Normalized defining polynomial
\( x^{21} - 126 x^{19} - 63 x^{18} + 6174 x^{17} + 5145 x^{16} - 151263 x^{15} - 153909 x^{14} + 2036391 x^{13} + 2323482 x^{12} - 15568182 x^{11} - 19772382 x^{10} + 67561690 x^{9} + 96070968 x^{8} - 159797400 x^{7} - 258336281 x^{6} + 184053996 x^{5} + 356224953 x^{4} - 78664992 x^{3} - 218834343 x^{2} + 6835059 x + 48473999 \)
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: | \(2972491714150324080426899160865869074720055489=3^{28}\cdot 7^{38}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $146.35$ | 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: | $3, 7$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(441=3^{2}\cdot 7^{2}\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{441}(256,·)$, $\chi_{441}(1,·)$, $\chi_{441}(130,·)$, $\chi_{441}(67,·)$, $\chi_{441}(4,·)$, $\chi_{441}(193,·)$, $\chi_{441}(64,·)$, $\chi_{441}(394,·)$, $\chi_{441}(331,·)$, $\chi_{441}(268,·)$, $\chi_{441}(205,·)$, $\chi_{441}(142,·)$, $\chi_{441}(79,·)$, $\chi_{441}(16,·)$, $\chi_{441}(319,·)$, $\chi_{441}(190,·)$, $\chi_{441}(379,·)$, $\chi_{441}(316,·)$, $\chi_{441}(253,·)$, $\chi_{441}(382,·)$, $\chi_{441}(127,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{1513160823031083794496902931981415267982533530886319675607646902592799} a^{20} - \frac{334010557596654123842216458148476795763477032817631878218350807560557}{1513160823031083794496902931981415267982533530886319675607646902592799} a^{19} + \frac{334191681479537972500457648512450923593543526163083512325303755590324}{1513160823031083794496902931981415267982533530886319675607646902592799} a^{18} + \frac{436240415758415031193514993762470917072709898244130808589292577521728}{1513160823031083794496902931981415267982533530886319675607646902592799} a^{17} + \frac{527851309494472822084202343851479964562351954922240376385657555021124}{1513160823031083794496902931981415267982533530886319675607646902592799} a^{16} - \frac{220867318152723924882542270488974669254073477151592634092605724296660}{1513160823031083794496902931981415267982533530886319675607646902592799} a^{15} - \frac{204578275208528431164943024251369479231796092145126343795989350984289}{1513160823031083794496902931981415267982533530886319675607646902592799} a^{14} + \frac{118212256139422197187516423795617211309457310273212389774105391987640}{1513160823031083794496902931981415267982533530886319675607646902592799} a^{13} - \frac{665121844625285445299504516547595195629609895643194795452201501520640}{1513160823031083794496902931981415267982533530886319675607646902592799} a^{12} + \frac{180808076949854740358218154702996939951741109033557085699751635459693}{1513160823031083794496902931981415267982533530886319675607646902592799} a^{11} - \frac{593343098995712050651715083317899687300587082339750274098341316826403}{1513160823031083794496902931981415267982533530886319675607646902592799} a^{10} - \frac{263434057299850928090435432594624711304327980293244676133120608659299}{1513160823031083794496902931981415267982533530886319675607646902592799} a^{9} + \frac{54291769699791046601961190096383951316033303329644171526025321308806}{1513160823031083794496902931981415267982533530886319675607646902592799} a^{8} + \frac{543119108617604702782940218338366148418223920712091235209148361171225}{1513160823031083794496902931981415267982533530886319675607646902592799} a^{7} - \frac{185765357423964131622402102586369549092518047016650761754286810689}{7681019406249156317243162091276219634429104217697054190901760926867} a^{6} + \frac{458098236024755846486530560077070524681908887704643866042705676052710}{1513160823031083794496902931981415267982533530886319675607646902592799} a^{5} + \frac{603177834022249989467764648361144251841789479863782065113235153617473}{1513160823031083794496902931981415267982533530886319675607646902592799} a^{4} - \frac{689763351622056486222876252812842315615231279322180772257018856196251}{1513160823031083794496902931981415267982533530886319675607646902592799} a^{3} - \frac{199298367592478810936510907750756452296723239197501710919184652584755}{1513160823031083794496902931981415267982533530886319675607646902592799} a^{2} + \frac{522076259881680801450320726249153537073251042881658998530194556713971}{1513160823031083794496902931981415267982533530886319675607646902592799} a - \frac{43026739164377982777662553385982952466916598115516109972391800}{218511490277861881407356560862038778271166253813807227442768407}$
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: | \( 2623927281262563.0 \) (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
| 3.3.3969.2, 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 | $21$ | R | ${\href{/LocalNumberField/5.7.0.1}{7} }^{3}$ | R | ${\href{/LocalNumberField/11.7.0.1}{7} }^{3}$ | $21$ | $21$ | ${\href{/LocalNumberField/19.3.0.1}{3} }^{7}$ | ${\href{/LocalNumberField/23.7.0.1}{7} }^{3}$ | $21$ | ${\href{/LocalNumberField/31.3.0.1}{3} }^{7}$ | $21$ | $21$ | $21$ | $21$ | $21$ | $21$ |
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 |
|---|---|---|---|---|---|---|---|
| 3 | Data not computed | ||||||
| 7 | Data not computed | ||||||