Normalized defining polynomial
\( x^{21} - 126 x^{19} - 126 x^{18} + 6174 x^{17} + 11760 x^{16} - 145530 x^{15} - 403956 x^{14} + 1638168 x^{13} + 6429192 x^{12} - 6395823 x^{11} - 48369027 x^{10} - 21758891 x^{9} + 148886451 x^{8} + 200914581 x^{7} - 81540704 x^{6} - 293313510 x^{5} - 130714458 x^{4} + 46911963 x^{3} + 32454954 x^{2} - 2195739 x - 1148581 \)
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$$\chi_{441}(64,·)$, $\chi_{441}(1,·)$, $\chi_{441}(403,·)$, $\chi_{441}(340,·)$, $\chi_{441}(277,·)$, $\chi_{441}(214,·)$, $\chi_{441}(151,·)$, $\chi_{441}(88,·)$, $\chi_{441}(25,·)$, $\chi_{441}(436,·)$, $\chi_{441}(373,·)$, $\chi_{441}(310,·)$, $\chi_{441}(247,·)$, $\chi_{441}(184,·)$, $\chi_{441}(121,·)$, $\chi_{441}(58,·)$, $\chi_{441}(379,·)$, $\chi_{441}(316,·)$, $\chi_{441}(253,·)$, $\chi_{441}(190,·)$, $\chi_{441}(127,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{31} a^{16} + \frac{2}{31} a^{15} + \frac{12}{31} a^{14} + \frac{2}{31} a^{13} + \frac{7}{31} a^{12} - \frac{13}{31} a^{11} - \frac{9}{31} a^{10} - \frac{15}{31} a^{9} + \frac{6}{31} a^{8} + \frac{10}{31} a^{7} + \frac{2}{31} a^{6} - \frac{5}{31} a^{5} - \frac{8}{31} a^{4} + \frac{13}{31} a^{3} - \frac{8}{31} a^{2} + \frac{4}{31} a$, $\frac{1}{31} a^{17} + \frac{8}{31} a^{15} + \frac{9}{31} a^{14} + \frac{3}{31} a^{13} + \frac{4}{31} a^{12} - \frac{14}{31} a^{11} + \frac{3}{31} a^{10} + \frac{5}{31} a^{9} - \frac{2}{31} a^{8} + \frac{13}{31} a^{7} - \frac{9}{31} a^{6} + \frac{2}{31} a^{5} - \frac{2}{31} a^{4} - \frac{3}{31} a^{3} - \frac{11}{31} a^{2} - \frac{8}{31} a$, $\frac{1}{31} a^{18} - \frac{7}{31} a^{15} - \frac{12}{31} a^{13} - \frac{8}{31} a^{12} + \frac{14}{31} a^{11} + \frac{15}{31} a^{10} - \frac{6}{31} a^{9} - \frac{4}{31} a^{8} + \frac{4}{31} a^{7} - \frac{14}{31} a^{6} + \frac{7}{31} a^{5} - \frac{1}{31} a^{4} + \frac{9}{31} a^{3} - \frac{6}{31} a^{2} - \frac{1}{31} a$, $\frac{1}{198568488193} a^{19} - \frac{1963093208}{198568488193} a^{18} - \frac{2130700570}{198568488193} a^{17} + \frac{1473783435}{198568488193} a^{16} - \frac{26134845722}{198568488193} a^{15} - \frac{53797383297}{198568488193} a^{14} + \frac{98012076260}{198568488193} a^{13} + \frac{85567697330}{198568488193} a^{12} - \frac{5188837808}{198568488193} a^{11} - \frac{14798208455}{198568488193} a^{10} + \frac{12222119564}{198568488193} a^{9} - \frac{10467406724}{198568488193} a^{8} - \frac{86431922343}{198568488193} a^{7} - \frac{30116099667}{198568488193} a^{6} - \frac{99189511776}{198568488193} a^{5} + \frac{51927877845}{198568488193} a^{4} + \frac{93673121835}{198568488193} a^{3} - \frac{97142112826}{198568488193} a^{2} + \frac{79880584004}{198568488193} a + \frac{104450}{1210171}$, $\frac{1}{8940244642488969252365538807324154343629835943867986075599} a^{20} - \frac{20367815696017094133511174665582685648613001313}{8940244642488969252365538807324154343629835943867986075599} a^{19} + \frac{137945854430075035418621858102871752540422765572785704573}{8940244642488969252365538807324154343629835943867986075599} a^{18} + \frac{63341264847491891401570909135885793855046873510595435150}{8940244642488969252365538807324154343629835943867986075599} a^{17} + \frac{61718884131662867621491173562899698268868835145793686465}{8940244642488969252365538807324154343629835943867986075599} a^{16} + \frac{2654340483816097214196244360867634911925714764823224560812}{8940244642488969252365538807324154343629835943867986075599} a^{15} + \frac{4238174020536199975561268422798142219140098230984728559772}{8940244642488969252365538807324154343629835943867986075599} a^{14} - \frac{244678583233946574185095169844881299658480505543553879392}{8940244642488969252365538807324154343629835943867986075599} a^{13} - \frac{154582601505407510432641053130554005671037848765863301099}{8940244642488969252365538807324154343629835943867986075599} a^{12} + \frac{2687486061307556111399267903974683591034324201646015345906}{8940244642488969252365538807324154343629835943867986075599} a^{11} + \frac{3321289714365877919503191899462563072460267807152387555583}{8940244642488969252365538807324154343629835943867986075599} a^{10} + \frac{2068617098257418401195933378405286242574728180409086756886}{8940244642488969252365538807324154343629835943867986075599} a^{9} - \frac{3543009014401994042657653796029616805622645480493430766296}{8940244642488969252365538807324154343629835943867986075599} a^{8} - \frac{1571451863259946491700818729475054773027701364981250381917}{8940244642488969252365538807324154343629835943867986075599} a^{7} + \frac{1277344648556015522829200543262406267671463659477574308869}{8940244642488969252365538807324154343629835943867986075599} a^{6} - \frac{606549628983664265039019151184997197099074691099800538651}{8940244642488969252365538807324154343629835943867986075599} a^{5} - \frac{1614569811380776793087589343033682813233615735258137382835}{8940244642488969252365538807324154343629835943867986075599} a^{4} - \frac{4015307268282246912184036131256073084654061399440791480898}{8940244642488969252365538807324154343629835943867986075599} a^{3} - \frac{3563842917468563274999190519240156383283421549704661972083}{8940244642488969252365538807324154343629835943867986075599} a^{2} + \frac{147585257150619152557534080226188622135123082286737132993}{8940244642488969252365538807324154343629835943867986075599} a - \frac{6944282224055127825938553230299957749273634488008955}{54486111556279256549219229337129101391550836734262453}$
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: | \( 7412107993664674.0 \) (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
| 3.3.3969.1, 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 | ${\href{/LocalNumberField/2.7.0.1}{7} }^{3}$ | R | $21$ | R | $21$ | $21$ | $21$ | ${\href{/LocalNumberField/19.3.0.1}{3} }^{7}$ | $21$ | $21$ | ${\href{/LocalNumberField/31.1.0.1}{1} }^{21}$ | $21$ | $21$ | $21$ | ${\href{/LocalNumberField/47.7.0.1}{7} }^{3}$ | $21$ | ${\href{/LocalNumberField/59.7.0.1}{7} }^{3}$ |
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 | ||||||