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Properties

Label 21.21.3129958177...4112.1
Degree $21$
Signature $[21, 0]$
Discriminant $2^{21}\cdot 71^{19}$
Root discriminant $94.62$
Ramified primes $2, 71$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_7\times S_3$ (as 21T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2329, -70834, 499880, 355302, -4148808, -336531, 8529803, -41098, -7736312, 323164, 3636823, -298102, -949109, 119845, 137697, -23312, -10377, 2160, 342, -85, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 3*x^20 - 85*x^19 + 342*x^18 + 2160*x^17 - 10377*x^16 - 23312*x^15 + 137697*x^14 + 119845*x^13 - 949109*x^12 - 298102*x^11 + 3636823*x^10 + 323164*x^9 - 7736312*x^8 - 41098*x^7 + 8529803*x^6 - 336531*x^5 - 4148808*x^4 + 355302*x^3 + 499880*x^2 - 70834*x + 2329)
 
gp: K = bnfinit(x^21 - 3*x^20 - 85*x^19 + 342*x^18 + 2160*x^17 - 10377*x^16 - 23312*x^15 + 137697*x^14 + 119845*x^13 - 949109*x^12 - 298102*x^11 + 3636823*x^10 + 323164*x^9 - 7736312*x^8 - 41098*x^7 + 8529803*x^6 - 336531*x^5 - 4148808*x^4 + 355302*x^3 + 499880*x^2 - 70834*x + 2329, 1)
 

Normalized defining polynomial

\( x^{21} - 3 x^{20} - 85 x^{19} + 342 x^{18} + 2160 x^{17} - 10377 x^{16} - 23312 x^{15} + 137697 x^{14} + 119845 x^{13} - 949109 x^{12} - 298102 x^{11} + 3636823 x^{10} + 323164 x^{9} - 7736312 x^{8} - 41098 x^{7} + 8529803 x^{6} - 336531 x^{5} - 4148808 x^{4} + 355302 x^{3} + 499880 x^{2} - 70834 x + 2329 \)

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:  \(312995817795780583119591509136121488474112=2^{21}\cdot 71^{19}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $94.62$
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:  $2, 71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{5} a^{16} + \frac{1}{5} a^{15} + \frac{1}{5} a^{14} + \frac{1}{5} a^{13} + \frac{2}{5} a^{12} - \frac{2}{5} a^{11} - \frac{1}{5} a^{10} + \frac{2}{5} a^{9} - \frac{1}{5} a^{7} - \frac{1}{5} a^{6} - \frac{1}{5} a^{5} - \frac{2}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2} - \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{17} + \frac{1}{5} a^{13} + \frac{1}{5} a^{12} + \frac{1}{5} a^{11} - \frac{2}{5} a^{10} - \frac{2}{5} a^{9} - \frac{1}{5} a^{8} - \frac{1}{5} a^{5} - \frac{1}{5} a^{4} - \frac{1}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{18} + \frac{1}{5} a^{14} + \frac{1}{5} a^{13} + \frac{1}{5} a^{12} - \frac{2}{5} a^{11} - \frac{2}{5} a^{10} - \frac{1}{5} a^{9} - \frac{1}{5} a^{6} - \frac{1}{5} a^{5} - \frac{1}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2} + \frac{1}{5} a$, $\frac{1}{85} a^{19} + \frac{7}{85} a^{18} - \frac{7}{85} a^{17} + \frac{1}{17} a^{16} - \frac{39}{85} a^{15} - \frac{1}{5} a^{14} + \frac{6}{85} a^{13} + \frac{23}{85} a^{12} - \frac{33}{85} a^{11} - \frac{26}{85} a^{10} - \frac{38}{85} a^{9} + \frac{37}{85} a^{8} + \frac{9}{85} a^{7} - \frac{8}{85} a^{6} + \frac{2}{5} a^{5} - \frac{28}{85} a^{4} + \frac{27}{85} a^{3} - \frac{31}{85} a^{2} + \frac{2}{17} a - \frac{1}{5}$, $\frac{1}{926479247647264622062073491648252207718246953433351165} a^{20} + \frac{10701497646182756164447485350691004131630140233349}{6762622245600471693883748114220819034439758784185045} a^{19} - \frac{89427166935351491536224473866831598626808237224427336}{926479247647264622062073491648252207718246953433351165} a^{18} + \frac{4139839582499636108453680423087333873049451263363358}{54498779273368507180121970096956012218720409025491245} a^{17} - \frac{30156469596133572419285882639912572466889683580847169}{926479247647264622062073491648252207718246953433351165} a^{16} - \frac{77977409183078487981096275428381162155344325318866611}{926479247647264622062073491648252207718246953433351165} a^{15} + \frac{157764581842157362079967006667128364065088986794515568}{926479247647264622062073491648252207718246953433351165} a^{14} + \frac{108162481275028570390935878686557046538334355849436896}{926479247647264622062073491648252207718246953433351165} a^{13} - \frac{217312108837233854487190827903329199355646854447496008}{926479247647264622062073491648252207718246953433351165} a^{12} + \frac{438537306009326614251883056372362578776746789882649081}{926479247647264622062073491648252207718246953433351165} a^{11} + \frac{353447884924246315391096339882031771755467528217011112}{926479247647264622062073491648252207718246953433351165} a^{10} + \frac{323746481390155566877970395301009086383133197129858154}{926479247647264622062073491648252207718246953433351165} a^{9} - \frac{445238425840049914929480298041321072968632170310200842}{926479247647264622062073491648252207718246953433351165} a^{8} + \frac{378639627719512862568229526313413980485165145062878906}{926479247647264622062073491648252207718246953433351165} a^{7} + \frac{151164244083988028100275196672631023570542455557461602}{926479247647264622062073491648252207718246953433351165} a^{6} + \frac{79902180191732678941051252572625878075988270401754594}{926479247647264622062073491648252207718246953433351165} a^{5} - \frac{10985286479037736928467561543126127885966468503665028}{926479247647264622062073491648252207718246953433351165} a^{4} - \frac{270872581153024878855628436197380648980889252583041959}{926479247647264622062073491648252207718246953433351165} a^{3} + \frac{2420871751243025892511257096142202216870174976679942}{54498779273368507180121970096956012218720409025491245} a^{2} - \frac{6255329844004264639149164353948173465409048490190673}{185295849529452924412414698329650441543649390686670233} a - \frac{60814490817075472534754745725996967979267016577416}{397801308564733629051985183189459943202338752010885}$

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:  \( 609186511830000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_7\times S_3$ (as 21T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 42
The 21 conjugacy class representatives for $C_7\times S_3$
Character table for $C_7\times S_3$ is not computed

Intermediate fields

3.3.568.1, 7.7.128100283921.1

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

Sibling fields

Galois closure: 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 $21$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{7}$ $21$ ${\href{/LocalNumberField/11.14.0.1}{14} }{,}\,{\href{/LocalNumberField/11.7.0.1}{7} }$ $21$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{7}$ $21$ $21$ ${\href{/LocalNumberField/29.14.0.1}{14} }{,}\,{\href{/LocalNumberField/29.7.0.1}{7} }$ $21$ ${\href{/LocalNumberField/37.14.0.1}{14} }{,}\,{\href{/LocalNumberField/37.7.0.1}{7} }$ ${\href{/LocalNumberField/41.14.0.1}{14} }{,}\,{\href{/LocalNumberField/41.7.0.1}{7} }$ $21$ $21$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/59.14.0.1}{14} }{,}\,{\href{/LocalNumberField/59.7.0.1}{7} }$

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
$2$2.7.0.1$x^{7} - x + 1$$1$$7$$0$$C_7$$[\ ]^{7}$
2.14.21.6$x^{14} + 4 x^{11} - 3 x^{10} + 4 x^{9} + 2 x^{8} + 2 x^{7} - 3 x^{6} + 2 x^{5} - 2 x^{4} - 2 x^{3} - x^{2} - 2 x + 1$$2$$7$$21$$C_{14}$$[3]^{7}$
$71$71.7.6.1$x^{7} - 71$$7$$1$$6$$C_7$$[\ ]_{7}$
71.14.13.1$x^{14} - 71$$14$$1$$13$$C_{14}$$[\ ]_{14}$