↑

Properties

Label 21.21.1681567772...8481.1
Degree $21$
Signature $[21, 0]$
Discriminant $19^{14}\cdot 29^{18}$
Root discriminant $127.64$
Ramified primes $19, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{21}$ (as 21T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![21401, -152624, -336298, 3268313, -1687630, -8733417, 7456241, 7775839, -8725818, -2409705, 4265962, 79957, -1050735, 103892, 138634, -22526, -9666, 1940, 326, -74, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 4*x^20 - 74*x^19 + 326*x^18 + 1940*x^17 - 9666*x^16 - 22526*x^15 + 138634*x^14 + 103892*x^13 - 1050735*x^12 + 79957*x^11 + 4265962*x^10 - 2409705*x^9 - 8725818*x^8 + 7775839*x^7 + 7456241*x^6 - 8733417*x^5 - 1687630*x^4 + 3268313*x^3 - 336298*x^2 - 152624*x + 21401)
 
gp: K = bnfinit(x^21 - 4*x^20 - 74*x^19 + 326*x^18 + 1940*x^17 - 9666*x^16 - 22526*x^15 + 138634*x^14 + 103892*x^13 - 1050735*x^12 + 79957*x^11 + 4265962*x^10 - 2409705*x^9 - 8725818*x^8 + 7775839*x^7 + 7456241*x^6 - 8733417*x^5 - 1687630*x^4 + 3268313*x^3 - 336298*x^2 - 152624*x + 21401, 1)
 

Normalized defining polynomial

\( x^{21} - 4 x^{20} - 74 x^{19} + 326 x^{18} + 1940 x^{17} - 9666 x^{16} - 22526 x^{15} + 138634 x^{14} + 103892 x^{13} - 1050735 x^{12} + 79957 x^{11} + 4265962 x^{10} - 2409705 x^{9} - 8725818 x^{8} + 7775839 x^{7} + 7456241 x^{6} - 8733417 x^{5} - 1687630 x^{4} + 3268313 x^{3} - 336298 x^{2} - 152624 x + 21401 \)

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:  \(168156777279482171397813050167498470631248481=19^{14}\cdot 29^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $127.64$
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:  $19, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(551=19\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{551}(1,·)$, $\chi_{551}(197,·)$, $\chi_{551}(7,·)$, $\chi_{551}(140,·)$, $\chi_{551}(429,·)$, $\chi_{551}(400,·)$, $\chi_{551}(210,·)$, $\chi_{551}(83,·)$, $\chi_{551}(20,·)$, $\chi_{551}(277,·)$, $\chi_{551}(343,·)$, $\chi_{551}(349,·)$, $\chi_{551}(286,·)$, $\chi_{551}(45,·)$, $\chi_{551}(239,·)$, $\chi_{551}(368,·)$, $\chi_{551}(49,·)$, $\chi_{551}(372,·)$, $\chi_{551}(30,·)$, $\chi_{551}(248,·)$, $\chi_{551}(315,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{17} a^{18} + \frac{5}{17} a^{17} + \frac{7}{17} a^{16} + \frac{1}{17} a^{15} + \frac{7}{17} a^{14} + \frac{6}{17} a^{13} - \frac{8}{17} a^{12} - \frac{8}{17} a^{11} - \frac{6}{17} a^{10} + \frac{4}{17} a^{9} + \frac{1}{17} a^{8} + \frac{7}{17} a^{7} - \frac{3}{17} a^{6} + \frac{7}{17} a^{5} - \frac{4}{17} a^{4} + \frac{6}{17} a^{3} + \frac{7}{17} a^{2} - \frac{5}{17} a + \frac{7}{17}$, $\frac{1}{17} a^{19} - \frac{1}{17} a^{17} + \frac{2}{17} a^{15} + \frac{5}{17} a^{14} - \frac{4}{17} a^{13} - \frac{2}{17} a^{12} - \frac{2}{17} a^{9} + \frac{2}{17} a^{8} - \frac{4}{17} a^{7} + \frac{5}{17} a^{6} - \frac{5}{17} a^{5} - \frac{8}{17} a^{4} - \frac{6}{17} a^{3} - \frac{6}{17} a^{2} - \frac{2}{17} a - \frac{1}{17}$, $\frac{1}{332136228022977283712378590606514320385514036099305329} a^{20} - \frac{8448367280233582194671637978188004938129318894342824}{332136228022977283712378590606514320385514036099305329} a^{19} + \frac{1052456353184180384567052479577063171359524560629931}{332136228022977283712378590606514320385514036099305329} a^{18} - \frac{124831461531128304751224274144359802986805491805891994}{332136228022977283712378590606514320385514036099305329} a^{17} - \frac{37718167606464148336905341006305228591401990381078998}{332136228022977283712378590606514320385514036099305329} a^{16} - \frac{107947375062968502908633358148560007356503521940421170}{332136228022977283712378590606514320385514036099305329} a^{15} - \frac{44997737068926640634866314180900380114297511539702825}{332136228022977283712378590606514320385514036099305329} a^{14} - \frac{70575008286826297236063973957623673893159340550603243}{332136228022977283712378590606514320385514036099305329} a^{13} + \frac{77756830528815082950653364311992518313480414463628488}{332136228022977283712378590606514320385514036099305329} a^{12} - \frac{5170366039080690889804340079976568258855547529862163}{332136228022977283712378590606514320385514036099305329} a^{11} + \frac{43767643597249141736994621222315768366606697071992786}{332136228022977283712378590606514320385514036099305329} a^{10} - \frac{8321066928532090328718128971533908806808724954660429}{19537425177822193159551681800383195316794943299959137} a^{9} - \frac{133462405361555139173021907263945674987662517903631613}{332136228022977283712378590606514320385514036099305329} a^{8} + \frac{132525045472420706804394880935523197731048438597611113}{332136228022977283712378590606514320385514036099305329} a^{7} + \frac{42794740401876801212920775497491785703536780376540997}{332136228022977283712378590606514320385514036099305329} a^{6} - \frac{73189454548401829645679379895005197524152170060525405}{332136228022977283712378590606514320385514036099305329} a^{5} + \frac{99238886596569307023684793809221928950860621354039545}{332136228022977283712378590606514320385514036099305329} a^{4} - \frac{75277709390753377134021616228633906856594338361627951}{332136228022977283712378590606514320385514036099305329} a^{3} - \frac{305270580584756271129752509954967551070512313720732}{332136228022977283712378590606514320385514036099305329} a^{2} - \frac{45527526969482990436173426744148859760495322625779598}{332136228022977283712378590606514320385514036099305329} a + \frac{157669681541630450972679913396758899003771444348299490}{332136228022977283712378590606514320385514036099305329}$

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:  \( 1584733822664509.5 \) (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

3.3.361.1, 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 $21$ $21$ $21$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{3}$ $21$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{7}$ R $21$ R ${\href{/LocalNumberField/31.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{7}$ $21$ $21$ $21$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{7}$

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

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
19Data not computed
29Data not computed