Normalized defining polynomial
\( x^{21} - x^{20} - 278 x^{19} - 325 x^{18} + 30572 x^{17} + 90706 x^{16} - 1619293 x^{15} - 7441717 x^{14} + 40623104 x^{13} + 278806912 x^{12} - 309578439 x^{11} - 5095633178 x^{10} - 5354762119 x^{9} + 40289615751 x^{8} + 110934247984 x^{7} - 35893590814 x^{6} - 526207863829 x^{5} - 789731658124 x^{4} - 300252270554 x^{3} + 314047983523 x^{2} + 333487382586 x + 88392158713 \)
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: | \(373181000690273382911499008901975594970451729658121=19^{14}\cdot 43^{20}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $255.97$ | 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: | $19, 43$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(817=19\cdot 43\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{817}(704,·)$, $\chi_{817}(1,·)$, $\chi_{817}(514,·)$, $\chi_{817}(68,·)$, $\chi_{817}(144,·)$, $\chi_{817}(742,·)$, $\chi_{817}(723,·)$, $\chi_{817}(666,·)$, $\chi_{817}(539,·)$, $\chi_{817}(353,·)$, $\chi_{817}(805,·)$, $\chi_{817}(486,·)$, $\chi_{817}(425,·)$, $\chi_{817}(619,·)$, $\chi_{817}(368,·)$, $\chi_{817}(305,·)$, $\chi_{817}(178,·)$, $\chi_{817}(83,·)$, $\chi_{817}(311,·)$, $\chi_{817}(315,·)$, $\chi_{817}(638,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{251} a^{18} + \frac{49}{251} a^{17} - \frac{41}{251} a^{16} - \frac{117}{251} a^{15} - \frac{60}{251} a^{14} + \frac{83}{251} a^{13} + \frac{77}{251} a^{12} + \frac{80}{251} a^{11} + \frac{102}{251} a^{10} - \frac{71}{251} a^{9} - \frac{103}{251} a^{8} - \frac{77}{251} a^{7} + \frac{8}{251} a^{6} + \frac{64}{251} a^{5} + \frac{116}{251} a^{4} + \frac{31}{251} a^{3} - \frac{43}{251} a^{2} - \frac{116}{251} a + \frac{8}{251}$, $\frac{1}{77057} a^{19} - \frac{116}{77057} a^{18} - \frac{345}{77057} a^{17} + \frac{8907}{77057} a^{16} - \frac{13636}{77057} a^{15} - \frac{38460}{77057} a^{14} + \frac{36582}{77057} a^{13} + \frac{5698}{77057} a^{12} - \frac{7827}{77057} a^{11} + \frac{418}{77057} a^{10} + \frac{33951}{77057} a^{9} - \frac{32529}{77057} a^{8} - \frac{33220}{77057} a^{7} + \frac{34888}{77057} a^{6} + \frac{8632}{77057} a^{5} - \frac{24129}{77057} a^{4} - \frac{7166}{77057} a^{3} + \frac{16768}{77057} a^{2} + \frac{11367}{77057} a - \frac{20396}{77057}$, $\frac{1}{4194853604551201106005518454150375409471970818190164225788510826070855560966773791628680423951169} a^{20} + \frac{707442174520291750891167006132724656865722624903656815896329151756962675573361865480317219}{4194853604551201106005518454150375409471970818190164225788510826070855560966773791628680423951169} a^{19} - \frac{4824858864186796328889517913795943931015039520543938016897953821876843783947660261127948654006}{4194853604551201106005518454150375409471970818190164225788510826070855560966773791628680423951169} a^{18} + \frac{1311169340176225480018050134536730195920267407986983684746071577066747030551808829957592611953282}{4194853604551201106005518454150375409471970818190164225788510826070855560966773791628680423951169} a^{17} - \frac{139146568753202905952729983431994010954969975677697499046762402971938619809080081571776845396778}{4194853604551201106005518454150375409471970818190164225788510826070855560966773791628680423951169} a^{16} - \frac{1933979188406088364683851661962261199998246225343169400795520880721599406246621206263035940098033}{4194853604551201106005518454150375409471970818190164225788510826070855560966773791628680423951169} a^{15} - \frac{481871968252631903072758498594805322761655625301290687696230393047708084043645887111908143779302}{4194853604551201106005518454150375409471970818190164225788510826070855560966773791628680423951169} a^{14} - \frac{736447417319680699559952761113715297584898713108303860872132161346736603736485025061822756574311}{4194853604551201106005518454150375409471970818190164225788510826070855560966773791628680423951169} a^{13} - \frac{1868740142933930516190315175779854513661388721158962445172680617480713471242798299380005904959058}{4194853604551201106005518454150375409471970818190164225788510826070855560966773791628680423951169} a^{12} + \frac{125759744291016761036145205870435147531409661897743659857168154183132761341794511762221726618430}{4194853604551201106005518454150375409471970818190164225788510826070855560966773791628680423951169} a^{11} + \frac{823586462225788960510025051662455684920400766110351021079809184888964288206577410323931446370860}{4194853604551201106005518454150375409471970818190164225788510826070855560966773791628680423951169} a^{10} - \frac{1811179184069771346069516730616632865920744543808966244959524938876122456651385088354595959627439}{4194853604551201106005518454150375409471970818190164225788510826070855560966773791628680423951169} a^{9} - \frac{1039446093338539564049991903482357565650003197169637422507375393149272963518493092753782899938940}{4194853604551201106005518454150375409471970818190164225788510826070855560966773791628680423951169} a^{8} - \frac{1677777016302970561367158201364168996746782407752105961106509465397362834422350115138131607470806}{4194853604551201106005518454150375409471970818190164225788510826070855560966773791628680423951169} a^{7} + \frac{1280732391167595838453019309040242043492879761912499190282607261274286792727849551019763895259502}{4194853604551201106005518454150375409471970818190164225788510826070855560966773791628680423951169} a^{6} + \frac{550796730238970322258460620318403239457589540786194000726642846239035914366833551903385576224395}{4194853604551201106005518454150375409471970818190164225788510826070855560966773791628680423951169} a^{5} + \frac{430944725837969476523551690271507885425960820002956008471409501352054941020871437613207254971312}{4194853604551201106005518454150375409471970818190164225788510826070855560966773791628680423951169} a^{4} + \frac{1835357465116083014122395781653904796059435932623555729124009091316177542404745884342753812467035}{4194853604551201106005518454150375409471970818190164225788510826070855560966773791628680423951169} a^{3} - \frac{1117990269178264179448884519446466373235422809844126995476515451892238125272861258411486196723320}{4194853604551201106005518454150375409471970818190164225788510826070855560966773791628680423951169} a^{2} - \frac{1187771659743594449308929526411027375841901132099720636192431168250183127881024044425029373179643}{4194853604551201106005518454150375409471970818190164225788510826070855560966773791628680423951169} a + \frac{54094880276802157832207617357675835061275554563995951488371586620342575668576418149336231154589}{4194853604551201106005518454150375409471970818190164225788510826070855560966773791628680423951169}$
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: | \( 725348753199178600 \) (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
| 3.3.667489.1, 7.7.6321363049.1 |
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 | $21$ | ${\href{/LocalNumberField/3.7.0.1}{7} }^{3}$ | ${\href{/LocalNumberField/5.7.0.1}{7} }^{3}$ | ${\href{/LocalNumberField/7.3.0.1}{3} }^{7}$ | ${\href{/LocalNumberField/11.7.0.1}{7} }^{3}$ | ${\href{/LocalNumberField/13.7.0.1}{7} }^{3}$ | $21$ | R | $21$ | ${\href{/LocalNumberField/29.7.0.1}{7} }^{3}$ | $21$ | ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ | $21$ | R | $21$ | $21$ | $21$ |
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 |
|---|---|---|---|---|---|---|---|
| 19 | Data not computed | ||||||
| 43 | Data not computed | ||||||