Normalized defining polynomial
\( x^{21} - 4 x^{20} - 280 x^{19} + 1434 x^{18} + 28400 x^{17} - 174768 x^{16} - 1259964 x^{15} + 9241088 x^{14} + 24220628 x^{13} - 224432817 x^{12} - 225043591 x^{11} + 2800292150 x^{10} + 1310509833 x^{9} - 18643709590 x^{8} - 7862755157 x^{7} + 62071423969 x^{6} + 36855850475 x^{5} - 78614032446 x^{4} - 59296509039 x^{3} + 18479082734 x^{2} + 15129157356 x - 692122129 \)
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: | \(466076016634532996579786361302348154829875801567723769=7^{14}\cdot 211^{18}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $359.45$ | 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: | $7, 211$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1477=7\cdot 211\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1477}(1,·)$, $\chi_{1477}(967,·)$, $\chi_{1477}(1226,·)$, $\chi_{1477}(781,·)$, $\chi_{1477}(144,·)$, $\chi_{1477}(212,·)$, $\chi_{1477}(1465,·)$, $\chi_{1477}(1437,·)$, $\chi_{1477}(1178,·)$, $\chi_{1477}(988,·)$, $\chi_{1477}(58,·)$, $\chi_{1477}(410,·)$, $\chi_{1477}(480,·)$, $\chi_{1477}(1254,·)$, $\chi_{1477}(359,·)$, $\chi_{1477}(1324,·)$, $\chi_{1477}(1199,·)$, $\chi_{1477}(148,·)$, $\chi_{1477}(634,·)$, $\chi_{1477}(123,·)$, $\chi_{1477}(382,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{19} a^{15} + \frac{2}{19} a^{14} - \frac{6}{19} a^{13} + \frac{1}{19} a^{12} + \frac{9}{19} a^{11} + \frac{9}{19} a^{10} - \frac{3}{19} a^{9} - \frac{1}{19} a^{8} + \frac{4}{19} a^{7} + \frac{4}{19} a^{6} - \frac{5}{19} a^{5} - \frac{9}{19} a^{4} - \frac{7}{19} a^{3} + \frac{3}{19} a^{2} - \frac{1}{19}$, $\frac{1}{19} a^{16} + \frac{9}{19} a^{14} - \frac{6}{19} a^{13} + \frac{7}{19} a^{12} - \frac{9}{19} a^{11} - \frac{2}{19} a^{10} + \frac{5}{19} a^{9} + \frac{6}{19} a^{8} - \frac{4}{19} a^{7} + \frac{6}{19} a^{6} + \frac{1}{19} a^{5} - \frac{8}{19} a^{4} - \frac{2}{19} a^{3} - \frac{6}{19} a^{2} - \frac{1}{19} a + \frac{2}{19}$, $\frac{1}{19} a^{17} - \frac{5}{19} a^{14} + \frac{4}{19} a^{13} + \frac{1}{19} a^{12} - \frac{7}{19} a^{11} - \frac{5}{19} a^{9} + \frac{5}{19} a^{8} + \frac{8}{19} a^{7} + \frac{3}{19} a^{6} - \frac{1}{19} a^{5} + \frac{3}{19} a^{4} - \frac{9}{19} a^{2} + \frac{2}{19} a + \frac{9}{19}$, $\frac{1}{1137511} a^{18} - \frac{590}{1137511} a^{17} - \frac{8479}{1137511} a^{16} + \frac{24185}{1137511} a^{15} + \frac{266008}{1137511} a^{14} - \frac{557242}{1137511} a^{13} - \frac{144782}{1137511} a^{12} + \frac{487106}{1137511} a^{11} + \frac{257957}{1137511} a^{10} + \frac{180723}{1137511} a^{9} - \frac{518901}{1137511} a^{8} + \frac{247996}{1137511} a^{7} - \frac{476447}{1137511} a^{6} + \frac{514941}{1137511} a^{5} + \frac{456998}{1137511} a^{4} + \frac{480794}{1137511} a^{3} - \frac{503146}{1137511} a^{2} + \frac{519396}{1137511} a - \frac{223728}{1137511}$, $\frac{1}{80763281} a^{19} + \frac{21}{80763281} a^{18} + \frac{1427101}{80763281} a^{17} + \frac{1728451}{80763281} a^{16} - \frac{1899884}{80763281} a^{15} - \frac{30505189}{80763281} a^{14} - \frac{23972503}{80763281} a^{13} + \frac{29248806}{80763281} a^{12} + \frac{6018348}{80763281} a^{11} - \frac{34626516}{80763281} a^{10} + \frac{15549308}{80763281} a^{9} + \frac{15472435}{80763281} a^{8} + \frac{31430847}{80763281} a^{7} + \frac{1147461}{80763281} a^{6} + \frac{1613070}{4250699} a^{5} + \frac{16223103}{80763281} a^{4} - \frac{7998820}{80763281} a^{3} + \frac{34111014}{80763281} a^{2} + \frac{1018908}{80763281} a + \frac{144797}{1137511}$, $\frac{1}{12422263428479961217920940031214311968146110647730233779849569241860932221011665899245908957} a^{20} + \frac{51164213101166551918533838097890503252646724297586715358908838069229087642179396434}{12422263428479961217920940031214311968146110647730233779849569241860932221011665899245908957} a^{19} + \frac{1688268726706608498747343768437912618674434285680665942337925721675864611942626192750}{12422263428479961217920940031214311968146110647730233779849569241860932221011665899245908957} a^{18} + \frac{298704402620816386853051179285013312672373362233822425151647714435681640564230767306374946}{12422263428479961217920940031214311968146110647730233779849569241860932221011665899245908957} a^{17} + \frac{257870884072483526482570574436998416845250114610102566246300334800923764206855957217233396}{12422263428479961217920940031214311968146110647730233779849569241860932221011665899245908957} a^{16} - \frac{186561106754283051922393927815528673996566525706281238134514138070708653854112042157638096}{12422263428479961217920940031214311968146110647730233779849569241860932221011665899245908957} a^{15} + \frac{4118607264805276051843254115014069530509592375115124579528044932452813042996915538162557013}{12422263428479961217920940031214311968146110647730233779849569241860932221011665899245908957} a^{14} + \frac{2158767482361338256971218803929867678840581482779804950279536207598617992111571102832723502}{12422263428479961217920940031214311968146110647730233779849569241860932221011665899245908957} a^{13} + \frac{4019065789847497613529768988780239608235837900928282287613186454149152026116773438784937185}{12422263428479961217920940031214311968146110647730233779849569241860932221011665899245908957} a^{12} + \frac{11617115940508852115246725291866186991793936345717191153583264686424939890152658849245190}{12422263428479961217920940031214311968146110647730233779849569241860932221011665899245908957} a^{11} - \frac{2405979759303938463046329735502280579149246455203449453596202902514256607008703783808444993}{12422263428479961217920940031214311968146110647730233779849569241860932221011665899245908957} a^{10} - \frac{1335633130259334357370618814970613152266141647864920432157913302471463976390600523394954642}{12422263428479961217920940031214311968146110647730233779849569241860932221011665899245908957} a^{9} + \frac{1813982917553901876452966442465899030753789817435195339526155784431452576193278123420142370}{12422263428479961217920940031214311968146110647730233779849569241860932221011665899245908957} a^{8} - \frac{1688413524197248866235499526228657369344255967040891810612294328895990748163391542409131}{4772287141175551754867821756133043399210952995670470142085889067176693131391343026986519} a^{7} - \frac{5186306973901290480035675149997942593402200462603956108091596817767674686732066115850326352}{12422263428479961217920940031214311968146110647730233779849569241860932221011665899245908957} a^{6} + \frac{3563715382846751281430076601544669261185417187137855330197236918220933712141630879947102634}{12422263428479961217920940031214311968146110647730233779849569241860932221011665899245908957} a^{5} - \frac{257951167228397864876351029783414726724449535686766113566127932979101303756942010845755052}{540098409933911357300910436139752694267222202075227555645633445298301400913550691271561259} a^{4} - \frac{1810394339056597662478948240879811772093077494671630864955256517407389053323702319970061287}{12422263428479961217920940031214311968146110647730233779849569241860932221011665899245908957} a^{3} - \frac{1324246513683177183643736836197371257722027836938436692215317206001714644356914520913052154}{12422263428479961217920940031214311968146110647730233779849569241860932221011665899245908957} a^{2} - \frac{2237268379479604310261764139957104685333118244523136985356805091557081598071007245185238212}{12422263428479961217920940031214311968146110647730233779849569241860932221011665899245908957} a - \frac{76821943221577318246196334384530079269494834980561763441882065777099855279444973568334524}{174961456739154383350999155369215661523184657010284982814782665378322989028333322524590267}$
Class group and class number
Trivial group, which has order $1$ (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: | \( 115303717001291800000 \) (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
| \(\Q(\zeta_{7})^+\), 7.7.88245939632761.1 |
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 | $21$ | $21$ | $21$ | R | $21$ | ${\href{/LocalNumberField/13.7.0.1}{7} }^{3}$ | $21$ | ${\href{/LocalNumberField/19.3.0.1}{3} }^{7}$ | ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ | ${\href{/LocalNumberField/29.7.0.1}{7} }^{3}$ | $21$ | $21$ | ${\href{/LocalNumberField/41.7.0.1}{7} }^{3}$ | ${\href{/LocalNumberField/43.7.0.1}{7} }^{3}$ | $21$ | $21$ | $21$ |
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 | ||||||
| 211 | Data not computed | ||||||