^{3} - \frac{1}{37} a^{2} - \frac{11}{37} a$, $\frac{1}{37} a^{16} + \frac{5}{37} a^{13} + \frac{5}{37} a^{12} - \frac{12}{37} a^{11} + \frac{5}{37} a^{10} - \frac{10}{37} a^{9} + \frac{1}{37} a^{8} - \frac{13}{37} a^{7} + \frac{4}{37} a^{6} + \frac{13}{37} a^{5} + \frac{3}{37} a^{4} - \frac{18}{37} a^{3} + \frac{17}{37} a$, $\frac{1}{37} a^{17} + \frac{17}{37} a^{13} + \frac{16}{37} a^{12} - \frac{3}{37} a^{11} + \frac{8}{37} a^{10} + \frac{11}{37} a^{9} - \frac{6}{37} a^{8} + \frac{5}{37} a^{7} + \frac{6}{37} a^{6} - \frac{5}{37} a^{5} + \frac{3}{37} a^{4} + \frac{11}{37} a^{3} - \frac{1}{37} a^{2} - \frac{11}{37} a$, $\frac{1}{83428229} a^{18} - \frac{973093}{83428229} a^{17} + \frac{1123538}{83428229} a^{16} + \frac{825068}{83428229} a^{15} - \frac{867799}{83428229} a^{14} - \frac{18353312}{83428229} a^{13} + \frac{4918030}{83428229} a^{12} + \frac{761478}{2254817} a^{11} - \frac{36625050}{83428229} a^{10} + \frac{32031497}{83428229} a^{9} - \frac{13961214}{83428229} a^{8} + \frac{12210671}{83428229} a^{7} - \frac{38598631}{83428229} a^{6} + \frac{3452607}{83428229} a^{5} - \frac{40605712}{83428229} a^{4} - \frac{13835179}{83428229} a^{3} - \frac{19565266}{83428229} a^{2} - \frac{30347869}{83428229} a + \frac{3088}{15133}$, $\frac{1}{7222706070056123} a^{19} + \frac{19974108}{7222706070056123} a^{18} - \frac{20744423177750}{7222706070056123} a^{17} + \frac{19642486555106}{7222706070056123} a^{16} - \frac{49564957906972}{7222706070056123} a^{15} + \frac{45189285770560}{7222706070056123} a^{14} + \frac{939418934205615}{7222706070056123} a^{13} + \frac{2845477719081034}{7222706070056123} a^{12} + \frac{1162129863956101}{7222706070056123} a^{11} + \frac{1462753327350497}{7222706070056123} a^{10} - \frac{21460729477524}{47832490530173} a^{9} + \frac{2836958857755509}{7222706070056123} a^{8} + \frac{1863414783836418}{7222706070056123} a^{7} + \frac{494667970305176}{7222706070056123} a^{6} - \frac{2968795684346199}{7222706070056123} a^{5} + \frac{295872636941181}{7222706070056123} a^{4} - \frac{197015580956911}{7222706070056123} a^{3} - \frac{1966526384920019}{7222706070056123} a^{2} - \frac{1436858755092640}{7222706070056123} a - \frac{580227798739}{1310122631971}$, $\frac{1}{1105117351040232582713630381952729400767840515704149207822916918345559029429092203} a^{20} + \frac{10504127762190704171631692019167137614829085124632306014699417534}{1105117351040232582713630381952729400767840515704149207822916918345559029429092203} a^{19} + \frac{1906062262930579273651506523794069742488416571166875623359239000857676962}{1105117351040232582713630381952729400767840515704149207822916918345559029429092203} a^{18} - \frac{1925352739390533952419741269989888689501859240903321537419472341807552489456465}{1105117351040232582713630381952729400767840515704149207822916918345559029429092203} a^{17} - \frac{1476119661641955731006288956863867466041591714962329217630444671000646983509402}{1105117351040232582713630381952729400767840515704149207822916918345559029429092203} a^{16} - \frac{6603512736382640081868975875820620637920065426491386826392868612609708900956022}{1105117351040232582713630381952729400767840515704149207822916918345559029429092203} a^{15} + \frac{6202271048269137237867759959309454972651640328167588917231425800209123289949227}{1105117351040232582713630381952729400767840515704149207822916918345559029429092203} a^{14} + \frac{515247007179746187490341931696854589358574686210144473296109773853556417205679990}{1105117351040232582713630381952729400767840515704149207822916918345559029429092203} a^{13} + \frac{360619098711781842961128161780471857753538528028091701697800557585182006562288062}{1105117351040232582713630381952729400767840515704149207822916918345559029429092203} a^{12} - \frac{232671074672872682633825081562959114164559348658322092026386373699448380162661690}{1105117351040232582713630381952729400767840515704149207822916918345559029429092203} a^{11} - \frac{248345465474395159744935538645520792742982321658943808583034309815197731805900704}{1105117351040232582713630381952729400767840515704149207822916918345559029429092203} a^{10} - \frac{369180711500388080925359116472476636765352394911021849193400710888626364058182101}{1105117351040232582713630381952729400767840515704149207822916918345559029429092203} a^{9} + \frac{439643304098979957417777777795053556131396797416202175530497187626676543073599745}{1105117351040232582713630381952729400767840515704149207822916918345559029429092203} a^{8} - \frac{3760137314362565725803630271045196218233695017583443583202011918808623376609759}{1105117351040232582713630381952729400767840515704149207822916918345559029429092203} a^{7} - \frac{267478168713281615537531331499144770966434118612496408068982161255245849551281557}{1105117351040232582713630381952729400767840515704149207822916918345559029429092203} a^{6} + \frac{35868223797762066387984076326875298439335646190265974111112451021771441817019755}{1105117351040232582713630381952729400767840515704149207822916918345559029429092203} a^{5} + \frac{352854692889378224649917464734796672121820281772994926568037815523395578480051637}{1105117351040232582713630381952729400767840515704149207822916918345559029429092203} a^{4} + \frac{105367298757715275724258140971885646028397397890588722092208882339195801941540004}{1105117351040232582713630381952729400767840515704149207822916918345559029429092203} a^{3} + \frac{346037462370474892883053367155441423099195279239068008038537900525329398832216262}{1105117351040232582713630381952729400767840515704149207822916918345559029429092203} a^{2} + \frac{547318530417878407884431720729373302482175211031166444580076155625575136486371138}{1105117351040232582713630381952729400767840515704149207822916918345559029429092203} a - \frac{21019340981941162324176084292808265913325104488194439449685204868256208528690}{200456620903361614858267800100259278209294488609495593655526377352722479490131}$
Class group and class number
$C_{3}$, which has order $3$ (assuming GRH)
Unit group
| Rank: | $20$ | 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: | \( 84153812913585720000 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 21 |
| The 21 conjugacy class representatives for $C_{21}$ |
| Character table for $C_{21}$ is not computed |
Intermediate fields
| 3.3.790321.1, 7.7.4195872914689.1 |
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 | $21$ | ${\href{/LocalNumberField/3.7.0.1}{7} }^{3}$ | $21$ | R | $21$ | $21$ | $21$ | ${\href{/LocalNumberField/19.3.0.1}{3} }^{7}$ | $21$ | $21$ | ${\href{/LocalNumberField/31.7.0.1}{7} }^{3}$ | ${\href{/LocalNumberField/37.1.0.1}{1} }^{21}$ | $21$ | $21$ | $21$ | $21$ | ${\href{/LocalNumberField/59.3.0.1}{3} }^{7}$ |
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 |
|---|---|---|---|---|---|---|---|
| 7 | Data not computed | ||||||
| 127 | Data not computed | ||||||