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Properties

Label 21.21.1299348114...8449.1
Degree $21$
Signature $[21, 0]$
Discriminant $7^{38}$
Root discriminant $33.82$
Ramified prime $7$
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![1, 14, -49, -371, 196, 2072, -294, -5147, 210, 7007, -77, -5733, 14, 2940, -1, -952, 0, 189, 0, -21, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 21*x^19 + 189*x^17 - 952*x^15 - x^14 + 2940*x^13 + 14*x^12 - 5733*x^11 - 77*x^10 + 7007*x^9 + 210*x^8 - 5147*x^7 - 294*x^6 + 2072*x^5 + 196*x^4 - 371*x^3 - 49*x^2 + 14*x + 1)
 
gp: K = bnfinit(x^21 - 21*x^19 + 189*x^17 - 952*x^15 - x^14 + 2940*x^13 + 14*x^12 - 5733*x^11 - 77*x^10 + 7007*x^9 + 210*x^8 - 5147*x^7 - 294*x^6 + 2072*x^5 + 196*x^4 - 371*x^3 - 49*x^2 + 14*x + 1, 1)
 

Normalized defining polynomial

\( x^{21} - 21 x^{19} + 189 x^{17} - 952 x^{15} - x^{14} + 2940 x^{13} + 14 x^{12} - 5733 x^{11} - 77 x^{10} + 7007 x^{9} + 210 x^{8} - 5147 x^{7} - 294 x^{6} + 2072 x^{5} + 196 x^{4} - 371 x^{3} - 49 x^{2} + 14 x + 1 \)

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:  \(129934811447123020117172145698449=7^{38}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.82$
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:  $7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(49=7^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{49}(1,·)$, $\chi_{49}(2,·)$, $\chi_{49}(4,·)$, $\chi_{49}(8,·)$, $\chi_{49}(9,·)$, $\chi_{49}(11,·)$, $\chi_{49}(15,·)$, $\chi_{49}(16,·)$, $\chi_{49}(18,·)$, $\chi_{49}(22,·)$, $\chi_{49}(23,·)$, $\chi_{49}(25,·)$, $\chi_{49}(29,·)$, $\chi_{49}(30,·)$, $\chi_{49}(32,·)$, $\chi_{49}(36,·)$, $\chi_{49}(37,·)$, $\chi_{49}(39,·)$, $\chi_{49}(43,·)$, $\chi_{49}(44,·)$, $\chi_{49}(46,·)$$\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:  \( 1694541833.65 \) (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

\(\Q(\zeta_{7})^+\), 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 $21$ $21$ $21$ R $21$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{3}$ $21$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{7}$ $21$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{7}$ $21$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{3}$ $21$ $21$ $21$

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
7Data not computed