Normalized defining polynomial
\( x^{21} - 3 x^{20} - 180 x^{19} + 246 x^{18} + 13731 x^{17} + 1647 x^{16} - 557982 x^{15} - 788016 x^{14} + 12481956 x^{13} + 32581845 x^{12} - 140147985 x^{11} - 580762089 x^{10} + 459856514 x^{9} + 4704604611 x^{8} + 3799626111 x^{7} - 13678950530 x^{6} - 26746513641 x^{5} - 2530942323 x^{4} + 29275076935 x^{3} + 22576339788 x^{2} + 2145270072 x - 1693276957 \)
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: | \(3265364406784528133678938964803591475967396373221129=3^{28}\cdot 7^{14}\cdot 29^{18}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $283.82$ | 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: | $3, 7, 29$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1827=3^{2}\cdot 7\cdot 29\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1827}(256,·)$, $\chi_{1827}(1,·)$, $\chi_{1827}(835,·)$, $\chi_{1827}(1654,·)$, $\chi_{1827}(268,·)$, $\chi_{1827}(16,·)$, $\chi_{1827}(442,·)$, $\chi_{1827}(1765,·)$, $\chi_{1827}(1702,·)$, $\chi_{1827}(1705,·)$, $\chi_{1827}(1387,·)$, $\chi_{1827}(1009,·)$, $\chi_{1827}(1138,·)$, $\chi_{1827}(886,·)$, $\chi_{1827}(1591,·)$, $\chi_{1827}(1528,·)$, $\chi_{1827}(697,·)$, $\chi_{1827}(634,·)$, $\chi_{1827}(571,·)$, $\chi_{1827}(1213,·)$, $\chi_{1827}(190,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{17} a^{18} + \frac{6}{17} a^{17} + \frac{4}{17} a^{16} + \frac{7}{17} a^{15} - \frac{6}{17} a^{14} + \frac{4}{17} a^{12} - \frac{7}{17} a^{11} - \frac{7}{17} a^{10} - \frac{6}{17} a^{9} - \frac{7}{17} a^{8} + \frac{2}{17} a^{5} + \frac{6}{17} a^{4} - \frac{6}{17} a^{3} - \frac{4}{17} a^{2} + \frac{8}{17} a + \frac{2}{17}$, $\frac{1}{17} a^{19} + \frac{2}{17} a^{17} + \frac{3}{17} a^{15} + \frac{2}{17} a^{14} + \frac{4}{17} a^{13} + \frac{3}{17} a^{12} + \frac{1}{17} a^{11} + \frac{2}{17} a^{10} - \frac{5}{17} a^{9} + \frac{8}{17} a^{8} + \frac{2}{17} a^{6} - \frac{6}{17} a^{5} - \frac{8}{17} a^{4} - \frac{2}{17} a^{3} - \frac{2}{17} a^{2} + \frac{5}{17} a + \frac{5}{17}$, $\frac{1}{1352545526336251861297614318658517447864307032751759129401612172768748969602148961903} a^{20} + \frac{591667683439965762468383817882428395844766969180633663698219731915981899479934238}{79561501549191285958683195215206908697900413691279948788330127809926409976596997759} a^{19} - \frac{30521589947021915226853440786493620794317405536497382581932732694395444632695662787}{1352545526336251861297614318658517447864307032751759129401612172768748969602148961903} a^{18} + \frac{159434569063990201893224305173480777960741838322678164078925386452976328694578915472}{1352545526336251861297614318658517447864307032751759129401612172768748969602148961903} a^{17} + \frac{306697761403581480184960436068391052026592072887608297586352133981593382510288749799}{1352545526336251861297614318658517447864307032751759129401612172768748969602148961903} a^{16} - \frac{537855522845234455351242854546585282907512788000019848907905544783758040776063145034}{1352545526336251861297614318658517447864307032751759129401612172768748969602148961903} a^{15} - \frac{593491167504108316459659829588341818249429988909042060524096798673957158910225329324}{1352545526336251861297614318658517447864307032751759129401612172768748969602148961903} a^{14} - \frac{489643129888101316384914647238322272968816459704436308696031341713390935410379308811}{1352545526336251861297614318658517447864307032751759129401612172768748969602148961903} a^{13} - \frac{136505351117912933759746992582147178176913720597610248397039087482794805860995454604}{1352545526336251861297614318658517447864307032751759129401612172768748969602148961903} a^{12} - \frac{422833564646701004164871662954432436309677031081186989631899961189947111952376241917}{1352545526336251861297614318658517447864307032751759129401612172768748969602148961903} a^{11} - \frac{323688969616998174005035698765213952091630251508693883480778093281297003510119559695}{1352545526336251861297614318658517447864307032751759129401612172768748969602148961903} a^{10} + \frac{408292176023361030860145973956478387270169675272347852127368851293439344422815871445}{1352545526336251861297614318658517447864307032751759129401612172768748969602148961903} a^{9} + \frac{233156590841241839848265336528791954020004542275572069249582297580618381888030002218}{1352545526336251861297614318658517447864307032751759129401612172768748969602148961903} a^{8} - \frac{553059647152200315111882834608120945727499698780217990844475320499938039880877600648}{1352545526336251861297614318658517447864307032751759129401612172768748969602148961903} a^{7} + \frac{118681025537984327436633055452096349525327960345760529142301283366520948761808265468}{1352545526336251861297614318658517447864307032751759129401612172768748969602148961903} a^{6} - \frac{101065295952068844119266056842353189610028531335335862741125584302683366890970801153}{1352545526336251861297614318658517447864307032751759129401612172768748969602148961903} a^{5} - \frac{460559404204718220259621729993282878903333172134415926719528358809696566326037956384}{1352545526336251861297614318658517447864307032751759129401612172768748969602148961903} a^{4} + \frac{7173973539181933865415337648217663704632395546992798631382390024977743611243253667}{79561501549191285958683195215206908697900413691279948788330127809926409976596997759} a^{3} + \frac{530084635808074082754714203102812119566199493673535889276243459626210115043140352615}{1352545526336251861297614318658517447864307032751759129401612172768748969602148961903} a^{2} + \frac{583825549343690938539004004504636574794861688716415828584275130130041676474134876499}{1352545526336251861297614318658517447864307032751759129401612172768748969602148961903} a + \frac{50335206477599951197082277228209034112873985483277860376551540897548181976343691439}{1352545526336251861297614318658517447864307032751759129401612172768748969602148961903}$
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: | \( 1679101587664927000 \) (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, 7.7.594823321.1 |
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 | ${\href{/LocalNumberField/11.7.0.1}{7} }^{3}$ | $21$ | ${\href{/LocalNumberField/17.3.0.1}{3} }^{7}$ | $21$ | ${\href{/LocalNumberField/23.7.0.1}{7} }^{3}$ | R | $21$ | $21$ | ${\href{/LocalNumberField/41.3.0.1}{3} }^{7}$ | $21$ | $21$ | $21$ | ${\href{/LocalNumberField/59.3.0.1}{3} }^{7}$ |
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 | ||||||
| 29 | Data not computed | ||||||