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Properties

Label 21.21.4670561677...4001.1
Degree $21$
Signature $[21, 0]$
Discriminant $43^{20}$
Root discriminant $35.95$
Ramified prime $43$
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![-1, 11, 55, -220, -495, 1287, 1716, -3432, -3003, 5005, 3003, -4368, -1820, 2380, 680, -816, -153, 171, 19, -20, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - x^20 - 20*x^19 + 19*x^18 + 171*x^17 - 153*x^16 - 816*x^15 + 680*x^14 + 2380*x^13 - 1820*x^12 - 4368*x^11 + 3003*x^10 + 5005*x^9 - 3003*x^8 - 3432*x^7 + 1716*x^6 + 1287*x^5 - 495*x^4 - 220*x^3 + 55*x^2 + 11*x - 1)
 
gp: K = bnfinit(x^21 - x^20 - 20*x^19 + 19*x^18 + 171*x^17 - 153*x^16 - 816*x^15 + 680*x^14 + 2380*x^13 - 1820*x^12 - 4368*x^11 + 3003*x^10 + 5005*x^9 - 3003*x^8 - 3432*x^7 + 1716*x^6 + 1287*x^5 - 495*x^4 - 220*x^3 + 55*x^2 + 11*x - 1, 1)
 

Normalized defining polynomial

\( x^{21} - x^{20} - 20 x^{19} + 19 x^{18} + 171 x^{17} - 153 x^{16} - 816 x^{15} + 680 x^{14} + 2380 x^{13} - 1820 x^{12} - 4368 x^{11} + 3003 x^{10} + 5005 x^{9} - 3003 x^{8} - 3432 x^{7} + 1716 x^{6} + 1287 x^{5} - 495 x^{4} - 220 x^{3} + 55 x^{2} + 11 x - 1 \)

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:  \(467056167777397914441056671494001=43^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.95$
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:  $43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(43\)
Dirichlet character group:    $\lbrace$$\chi_{43}(1,·)$, $\chi_{43}(4,·)$, $\chi_{43}(6,·)$, $\chi_{43}(9,·)$, $\chi_{43}(10,·)$, $\chi_{43}(11,·)$, $\chi_{43}(13,·)$, $\chi_{43}(14,·)$, $\chi_{43}(15,·)$, $\chi_{43}(16,·)$, $\chi_{43}(17,·)$, $\chi_{43}(21,·)$, $\chi_{43}(23,·)$, $\chi_{43}(24,·)$, $\chi_{43}(25,·)$, $\chi_{43}(31,·)$, $\chi_{43}(35,·)$, $\chi_{43}(36,·)$, $\chi_{43}(38,·)$, $\chi_{43}(40,·)$, $\chi_{43}(41,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$

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:  \( 2620717953.48 \) (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

3.3.1849.1, 7.7.6321363049.1

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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{3}$ $21$ $21$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{3}$ $21$ $21$ $21$ $21$ $21$ $21$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{3}$ R ${\href{/LocalNumberField/47.7.0.1}{7} }^{3}$ $21$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{3}$

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

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
43Data not computed