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Properties

Label 21.21.1936286121...5456.1
Degree $21$
Signature $[21, 0]$
Discriminant $2^{27}\cdot 3^{34}\cdot 13^{17}$
Root discriminant $115.16$
Ramified primes $2, 3, 13$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_3\times F_7$ (as 21T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![17535232, 23653968, -62360064, -75336041, 84833856, 95913018, -59521236, -64326231, 24195240, 25202042, -5993106, -6031368, 906992, 890316, -79758, -79386, 3564, 4023, -56, -102, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 102*x^19 - 56*x^18 + 4023*x^17 + 3564*x^16 - 79386*x^15 - 79758*x^14 + 890316*x^13 + 906992*x^12 - 6031368*x^11 - 5993106*x^10 + 25202042*x^9 + 24195240*x^8 - 64326231*x^7 - 59521236*x^6 + 95913018*x^5 + 84833856*x^4 - 75336041*x^3 - 62360064*x^2 + 23653968*x + 17535232)
 
gp: K = bnfinit(x^21 - 102*x^19 - 56*x^18 + 4023*x^17 + 3564*x^16 - 79386*x^15 - 79758*x^14 + 890316*x^13 + 906992*x^12 - 6031368*x^11 - 5993106*x^10 + 25202042*x^9 + 24195240*x^8 - 64326231*x^7 - 59521236*x^6 + 95913018*x^5 + 84833856*x^4 - 75336041*x^3 - 62360064*x^2 + 23653968*x + 17535232, 1)
 

Normalized defining polynomial

\( x^{21} - 102 x^{19} - 56 x^{18} + 4023 x^{17} + 3564 x^{16} - 79386 x^{15} - 79758 x^{14} + 890316 x^{13} + 906992 x^{12} - 6031368 x^{11} - 5993106 x^{10} + 25202042 x^{9} + 24195240 x^{8} - 64326231 x^{7} - 59521236 x^{6} + 95913018 x^{5} + 84833856 x^{4} - 75336041 x^{3} - 62360064 x^{2} + 23653968 x + 17535232 \)

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:  \(19362861214436670419310223517907912235155456=2^{27}\cdot 3^{34}\cdot 13^{17}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $115.16$
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:  $2, 3, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{9} + \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{4}$, $\frac{1}{9} a^{14} - \frac{1}{9} a^{13} + \frac{1}{9} a^{12} + \frac{1}{9} a^{11} - \frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{1}{9} a^{5} - \frac{1}{9} a^{4} + \frac{1}{9} a^{3} + \frac{1}{9} a^{2} - \frac{1}{9} a + \frac{1}{9}$, $\frac{1}{36} a^{15} - \frac{1}{9} a^{12} - \frac{1}{6} a^{11} - \frac{1}{18} a^{9} - \frac{1}{6} a^{8} - \frac{1}{6} a^{7} - \frac{2}{9} a^{6} + \frac{1}{6} a^{5} - \frac{1}{3} a^{4} - \frac{1}{9} a^{3} - \frac{1}{3} a^{2} + \frac{1}{12} a + \frac{4}{9}$, $\frac{1}{36} a^{16} - \frac{1}{9} a^{13} - \frac{1}{6} a^{12} - \frac{1}{18} a^{10} - \frac{1}{6} a^{9} - \frac{1}{6} a^{8} + \frac{1}{9} a^{7} - \frac{1}{2} a^{6} - \frac{1}{3} a^{5} - \frac{4}{9} a^{4} + \frac{1}{3} a^{3} + \frac{1}{12} a^{2} - \frac{2}{9} a + \frac{1}{3}$, $\frac{1}{36} a^{17} + \frac{1}{18} a^{13} + \frac{1}{9} a^{12} + \frac{1}{18} a^{11} + \frac{1}{18} a^{10} - \frac{1}{18} a^{9} + \frac{1}{9} a^{8} - \frac{1}{6} a^{7} - \frac{1}{3} a^{5} + \frac{2}{9} a^{4} - \frac{5}{36} a^{3} - \frac{1}{9} a^{2} - \frac{1}{9} a + \frac{4}{9}$, $\frac{1}{144} a^{18} - \frac{1}{72} a^{17} - \frac{1}{144} a^{16} + \frac{1}{24} a^{14} - \frac{1}{6} a^{13} - \frac{5}{36} a^{12} + \frac{1}{72} a^{11} + \frac{1}{36} a^{10} - \frac{1}{24} a^{9} - \frac{1}{18} a^{8} + \frac{1}{18} a^{7} - \frac{5}{24} a^{6} + \frac{1}{12} a^{5} + \frac{1}{48} a^{4} - \frac{31}{72} a^{3} + \frac{41}{144} a^{2} + \frac{5}{18} a - \frac{4}{9}$, $\frac{1}{144} a^{19} - \frac{1}{144} a^{17} - \frac{1}{72} a^{16} - \frac{1}{72} a^{15} + \frac{1}{36} a^{14} + \frac{5}{36} a^{13} - \frac{11}{72} a^{12} - \frac{1}{9} a^{11} - \frac{1}{24} a^{10} + \frac{1}{36} a^{9} + \frac{1}{18} a^{8} + \frac{5}{72} a^{7} + \frac{4}{9} a^{6} - \frac{5}{144} a^{5} + \frac{1}{18} a^{4} - \frac{7}{144} a^{3} - \frac{35}{72} a^{2} - \frac{5}{18} a + \frac{1}{9}$, $\frac{1}{7882568670461068902011590860268417149003111648} a^{20} + \frac{56475698544042525615284891942863184685580}{246330270951908403187862214383388035906347239} a^{19} + \frac{5922295383536016364639704210332893580052611}{3941284335230534451005795430134208574501555824} a^{18} + \frac{284014756487197271966248052140132667297575}{656880722538422408500965905022368095750259304} a^{17} - \frac{17036018432618866925444555707508854336121927}{2627522890153689634003863620089472383001037216} a^{16} + \frac{35043764527068432144976430043592433804105}{164220180634605602125241476255592023937564826} a^{15} + \frac{19226638036943167875524408796685879405873573}{1313761445076844817001931810044736191500518608} a^{14} + \frac{28243458419592738956937079296798822216962039}{1313761445076844817001931810044736191500518608} a^{13} + \frac{13197240058690633277044452053101852829995019}{218960240846140802833655301674122698583419768} a^{12} + \frac{33758846243699195651454405529861443697841377}{985321083807633612751448857533552143625388956} a^{11} - \frac{20971982061046891534485373860455503174534006}{246330270951908403187862214383388035906347239} a^{10} - \frac{654370624988002153812865395595261169556596165}{3941284335230534451005795430134208574501555824} a^{9} - \frac{131854258728763933211398221428794758539310565}{1313761445076844817001931810044736191500518608} a^{8} + \frac{17760339054143691777186858670037621741442125}{328440361269211204250482952511184047875129652} a^{7} + \frac{306253558781343196861507026695019285928532411}{2627522890153689634003863620089472383001037216} a^{6} - \frac{83319929448435253876911588862420066620182329}{656880722538422408500965905022368095750259304} a^{5} - \frac{61693986825396776244392638105036322075037427}{1313761445076844817001931810044736191500518608} a^{4} + \frac{10017041304774394292168489825377547351020739}{72986746948713600944551767224707566194473256} a^{3} + \frac{3592100687645427911506422146981810794912225123}{7882568670461068902011590860268417149003111648} a^{2} + \frac{261708884324109573937895632052268891435325415}{1970642167615267225502897715067104287250777912} a - \frac{108948636264566559952997432758132322966001268}{246330270951908403187862214383388035906347239}$

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:  \( 5367165978220000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times F_7$ (as 21T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 126
The 21 conjugacy class representatives for $C_3\times F_7$
Character table for $C_3\times F_7$ is not computed

Intermediate fields

3.3.13689.1, 7.7.138584369664.1

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

Sibling fields

Degree 21 siblings: data not computed
Degree 42 siblings: 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 $21$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{3}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.6.9.3$x^{6} - 4 x^{4} + 4 x^{2} + 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.3$x^{6} - 4 x^{4} + 4 x^{2} + 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.3$x^{6} - 4 x^{4} + 4 x^{2} + 24$$2$$3$$9$$C_6$$[3]^{3}$
3Data not computed
13Data not computed