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Properties

Label 21.21.1313623352...6944.1
Degree $21$
Signature $[21, 0]$
Discriminant $2^{18}\cdot 7^{14}\cdot 43^{14}$
Root discriminant $81.36$
Ramified primes $2, 7, 43$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_7:C_3$ (as 21T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![343, 1372, -15141, -81417, 68775, 1042524, 1604980, -1827259, -7903686, -8884217, -3442335, 1063163, 1279532, 189850, -131957, -41166, 5152, 2809, -44, -85, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - x^20 - 85*x^19 - 44*x^18 + 2809*x^17 + 5152*x^16 - 41166*x^15 - 131957*x^14 + 189850*x^13 + 1279532*x^12 + 1063163*x^11 - 3442335*x^10 - 8884217*x^9 - 7903686*x^8 - 1827259*x^7 + 1604980*x^6 + 1042524*x^5 + 68775*x^4 - 81417*x^3 - 15141*x^2 + 1372*x + 343)
 
gp: K = bnfinit(x^21 - x^20 - 85*x^19 - 44*x^18 + 2809*x^17 + 5152*x^16 - 41166*x^15 - 131957*x^14 + 189850*x^13 + 1279532*x^12 + 1063163*x^11 - 3442335*x^10 - 8884217*x^9 - 7903686*x^8 - 1827259*x^7 + 1604980*x^6 + 1042524*x^5 + 68775*x^4 - 81417*x^3 - 15141*x^2 + 1372*x + 343, 1)
 

Normalized defining polynomial

\( x^{21} - x^{20} - 85 x^{19} - 44 x^{18} + 2809 x^{17} + 5152 x^{16} - 41166 x^{15} - 131957 x^{14} + 189850 x^{13} + 1279532 x^{12} + 1063163 x^{11} - 3442335 x^{10} - 8884217 x^{9} - 7903686 x^{8} - 1827259 x^{7} + 1604980 x^{6} + 1042524 x^{5} + 68775 x^{4} - 81417 x^{3} - 15141 x^{2} + 1372 x + 343 \)

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:  \(13136233521869762226411268456105692626944=2^{18}\cdot 7^{14}\cdot 43^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $81.36$
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:  $2, 7, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7} a^{12} - \frac{3}{7} a^{11} - \frac{3}{7} a^{10} + \frac{3}{7} a^{9} + \frac{1}{7} a^{8} - \frac{2}{7} a^{7} - \frac{2}{7} a^{5} - \frac{3}{7} a^{4} + \frac{1}{7} a^{3}$, $\frac{1}{7} a^{13} + \frac{2}{7} a^{11} + \frac{1}{7} a^{10} + \frac{3}{7} a^{9} + \frac{1}{7} a^{8} + \frac{1}{7} a^{7} - \frac{2}{7} a^{6} - \frac{2}{7} a^{5} - \frac{1}{7} a^{4} + \frac{3}{7} a^{3}$, $\frac{1}{7} a^{14} + \frac{2}{7} a^{10} + \frac{2}{7} a^{9} - \frac{1}{7} a^{8} + \frac{2}{7} a^{7} - \frac{2}{7} a^{6} + \frac{3}{7} a^{5} + \frac{2}{7} a^{4} - \frac{2}{7} a^{3}$, $\frac{1}{7} a^{15} + \frac{2}{7} a^{11} + \frac{2}{7} a^{10} - \frac{1}{7} a^{9} + \frac{2}{7} a^{8} - \frac{2}{7} a^{7} + \frac{3}{7} a^{6} + \frac{2}{7} a^{5} - \frac{2}{7} a^{4}$, $\frac{1}{7} a^{16} + \frac{1}{7} a^{11} - \frac{2}{7} a^{10} + \frac{3}{7} a^{9} + \frac{3}{7} a^{8} + \frac{2}{7} a^{6} + \frac{2}{7} a^{5} - \frac{1}{7} a^{4} - \frac{2}{7} a^{3}$, $\frac{1}{7} a^{17} + \frac{1}{7} a^{11} - \frac{1}{7} a^{10} - \frac{1}{7} a^{8} - \frac{3}{7} a^{7} + \frac{2}{7} a^{6} + \frac{1}{7} a^{5} + \frac{1}{7} a^{4} - \frac{1}{7} a^{3}$, $\frac{1}{7} a^{18} + \frac{2}{7} a^{11} + \frac{3}{7} a^{10} + \frac{3}{7} a^{9} + \frac{3}{7} a^{8} - \frac{3}{7} a^{7} + \frac{1}{7} a^{6} + \frac{3}{7} a^{5} + \frac{2}{7} a^{4} - \frac{1}{7} a^{3}$, $\frac{1}{2177} a^{19} - \frac{81}{2177} a^{18} + \frac{15}{2177} a^{17} - \frac{102}{2177} a^{16} - \frac{139}{2177} a^{15} - \frac{9}{311} a^{14} + \frac{109}{2177} a^{13} + \frac{23}{2177} a^{12} + \frac{142}{2177} a^{11} + \frac{277}{2177} a^{10} - \frac{864}{2177} a^{9} - \frac{267}{2177} a^{8} + \frac{488}{2177} a^{7} + \frac{16}{2177} a^{6} + \frac{1055}{2177} a^{5} + \frac{340}{2177} a^{4} + \frac{604}{2177} a^{3} - \frac{51}{311} a^{2} - \frac{31}{311} a + \frac{80}{311}$, $\frac{1}{6624640578226338459736644172412571415427} a^{20} - \frac{1137511897984629778056102897043446602}{6624640578226338459736644172412571415427} a^{19} - \frac{449790613349700985511517407888941718760}{6624640578226338459736644172412571415427} a^{18} + \frac{369828105122495367809276168133492196693}{6624640578226338459736644172412571415427} a^{17} + \frac{455830257529664460037092278533942642210}{6624640578226338459736644172412571415427} a^{16} + \frac{4431694836456396685184019651336368030}{135196746494415070606870289232909620723} a^{15} + \frac{317502059485304022800361001738381337878}{6624640578226338459736644172412571415427} a^{14} + \frac{6856069833837186771797111166990529447}{946377225460905494248092024630367345061} a^{13} + \frac{308522980683302477826713080103404708316}{6624640578226338459736644172412571415427} a^{12} - \frac{278642145619552269349133170419168978780}{6624640578226338459736644172412571415427} a^{11} + \frac{1935034202776364747037316108180487300119}{6624640578226338459736644172412571415427} a^{10} + \frac{1958622590453281158331065986743296445192}{6624640578226338459736644172412571415427} a^{9} + \frac{3232928972742926450525845901176819352613}{6624640578226338459736644172412571415427} a^{8} - \frac{1463172671210166183188448247379398431}{3598392492246788951513657888328392947} a^{7} + \frac{194999143371270036282007479397684488597}{946377225460905494248092024630367345061} a^{6} + \frac{1778442526013557887709210832203949942719}{6624640578226338459736644172412571415427} a^{5} + \frac{409146508204983009559320118834390287553}{946377225460905494248092024630367345061} a^{4} - \frac{231647664517699491499192935476356385999}{946377225460905494248092024630367345061} a^{3} - \frac{9153015918259673906127639590717232845}{946377225460905494248092024630367345061} a^{2} + \frac{3811840426315496436181411545717676269}{135196746494415070606870289232909620723} a - \frac{11548581901467198574078886755297599558}{135196746494415070606870289232909620723}$

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:  \( 18989130461400 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_7:C_3$ (as 21T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 21
The 5 conjugacy class representatives for $C_7:C_3$
Character table for $C_7:C_3$

Intermediate fields

3.3.90601.1, 7.7.525346636864.1 x7

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

Sibling fields

Degree 7 sibling: 7.7.525346636864.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{7}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{3}$ R ${\href{/LocalNumberField/47.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{7}$

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
2Data not computed
$7$7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
$43$43.3.2.1$x^{3} - 43$$3$$1$$2$$C_3$$[\ ]_{3}$
43.3.2.1$x^{3} - 43$$3$$1$$2$$C_3$$[\ ]_{3}$
43.3.2.1$x^{3} - 43$$3$$1$$2$$C_3$$[\ ]_{3}$
43.3.2.1$x^{3} - 43$$3$$1$$2$$C_3$$[\ ]_{3}$
43.3.2.1$x^{3} - 43$$3$$1$$2$$C_3$$[\ ]_{3}$
43.3.2.1$x^{3} - 43$$3$$1$$2$$C_3$$[\ ]_{3}$
43.3.2.1$x^{3} - 43$$3$$1$$2$$C_3$$[\ ]_{3}$