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Properties

Label 21.21.6623268093...9184.1
Degree $21$
Signature $[21, 0]$
Discriminant $2^{18}\cdot 3^{28}\cdot 7^{32}$
Root discriminant $152.04$
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![-577536, 10063872, -27697152, -94947328, 27697152, 135502080, -10386432, -84318144, 1854720, 28700672, -170016, -5870592, 7728, 752640, -138, -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 - 138*x^14 + 752640*x^13 + 7728*x^12 - 5870592*x^11 - 170016*x^10 + 28700672*x^9 + 1854720*x^8 - 84318144*x^7 - 10386432*x^6 + 135502080*x^5 + 27697152*x^4 - 94947328*x^3 - 27697152*x^2 + 10063872*x - 577536)
 
gp: K = bnfinit(x^21 - 84*x^19 + 3024*x^17 - 60928*x^15 - 138*x^14 + 752640*x^13 + 7728*x^12 - 5870592*x^11 - 170016*x^10 + 28700672*x^9 + 1854720*x^8 - 84318144*x^7 - 10386432*x^6 + 135502080*x^5 + 27697152*x^4 - 94947328*x^3 - 27697152*x^2 + 10063872*x - 577536, 1)
 

Normalized defining polynomial

\( x^{21} - 84 x^{19} + 3024 x^{17} - 60928 x^{15} - 138 x^{14} + 752640 x^{13} + 7728 x^{12} - 5870592 x^{11} - 170016 x^{10} + 28700672 x^{9} + 1854720 x^{8} - 84318144 x^{7} - 10386432 x^{6} + 135502080 x^{5} + 27697152 x^{4} - 94947328 x^{3} - 27697152 x^{2} + 10063872 x - 577536 \)

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:  \(6623268093347351492485520944725602280711389184=2^{18}\cdot 3^{28}\cdot 7^{32}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $152.04$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{14} a^{7} - \frac{2}{7}$, $\frac{1}{14} a^{8} - \frac{2}{7} a$, $\frac{1}{28} a^{9} + \frac{5}{14} a^{2}$, $\frac{1}{56} a^{10} - \frac{9}{28} a^{3}$, $\frac{1}{56} a^{11} - \frac{9}{28} a^{4}$, $\frac{1}{784} a^{12} - \frac{3}{392} a^{11} + \frac{3}{392} a^{10} + \frac{1}{196} a^{9} - \frac{1}{98} a^{8} - \frac{1}{49} a^{7} + \frac{3}{7} a^{6} + \frac{131}{392} a^{5} + \frac{27}{196} a^{4} - \frac{69}{196} a^{3} + \frac{19}{98} a^{2} - \frac{5}{49} a - \frac{24}{49}$, $\frac{1}{1568} a^{13} + \frac{3}{392} a^{11} + \frac{3}{392} a^{10} + \frac{1}{98} a^{9} + \frac{3}{98} a^{8} + \frac{1}{98} a^{7} - \frac{37}{784} a^{6} - \frac{3}{7} a^{5} + \frac{25}{98} a^{4} - \frac{27}{196} a^{3} - \frac{23}{49} a^{2} + \frac{8}{49} a + \frac{5}{49}$, $\frac{1}{3136} a^{14} - \frac{53}{1568} a^{7} - \frac{3}{8} a^{5} - \frac{1}{2} a^{3} - \frac{12}{49}$, $\frac{1}{3136} a^{15} - \frac{53}{1568} a^{8} - \frac{3}{8} a^{6} - \frac{1}{2} a^{4} - \frac{12}{49} a$, $\frac{1}{6272} a^{16} - \frac{53}{3136} a^{9} + \frac{3}{112} a^{7} + \frac{1}{4} a^{5} - \frac{6}{49} a^{2} + \frac{1}{7}$, $\frac{1}{87808} a^{17} + \frac{1}{43904} a^{16} - \frac{1}{10976} a^{15} - \frac{1}{10976} a^{14} + \frac{3}{392} a^{11} + \frac{59}{43904} a^{10} - \frac{389}{21952} a^{9} - \frac{377}{10976} a^{8} + \frac{37}{2744} a^{7} + \frac{1}{8} a^{6} + \frac{2}{7} a^{5} + \frac{1}{196} a^{4} - \frac{271}{1372} a^{3} + \frac{275}{686} a^{2} + \frac{164}{343} a + \frac{129}{343}$, $\frac{1}{12468736} a^{18} + \frac{1}{6234368} a^{17} + \frac{31}{779296} a^{16} + \frac{125}{1558592} a^{15} - \frac{1}{111328} a^{14} + \frac{17}{55664} a^{13} - \frac{1}{3976} a^{12} + \frac{37467}{6234368} a^{11} - \frac{26821}{3117184} a^{10} + \frac{4929}{1558592} a^{9} - \frac{1945}{194824} a^{8} - \frac{62}{3479} a^{7} - \frac{7447}{27832} a^{6} - \frac{19}{71} a^{5} + \frac{74335}{194824} a^{4} + \frac{16025}{97412} a^{3} - \frac{16027}{48706} a^{2} - \frac{737}{24353} a + \frac{29}{497}$, $\frac{1}{24937472} a^{19} - \frac{19}{6234368} a^{17} - \frac{141}{3117184} a^{16} + \frac{19}{194824} a^{15} + \frac{39}{1558592} a^{14} + \frac{23}{111328} a^{13} - \frac{7109}{12468736} a^{12} + \frac{139}{27832} a^{11} - \frac{6085}{779296} a^{10} - \frac{297}{21952} a^{9} - \frac{4623}{389648} a^{8} + \frac{26283}{779296} a^{7} - \frac{25001}{55664} a^{6} + \frac{128907}{389648} a^{5} + \frac{225}{6958} a^{4} - \frac{761}{48706} a^{3} - \frac{19287}{48706} a^{2} - \frac{8670}{24353} a - \frac{6817}{24353}$, $\frac{1}{49874944} a^{20} + \frac{5}{1558592} a^{17} + \frac{165}{3117184} a^{16} - \frac{9}{111328} a^{15} + \frac{27}{389648} a^{14} - \frac{5541}{24937472} a^{13} + \frac{15}{55664} a^{12} - \frac{2885}{890624} a^{11} - \frac{5333}{779296} a^{10} - \frac{2753}{194824} a^{9} + \frac{419}{55664} a^{8} + \frac{8009}{389648} a^{7} - \frac{155097}{779296} a^{6} + \frac{11233}{27832} a^{5} - \frac{748}{3479} a^{4} - \frac{28135}{97412} a^{3} + \frac{5821}{24353} a^{2} - \frac{1397}{3479} a - \frac{10347}{24353}$

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:  \( 72918598906000000 \) (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$2.7.6.1$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}$
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