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Properties

Label 21.21.1034885639...5456.1
Degree $21$
Signature $[21, 0]$
Discriminant $2^{12}\cdot 3^{28}\cdot 7^{32}$
Root discriminant $124.72$
Ramified primes $2, 3, 7$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_7^2:C_3:C_3$ (as 21T21)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3818312, -38914176, -107778048, -70458304, 107778048, 132440952, -40416768, -84208818, 7217280, 28700672, -661584, -5870592, 30072, 752640, -537, -60928, 0, 3024, 0, -84, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 84*x^19 + 3024*x^17 - 60928*x^15 - 537*x^14 + 752640*x^13 + 30072*x^12 - 5870592*x^11 - 661584*x^10 + 28700672*x^9 + 7217280*x^8 - 84208818*x^7 - 40416768*x^6 + 132440952*x^5 + 107778048*x^4 - 70458304*x^3 - 107778048*x^2 - 38914176*x - 3818312)
 
gp: K = bnfinit(x^21 - 84*x^19 + 3024*x^17 - 60928*x^15 - 537*x^14 + 752640*x^13 + 30072*x^12 - 5870592*x^11 - 661584*x^10 + 28700672*x^9 + 7217280*x^8 - 84208818*x^7 - 40416768*x^6 + 132440952*x^5 + 107778048*x^4 - 70458304*x^3 - 107778048*x^2 - 38914176*x - 3818312, 1)
 

Normalized defining polynomial

\( x^{21} - 84 x^{19} + 3024 x^{17} - 60928 x^{15} - 537 x^{14} + 752640 x^{13} + 30072 x^{12} - 5870592 x^{11} - 661584 x^{10} + 28700672 x^{9} + 7217280 x^{8} - 84208818 x^{7} - 40416768 x^{6} + 132440952 x^{5} + 107778048 x^{4} - 70458304 x^{3} - 107778048 x^{2} - 38914176 x - 3818312 \)

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:  \(103488563958552367070086264761337535636115456=2^{12}\cdot 3^{28}\cdot 7^{32}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $124.72$
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:  $2, 3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{21} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a + \frac{10}{21}$, $\frac{1}{21} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{10}{21} a$, $\frac{1}{21} a^{9} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{10}{21} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{21} a^{10} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{10}{21} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{63} a^{11} - \frac{1}{63} a^{10} + \frac{1}{63} a^{9} + \frac{1}{63} a^{8} - \frac{1}{63} a^{7} + \frac{4}{9} a^{6} + \frac{1}{9} a^{5} - \frac{25}{63} a^{4} + \frac{25}{63} a^{3} + \frac{17}{63} a^{2} + \frac{31}{63} a - \frac{10}{63}$, $\frac{1}{441} a^{12} + \frac{1}{441} a^{11} - \frac{1}{441} a^{10} - \frac{1}{147} a^{9} - \frac{8}{441} a^{8} + \frac{5}{441} a^{7} + \frac{3}{7} a^{6} + \frac{31}{441} a^{5} - \frac{151}{441} a^{4} - \frac{17}{441} a^{3} - \frac{58}{441} a^{2} + \frac{151}{441} a + \frac{43}{441}$, $\frac{1}{441} a^{13} - \frac{2}{441} a^{11} - \frac{2}{441} a^{10} - \frac{5}{441} a^{9} - \frac{8}{441} a^{8} - \frac{5}{441} a^{7} - \frac{11}{441} a^{6} - \frac{26}{63} a^{5} - \frac{160}{441} a^{4} - \frac{41}{441} a^{3} - \frac{85}{441} a^{2} + \frac{41}{147} a - \frac{169}{441}$, $\frac{1}{882} a^{14} + \frac{1}{63} a^{10} - \frac{1}{63} a^{9} - \frac{1}{882} a^{7} + \frac{2}{9} a^{6} - \frac{1}{9} a^{5} + \frac{1}{9} a^{4} - \frac{25}{63} a^{3} + \frac{11}{63} a^{2} - \frac{4}{9} a + \frac{43}{441}$, $\frac{1}{882} a^{15} - \frac{1}{63} a^{9} - \frac{5}{294} a^{8} + \frac{4}{9} a^{6} - \frac{1}{3} a^{5} + \frac{4}{9} a^{3} + \frac{2}{7} a^{2} + \frac{40}{147} a - \frac{2}{9}$, $\frac{1}{882} a^{16} - \frac{1}{63} a^{10} - \frac{5}{294} a^{9} + \frac{1}{63} a^{7} - \frac{1}{3} a^{6} + \frac{4}{9} a^{4} + \frac{2}{7} a^{3} + \frac{40}{147} a^{2} - \frac{2}{9} a - \frac{2}{7}$, $\frac{1}{6174} a^{17} + \frac{1}{3087} a^{16} - \frac{1}{6174} a^{15} - \frac{1}{6174} a^{14} - \frac{1}{441} a^{11} - \frac{11}{686} a^{10} - \frac{64}{3087} a^{9} - \frac{13}{6174} a^{8} + \frac{71}{6174} a^{7} - \frac{1}{3} a^{6} + \frac{11}{63} a^{5} + \frac{67}{441} a^{4} + \frac{32}{343} a^{3} + \frac{44}{3087} a^{2} - \frac{299}{1029} a + \frac{1042}{3087}$, $\frac{1}{12854268} a^{18} + \frac{1}{6427134} a^{17} - \frac{353}{1071189} a^{16} + \frac{3055}{6427134} a^{15} + \frac{101}{306054} a^{14} + \frac{136}{459081} a^{13} - \frac{23}{51009} a^{12} - \frac{3109}{12854268} a^{11} + \frac{43315}{6427134} a^{10} - \frac{8335}{3213567} a^{9} + \frac{137519}{6427134} a^{8} - \frac{3203}{918162} a^{7} + \frac{9592}{459081} a^{6} + \frac{194770}{459081} a^{5} - \frac{330697}{714126} a^{4} - \frac{5118}{119021} a^{3} - \frac{313394}{3213567} a^{2} + \frac{857216}{3213567} a - \frac{153826}{459081}$, $\frac{1}{12854268} a^{19} - \frac{19}{3213567} a^{17} - \frac{3119}{6427134} a^{16} + \frac{608}{3213567} a^{15} + \frac{2867}{6427134} a^{14} - \frac{479}{459081} a^{13} + \frac{8483}{12854268} a^{12} + \frac{1234}{459081} a^{11} - \frac{1682}{3213567} a^{10} - \frac{9007}{714126} a^{9} - \frac{21296}{1071189} a^{8} + \frac{38543}{2142378} a^{7} - \frac{130468}{459081} a^{6} - \frac{2614807}{6427134} a^{5} + \frac{11615}{153027} a^{4} - \frac{1120703}{3213567} a^{3} + \frac{78268}{1071189} a^{2} + \frac{959509}{3213567} a - \frac{668587}{3213567}$, $\frac{1}{12854268} a^{20} + \frac{40}{3213567} a^{17} - \frac{479}{6427134} a^{16} - \frac{10}{51009} a^{15} - \frac{830}{3213567} a^{14} + \frac{2137}{4284756} a^{13} + \frac{76}{459081} a^{12} + \frac{577}{153027} a^{11} - \frac{35698}{3213567} a^{10} - \frac{96569}{6427134} a^{9} + \frac{6535}{459081} a^{8} + \frac{61681}{3213567} a^{7} - \frac{2945921}{6427134} a^{6} + \frac{132199}{459081} a^{5} + \frac{124543}{459081} a^{4} + \frac{525722}{1071189} a^{3} - \frac{1479418}{3213567} a^{2} - \frac{212174}{459081} a - \frac{1535815}{3213567}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 9409649138830000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_7^2:C_3:C_3$ (as 21T21):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 441
The 25 conjugacy class representatives for $C_7^2:C_3:C_3$
Character table for $C_7^2:C_3:C_3$ is not computed

Intermediate fields

3.3.3969.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 21 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R $21$ R $21$ $21$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{7}$ $21$ $21$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ $21$ $21$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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
$2$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.7.6.1$x^{7} - 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
2.7.6.1$x^{7} - 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
$3$3.3.4.1$x^{3} - 3 x^{2} + 21$$3$$1$$4$$C_3$$[2]$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
7Data not computed