Normalized defining polynomial
\( x^{21} - 273 x^{19} - 707 x^{18} + 28329 x^{17} + 138600 x^{16} - 1256010 x^{15} - 9720649 x^{14} + 15415400 x^{13} + 295555148 x^{12} + 459394712 x^{11} - 3492190380 x^{10} - 14219288965 x^{9} - 1746076850 x^{8} + 95156654696 x^{7} + 227577402612 x^{6} + 180231411857 x^{5} - 72622185076 x^{4} - 206563651595 x^{3} - 96402059663 x^{2} + 11414547333 x + 12861176609 \)
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: | \(103818783062189717738091671292152377422379176428329=7^{38}\cdot 19^{14}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $240.84$ | 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: | $7, 19$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(931=7^{2}\cdot 19\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{931}(1,·)$, $\chi_{931}(387,·)$, $\chi_{931}(134,·)$, $\chi_{931}(520,·)$, $\chi_{931}(267,·)$, $\chi_{931}(653,·)$, $\chi_{931}(400,·)$, $\chi_{931}(11,·)$, $\chi_{931}(786,·)$, $\chi_{931}(533,·)$, $\chi_{931}(919,·)$, $\chi_{931}(666,·)$, $\chi_{931}(410,·)$, $\chi_{931}(799,·)$, $\chi_{931}(144,·)$, $\chi_{931}(676,·)$, $\chi_{931}(809,·)$, $\chi_{931}(121,·)$, $\chi_{931}(543,·)$, $\chi_{931}(254,·)$, $\chi_{931}(277,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{79} a^{16} - \frac{34}{79} a^{15} + \frac{18}{79} a^{14} - \frac{7}{79} a^{13} + \frac{19}{79} a^{12} - \frac{4}{79} a^{11} + \frac{4}{79} a^{10} - \frac{15}{79} a^{9} - \frac{39}{79} a^{8} + \frac{21}{79} a^{7} + \frac{14}{79} a^{6} + \frac{37}{79} a^{5} - \frac{9}{79} a^{4} + \frac{35}{79} a^{3} + \frac{17}{79} a^{2} + \frac{12}{79} a - \frac{8}{79}$, $\frac{1}{79} a^{17} - \frac{32}{79} a^{15} - \frac{27}{79} a^{14} + \frac{18}{79} a^{13} + \frac{10}{79} a^{12} + \frac{26}{79} a^{11} - \frac{37}{79} a^{10} + \frac{4}{79} a^{9} + \frac{38}{79} a^{8} + \frac{17}{79} a^{7} + \frac{39}{79} a^{6} - \frac{15}{79} a^{5} - \frac{34}{79} a^{4} + \frac{22}{79} a^{3} + \frac{37}{79} a^{2} + \frac{5}{79} a - \frac{35}{79}$, $\frac{1}{100567} a^{18} + \frac{27}{100567} a^{17} + \frac{441}{100567} a^{16} - \frac{20054}{100567} a^{15} - \frac{1714}{5293} a^{14} - \frac{31097}{100567} a^{13} - \frac{7307}{100567} a^{12} + \frac{39853}{100567} a^{11} - \frac{1236}{100567} a^{10} - \frac{20221}{100567} a^{9} - \frac{21196}{100567} a^{8} + \frac{14065}{100567} a^{7} - \frac{7824}{100567} a^{6} - \frac{15407}{100567} a^{5} + \frac{49752}{100567} a^{4} - \frac{46962}{100567} a^{3} + \frac{44200}{100567} a^{2} - \frac{36094}{100567} a - \frac{30088}{100567}$, $\frac{1}{100567} a^{19} - \frac{288}{100567} a^{17} - \frac{136}{100567} a^{16} + \frac{30244}{100567} a^{15} + \frac{13097}{100567} a^{14} + \frac{6135}{100567} a^{13} + \frac{37281}{100567} a^{12} + \frac{2237}{100567} a^{11} + \frac{39884}{100567} a^{10} + \frac{47396}{100567} a^{9} + \frac{49151}{100567} a^{8} - \frac{20955}{100567} a^{7} + \frac{567}{1501} a^{6} + \frac{34194}{100567} a^{5} + \frac{32948}{100567} a^{4} + \frac{12441}{100567} a^{3} + \frac{15500}{100567} a^{2} + \frac{18979}{100567} a - \frac{45626}{100567}$, $\frac{1}{50626358721450937930991743036116129447035171638601239800431133696640646100790497321509} a^{20} - \frac{120001436932845487503212976081358737730748963844918563452798847203612926868994391}{50626358721450937930991743036116129447035171638601239800431133696640646100790497321509} a^{19} - \frac{241631600196875453834916612418059119204986515720496272680077593612149759210574340}{50626358721450937930991743036116129447035171638601239800431133696640646100790497321509} a^{18} + \frac{236382854887319064220547063190412956606415733628212711493776787839904131438807085462}{50626358721450937930991743036116129447035171638601239800431133696640646100790497321509} a^{17} - \frac{94896415076280531707492652189357156784671040759154446791508161307359784749349312863}{50626358721450937930991743036116129447035171638601239800431133696640646100790497321509} a^{16} + \frac{3265220424323050159316479962612102996581141692796099183738583635251842294214761295369}{50626358721450937930991743036116129447035171638601239800431133696640646100790497321509} a^{15} + \frac{23550408101498163067128593601632887421117864063808974915723263764864613091704371745167}{50626358721450937930991743036116129447035171638601239800431133696640646100790497321509} a^{14} - \frac{1422989744177178680948414418339582528849592703293429082029117125377344853050892734555}{50626358721450937930991743036116129447035171638601239800431133696640646100790497321509} a^{13} - \frac{22136437570277715112778873214514790089313758606721802762162678954856257116965216471311}{50626358721450937930991743036116129447035171638601239800431133696640646100790497321509} a^{12} + \frac{77510828410019691664470810542989318810626289031783546583523595462140332183112837607}{2664545195865838838473249633479796286686061665189538936864796510349507689515289332711} a^{11} + \frac{13344014324137340899354784362936575831564833089738034250394744452830990752159703397441}{50626358721450937930991743036116129447035171638601239800431133696640646100790497321509} a^{10} + \frac{3652518745857581931223654032479678777002714261546986944850232978520291067068845962711}{50626358721450937930991743036116129447035171638601239800431133696640646100790497321509} a^{9} + \frac{15837615760551688700850677904729653785028498153098456007086303903944875825507051347200}{50626358721450937930991743036116129447035171638601239800431133696640646100790497321509} a^{8} - \frac{13570419443260203770881938026911487478480719757605625217188014485200219137778954992471}{50626358721450937930991743036116129447035171638601239800431133696640646100790497321509} a^{7} - \frac{20762963596101642909996211085372963584503942376656699451330883470873307025425298198313}{50626358721450937930991743036116129447035171638601239800431133696640646100790497321509} a^{6} - \frac{23644077478015791811338270415182594622605615995554458968940917780579245405845182762410}{50626358721450937930991743036116129447035171638601239800431133696640646100790497321509} a^{5} + \frac{19271184164190112914975554637624745045624647391882108726556841942450789049910965832417}{50626358721450937930991743036116129447035171638601239800431133696640646100790497321509} a^{4} - \frac{62910492358495884676658349013348275540097802431259185065172439648849038784198677419}{640839983815834657354325861216659866418166729602547339245963717678995520263170852171} a^{3} + \frac{12094683079957741106102845627983024110159502176794659764612795535972036270610493869293}{50626358721450937930991743036116129447035171638601239800431133696640646100790497321509} a^{2} + \frac{10839260613217096330070162580057055822554398938012723539010118582023812449149499262229}{50626358721450937930991743036116129447035171638601239800431133696640646100790497321509} a - \frac{3017251367010048237430094762218439116413492312087031555288651440765302793720223098433}{50626358721450937930991743036116129447035171638601239800431133696640646100790497321509}$
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: | \( 1030828564219823600 \) (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
| 3.3.17689.2, 7.7.13841287201.1 |
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 | ${\href{/LocalNumberField/2.7.0.1}{7} }^{3}$ | $21$ | ${\href{/LocalNumberField/5.7.0.1}{7} }^{3}$ | R | $21$ | $21$ | $21$ | R | $21$ | $21$ | ${\href{/LocalNumberField/31.3.0.1}{3} }^{7}$ | $21$ | $21$ | $21$ | $21$ | ${\href{/LocalNumberField/53.7.0.1}{7} }^{3}$ | $21$ |
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 | ||||||
| $19$ | 19.3.2.1 | $x^{3} + 76$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 19.3.2.1 | $x^{3} + 76$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 19.3.2.1 | $x^{3} + 76$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 19.3.2.1 | $x^{3} + 76$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 19.3.2.1 | $x^{3} + 76$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 19.3.2.1 | $x^{3} + 76$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 19.3.2.1 | $x^{3} + 76$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |