Normalized defining polynomial
\( x^{21} - 7 x^{20} - 70 x^{19} + 462 x^{18} + 2135 x^{17} - 12411 x^{16} - 36610 x^{15} + 175044 x^{14} + 373940 x^{13} - 1403661 x^{12} - 2218069 x^{11} + 6566007 x^{10} + 6982234 x^{9} - 17907827 x^{8} - 9448729 x^{7} + 26548844 x^{6} + 686581 x^{5} - 16732429 x^{4} + 5281647 x^{3} + 1717044 x^{2} - 573440 x - 71167 \)
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: | \(10440862413718200815536341817705502373881381689=7^{36}\cdot 13^{14}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $155.37$ | 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: | $7, 13$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(637=7^{2}\cdot 13\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{637}(1,·)$, $\chi_{637}(386,·)$, $\chi_{637}(456,·)$, $\chi_{637}(393,·)$, $\chi_{637}(204,·)$, $\chi_{637}(120,·)$, $\chi_{637}(274,·)$, $\chi_{637}(211,·)$, $\chi_{637}(22,·)$, $\chi_{637}(92,·)$, $\chi_{637}(477,·)$, $\chi_{637}(547,·)$, $\chi_{637}(484,·)$, $\chi_{637}(295,·)$, $\chi_{637}(365,·)$, $\chi_{637}(302,·)$, $\chi_{637}(29,·)$, $\chi_{637}(113,·)$, $\chi_{637}(183,·)$, $\chi_{637}(568,·)$, $\chi_{637}(575,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{31} a^{14} - \frac{12}{31} a^{13} - \frac{14}{31} a^{12} + \frac{13}{31} a^{11} - \frac{11}{31} a^{10} + \frac{5}{31} a^{9} + \frac{15}{31} a^{8} + \frac{6}{31} a^{7} - \frac{7}{31} a^{6} - \frac{11}{31} a^{5} + \frac{12}{31} a^{4} - \frac{9}{31} a^{3} - \frac{14}{31} a^{2} + \frac{7}{31} a - \frac{14}{31}$, $\frac{1}{31} a^{15} - \frac{3}{31} a^{13} - \frac{10}{31} a^{11} - \frac{3}{31} a^{10} + \frac{13}{31} a^{9} + \frac{3}{31} a^{7} - \frac{2}{31} a^{6} + \frac{4}{31} a^{5} + \frac{11}{31} a^{4} + \frac{2}{31} a^{3} - \frac{6}{31} a^{2} + \frac{8}{31} a - \frac{13}{31}$, $\frac{1}{31} a^{16} - \frac{5}{31} a^{13} + \frac{10}{31} a^{12} + \frac{5}{31} a^{11} + \frac{11}{31} a^{10} + \frac{15}{31} a^{9} - \frac{14}{31} a^{8} - \frac{15}{31} a^{7} + \frac{14}{31} a^{6} + \frac{9}{31} a^{5} + \frac{7}{31} a^{4} - \frac{2}{31} a^{3} - \frac{3}{31} a^{2} + \frac{8}{31} a - \frac{11}{31}$, $\frac{1}{31} a^{17} + \frac{12}{31} a^{13} - \frac{3}{31} a^{12} + \frac{14}{31} a^{11} - \frac{9}{31} a^{10} + \frac{11}{31} a^{9} - \frac{2}{31} a^{8} + \frac{13}{31} a^{7} + \frac{5}{31} a^{6} + \frac{14}{31} a^{5} - \frac{4}{31} a^{4} + \frac{14}{31} a^{3} - \frac{7}{31} a - \frac{8}{31}$, $\frac{1}{46531} a^{18} - \frac{258}{46531} a^{17} - \frac{692}{46531} a^{16} - \frac{423}{46531} a^{15} + \frac{363}{46531} a^{14} - \frac{20779}{46531} a^{13} + \frac{15025}{46531} a^{12} + \frac{14205}{46531} a^{11} - \frac{16210}{46531} a^{10} + \frac{8673}{46531} a^{9} - \frac{11767}{46531} a^{8} + \frac{12115}{46531} a^{7} - \frac{12327}{46531} a^{6} + \frac{19292}{46531} a^{5} - \frac{22312}{46531} a^{4} + \frac{3935}{46531} a^{3} + \frac{20897}{46531} a^{2} + \frac{19825}{46531} a - \frac{177}{1501}$, $\frac{1}{46531} a^{19} + \frac{289}{46531} a^{17} - \frac{340}{46531} a^{16} - \frac{699}{46531} a^{15} - \frac{674}{46531} a^{14} + \frac{15615}{46531} a^{13} - \frac{10444}{46531} a^{12} + \frac{234}{1501} a^{11} - \frac{5230}{46531} a^{10} - \frac{15126}{46531} a^{9} - \frac{15767}{46531} a^{8} - \frac{14741}{46531} a^{7} + \frac{22547}{46531} a^{6} + \frac{1724}{46531} a^{5} + \frac{772}{46531} a^{4} - \frac{530}{2449} a^{3} - \frac{1355}{46531} a^{2} - \frac{195}{1501} a - \frac{15213}{46531}$, $\frac{1}{17499943721807387295556364023404215484007169} a^{20} - \frac{2230153570307829431371570607549900556}{221518274959587180956409671182331841569711} a^{19} - \frac{48329008573369209854778944232790529330}{17499943721807387295556364023404215484007169} a^{18} + \frac{95348143569014916097128887722367256574041}{17499943721807387295556364023404215484007169} a^{17} - \frac{20447872305634701528538876702568101131297}{17499943721807387295556364023404215484007169} a^{16} - \frac{1239129931223673264961973426278467479095}{564514313606689912759882710432394047871199} a^{15} - \frac{43877519317517923380627483678834993331391}{17499943721807387295556364023404215484007169} a^{14} + \frac{8588265837762619960712139190349195599217672}{17499943721807387295556364023404215484007169} a^{13} + \frac{70058821897554449023069800760252789985316}{221518274959587180956409671182331841569711} a^{12} + \frac{1517581196754588641554318652575523500964997}{17499943721807387295556364023404215484007169} a^{11} + \frac{7548546162624326843340188326993305360424790}{17499943721807387295556364023404215484007169} a^{10} + \frac{5266390417504098960166537584808608445478068}{17499943721807387295556364023404215484007169} a^{9} + \frac{355332273425523448198599656636998507847133}{921049669568809857660861264389695551789851} a^{8} + \frac{7303991305364065321997390358712776994622891}{17499943721807387295556364023404215484007169} a^{7} + \frac{5577454272122423029644543953430626455968267}{17499943721807387295556364023404215484007169} a^{6} - \frac{2698682101677975441122795489962028385558595}{17499943721807387295556364023404215484007169} a^{5} - \frac{397734088208613470464866367753470256766452}{921049669568809857660861264389695551789851} a^{4} - \frac{2960462910602866741878496321273349903377295}{17499943721807387295556364023404215484007169} a^{3} - \frac{5060663329685689660822334704547820603670668}{17499943721807387295556364023404215484007169} a^{2} + \frac{8440126751481265800150765232297941202195527}{17499943721807387295556364023404215484007169} a + \frac{2912881243876993043840258254932085717031299}{17499943721807387295556364023404215484007169}$
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: | \( 24318621275742050 \) (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
| 3.3.169.1, 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 | $21$ | $21$ | ${\href{/LocalNumberField/5.7.0.1}{7} }^{3}$ | R | $21$ | R | $21$ | ${\href{/LocalNumberField/19.3.0.1}{3} }^{7}$ | $21$ | $21$ | ${\href{/LocalNumberField/31.1.0.1}{1} }^{21}$ | $21$ | $21$ | $21$ | ${\href{/LocalNumberField/47.7.0.1}{7} }^{3}$ | ${\href{/LocalNumberField/53.7.0.1}{7} }^{3}$ | $21$ |
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 | ||||||
| 13 | Data not computed | ||||||