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Properties

Label 21.21.6066309620...2561.1
Degree $21$
Signature $[21, 0]$
Discriminant $3^{28}\cdot 7^{36}$
Root discriminant $121.59$
Ramified primes $3, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{21}$ (as 21T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-38809, -66192, 948738, 3366867, 1051050, -9277107, -11269811, 3233979, 10909626, 2152563, -4067070, -1518279, 743141, 365904, -70242, -43722, 3234, 2730, -56, -84, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 84*x^19 - 56*x^18 + 2730*x^17 + 3234*x^16 - 43722*x^15 - 70242*x^14 + 365904*x^13 + 743141*x^12 - 1518279*x^11 - 4067070*x^10 + 2152563*x^9 + 10909626*x^8 + 3233979*x^7 - 11269811*x^6 - 9277107*x^5 + 1051050*x^4 + 3366867*x^3 + 948738*x^2 - 66192*x - 38809)
 
gp: K = bnfinit(x^21 - 84*x^19 - 56*x^18 + 2730*x^17 + 3234*x^16 - 43722*x^15 - 70242*x^14 + 365904*x^13 + 743141*x^12 - 1518279*x^11 - 4067070*x^10 + 2152563*x^9 + 10909626*x^8 + 3233979*x^7 - 11269811*x^6 - 9277107*x^5 + 1051050*x^4 + 3366867*x^3 + 948738*x^2 - 66192*x - 38809, 1)
 

Normalized defining polynomial

\( x^{21} - 84 x^{19} - 56 x^{18} + 2730 x^{17} + 3234 x^{16} - 43722 x^{15} - 70242 x^{14} + 365904 x^{13} + 743141 x^{12} - 1518279 x^{11} - 4067070 x^{10} + 2152563 x^{9} + 10909626 x^{8} + 3233979 x^{7} - 11269811 x^{6} - 9277107 x^{5} + 1051050 x^{4} + 3366867 x^{3} + 948738 x^{2} - 66192 x - 38809 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

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:  \(60663096207149471029120391038078960708572561=3^{28}\cdot 7^{36}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $121.59$
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$$\chi_{441}(64,·)$, $\chi_{441}(1,·)$, $\chi_{441}(400,·)$, $\chi_{441}(337,·)$, $\chi_{441}(274,·)$, $\chi_{441}(211,·)$, $\chi_{441}(148,·)$, $\chi_{441}(85,·)$, $\chi_{441}(22,·)$, $\chi_{441}(421,·)$, $\chi_{441}(358,·)$, $\chi_{441}(295,·)$, $\chi_{441}(232,·)$, $\chi_{441}(169,·)$, $\chi_{441}(106,·)$, $\chi_{441}(43,·)$, $\chi_{441}(379,·)$, $\chi_{441}(316,·)$, $\chi_{441}(253,·)$, $\chi_{441}(190,·)$, $\chi_{441}(127,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{19} a^{13} - \frac{1}{19} a^{12} - \frac{4}{19} a^{11} - \frac{8}{19} a^{10} - \frac{8}{19} a^{9} + \frac{9}{19} a^{8} + \frac{7}{19} a^{7} + \frac{8}{19} a^{6} + \frac{7}{19} a^{5} - \frac{9}{19} a^{4} + \frac{3}{19} a^{3} - \frac{6}{19} a^{2} - \frac{6}{19} a + \frac{7}{19}$, $\frac{1}{19} a^{14} - \frac{5}{19} a^{12} + \frac{7}{19} a^{11} + \frac{3}{19} a^{10} + \frac{1}{19} a^{9} - \frac{3}{19} a^{8} - \frac{4}{19} a^{7} - \frac{4}{19} a^{6} - \frac{2}{19} a^{5} - \frac{6}{19} a^{4} - \frac{3}{19} a^{3} + \frac{7}{19} a^{2} + \frac{1}{19} a + \frac{7}{19}$, $\frac{1}{19} a^{15} + \frac{2}{19} a^{12} + \frac{2}{19} a^{11} - \frac{1}{19} a^{10} - \frac{5}{19} a^{9} + \frac{3}{19} a^{8} - \frac{7}{19} a^{7} - \frac{9}{19} a^{5} + \frac{9}{19} a^{4} + \frac{3}{19} a^{3} + \frac{9}{19} a^{2} - \frac{4}{19} a - \frac{3}{19}$, $\frac{1}{19} a^{16} + \frac{4}{19} a^{12} + \frac{7}{19} a^{11} - \frac{8}{19} a^{10} - \frac{6}{19} a^{8} + \frac{5}{19} a^{7} - \frac{6}{19} a^{6} - \frac{5}{19} a^{5} + \frac{2}{19} a^{4} + \frac{3}{19} a^{3} + \frac{8}{19} a^{2} + \frac{9}{19} a + \frac{5}{19}$, $\frac{1}{19} a^{17} - \frac{8}{19} a^{12} + \frac{8}{19} a^{11} - \frac{6}{19} a^{10} + \frac{7}{19} a^{9} + \frac{7}{19} a^{8} + \frac{4}{19} a^{7} + \frac{1}{19} a^{6} - \frac{7}{19} a^{5} + \frac{1}{19} a^{4} - \frac{4}{19} a^{3} - \frac{5}{19} a^{2} - \frac{9}{19} a - \frac{9}{19}$, $\frac{1}{589} a^{18} + \frac{4}{589} a^{17} + \frac{2}{589} a^{16} - \frac{6}{589} a^{15} + \frac{3}{589} a^{14} - \frac{6}{589} a^{13} + \frac{145}{589} a^{12} - \frac{54}{589} a^{11} + \frac{213}{589} a^{10} - \frac{43}{589} a^{9} + \frac{163}{589} a^{8} - \frac{5}{589} a^{7} - \frac{277}{589} a^{6} - \frac{32}{589} a^{5} - \frac{10}{589} a^{4} + \frac{211}{589} a^{3} + \frac{56}{589} a^{2} + \frac{102}{589} a - \frac{239}{589}$, $\frac{1}{116033} a^{19} - \frac{40}{116033} a^{18} + \frac{2895}{116033} a^{17} + \frac{2541}{116033} a^{16} + \frac{1166}{116033} a^{15} - \frac{159}{6107} a^{14} + \frac{2455}{116033} a^{13} + \frac{3145}{116033} a^{12} - \frac{2843}{6107} a^{11} + \frac{39441}{116033} a^{10} - \frac{42616}{116033} a^{9} - \frac{2961}{116033} a^{8} - \frac{55764}{116033} a^{7} + \frac{39095}{116033} a^{6} - \frac{21263}{116033} a^{5} - \frac{1651}{3743} a^{4} - \frac{22031}{116033} a^{3} - \frac{29239}{116033} a^{2} - \frac{2650}{116033} a + \frac{119}{589}$, $\frac{1}{1858320355657315475664462402980699399450671} a^{20} - \frac{39584982186650592557178487481317592}{9433098252067591247027727933912179692643} a^{19} - \frac{309362821494389800820393123903808506018}{1858320355657315475664462402980699399450671} a^{18} + \frac{10688522418932730557002895642384625472143}{1858320355657315475664462402980699399450671} a^{17} - \frac{22120405799639365758699350705633320262093}{1858320355657315475664462402980699399450671} a^{16} + \frac{45728029223111450334583607158440831597451}{1858320355657315475664462402980699399450671} a^{15} + \frac{6835764372784208544590678710638840769915}{1858320355657315475664462402980699399450671} a^{14} - \frac{7649435022823163663798453165785325952543}{1858320355657315475664462402980699399450671} a^{13} - \frac{233628356807572306710676146574880890105007}{1858320355657315475664462402980699399450671} a^{12} - \frac{708187196421106442715080347607717481521114}{1858320355657315475664462402980699399450671} a^{11} - \frac{4530493948994997683373806747778836420583}{9433098252067591247027727933912179692643} a^{10} - \frac{617056579432262974106824383767556775033818}{1858320355657315475664462402980699399450671} a^{9} - \frac{387417806331498198351684651044308680462982}{1858320355657315475664462402980699399450671} a^{8} + \frac{384239701921840810973782414836604150489374}{1858320355657315475664462402980699399450671} a^{7} - \frac{818048192439091202730913162218797834419825}{1858320355657315475664462402980699399450671} a^{6} + \frac{906083619834242344667518534682811718805011}{1858320355657315475664462402980699399450671} a^{5} - \frac{544282749145848015687343384980646430701507}{1858320355657315475664462402980699399450671} a^{4} + \frac{421934760100903668733369745105392600263077}{1858320355657315475664462402980699399450671} a^{3} + \frac{903745477666883766385262512224541096912701}{1858320355657315475664462402980699399450671} a^{2} - \frac{248829195308469065993127704373079694215994}{1858320355657315475664462402980699399450671} a + \frac{3409780452903863212691204228737528036444}{9433098252067591247027727933912179692643}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.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:  \( 2419977962290507.0 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{21}$ (as 21T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 21
The 21 conjugacy class representatives for $C_{21}$
Character table for $C_{21}$ is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 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 $21$ R $21$ $21$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{21}$ $21$ $21$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{3}$ $21$ $21$ $21$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{3}$ $21$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
3Data not computed
7Data not computed