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Properties

Label 21.21.1951555699...5568.2
Degree $21$
Signature $[21, 0]$
Discriminant $2^{18}\cdot 3^{28}\cdot 7^{21}\cdot 17^{12}$
Root discriminant $276.95$
Ramified primes $2, 3, 7, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 21T24

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-295133736, 1737020628, 3290393316, -9447929911, -1184426040, 8925763266, -137390022, -3701547615, 81723726, 817061378, -10035984, -105527772, 497658, 8311674, -8790, -399742, 0, 11277, 0, -168, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 168*x^19 + 11277*x^17 - 399742*x^15 - 8790*x^14 + 8311674*x^13 + 497658*x^12 - 105527772*x^11 - 10035984*x^10 + 817061378*x^9 + 81723726*x^8 - 3701547615*x^7 - 137390022*x^6 + 8925763266*x^5 - 1184426040*x^4 - 9447929911*x^3 + 3290393316*x^2 + 1737020628*x - 295133736)
 
gp: K = bnfinit(x^21 - 168*x^19 + 11277*x^17 - 399742*x^15 - 8790*x^14 + 8311674*x^13 + 497658*x^12 - 105527772*x^11 - 10035984*x^10 + 817061378*x^9 + 81723726*x^8 - 3701547615*x^7 - 137390022*x^6 + 8925763266*x^5 - 1184426040*x^4 - 9447929911*x^3 + 3290393316*x^2 + 1737020628*x - 295133736, 1)
 

Normalized defining polynomial

\( x^{21} - 168 x^{19} + 11277 x^{17} - 399742 x^{15} - 8790 x^{14} + 8311674 x^{13} + 497658 x^{12} - 105527772 x^{11} - 10035984 x^{10} + 817061378 x^{9} + 81723726 x^{8} - 3701547615 x^{7} - 137390022 x^{6} + 8925763266 x^{5} - 1184426040 x^{4} - 9447929911 x^{3} + 3290393316 x^{2} + 1737020628 x - 295133736 \)

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:  \(1951555699117213348169834432463016320890103201005568=2^{18}\cdot 3^{28}\cdot 7^{21}\cdot 17^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $276.95$
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:  $2, 3, 7, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{17} a^{14} + \frac{2}{17} a^{12} + \frac{6}{17} a^{10} - \frac{4}{17} a^{8} - \frac{1}{17} a^{7}$, $\frac{1}{34} a^{15} + \frac{1}{17} a^{13} + \frac{3}{17} a^{11} - \frac{2}{17} a^{9} + \frac{8}{17} a^{8} - \frac{1}{2} a$, $\frac{1}{34} a^{16} + \frac{1}{17} a^{12} - \frac{8}{17} a^{10} + \frac{8}{17} a^{9} + \frac{4}{17} a^{8} + \frac{1}{17} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{34} a^{17} + \frac{1}{17} a^{13} - \frac{8}{17} a^{11} + \frac{8}{17} a^{10} + \frac{4}{17} a^{9} + \frac{1}{17} a^{8} - \frac{1}{2} a^{3}$, $\frac{1}{11214356} a^{18} - \frac{1415}{329834} a^{17} - \frac{114153}{11214356} a^{16} + \frac{142}{9701} a^{15} - \frac{105159}{5607178} a^{14} + \frac{26448}{164917} a^{13} + \frac{339030}{2803589} a^{12} + \frac{30415}{63002} a^{11} + \frac{100441}{329834} a^{10} + \frac{56430}{164917} a^{9} - \frac{117157}{329834} a^{8} + \frac{69047}{164917} a^{7} + \frac{2036}{9701} a^{6} - \frac{133067}{329834} a^{5} - \frac{118375}{659668} a^{4} - \frac{3067}{19402} a^{3} + \frac{6713}{38804} a^{2} - \frac{200}{9701} a + \frac{1583}{9701}$, $\frac{1}{11214356} a^{19} + \frac{88759}{11214356} a^{17} - \frac{1}{19402} a^{16} - \frac{18265}{2803589} a^{15} + \frac{999}{164917} a^{14} + \frac{26485}{2803589} a^{13} - \frac{367073}{5607178} a^{12} + \frac{60213}{329834} a^{11} - \frac{34282}{164917} a^{10} - \frac{14051}{329834} a^{9} + \frac{75520}{164917} a^{8} + \frac{11156}{164917} a^{7} - \frac{100325}{329834} a^{6} - \frac{328903}{659668} a^{5} - \frac{3113}{9701} a^{4} + \frac{4393}{38804} a^{3} - \frac{2031}{19402} a^{2} - \frac{3751}{19402} a - \frac{4421}{9701}$, $\frac{1}{230631922189406639807685212979381314101735360745020138073630284} a^{20} + \frac{2564752472105138161967201523534118383394397824499705973}{57657980547351659951921303244845328525433840186255034518407571} a^{19} - \frac{298651290897214963106194248922114942624659090326062743}{115315961094703319903842606489690657050867680372510069036815142} a^{18} + \frac{1173459216796974240722608730734946354336124476512369919224417}{115315961094703319903842606489690657050867680372510069036815142} a^{17} + \frac{2846559263518440653142828454025991018497691986515713240928397}{230631922189406639807685212979381314101735360745020138073630284} a^{16} + \frac{251057436640728770423120868980715035548381972952149207308741}{57657980547351659951921303244845328525433840186255034518407571} a^{15} - \frac{2707281717580253816309260467721370723874265790291041182221547}{115315961094703319903842606489690657050867680372510069036815142} a^{14} - \frac{40713133004519896721151742638346966000021841908611065174346315}{115315961094703319903842606489690657050867680372510069036815142} a^{13} + \frac{28449389750549215608578865029224838091805379724966695470698447}{115315961094703319903842606489690657050867680372510069036815142} a^{12} - \frac{55382271862521450480258756688115665979637135621054472784910533}{115315961094703319903842606489690657050867680372510069036815142} a^{11} + \frac{1438211629458835654628227389195521824127661266765243487613962}{3391645914550097644230664896755607560319637658015002030494563} a^{10} - \frac{991504432088780948531459929481818072290410386931929683635889}{3391645914550097644230664896755607560319637658015002030494563} a^{9} + \frac{2827021023537777687978299436847828042345628814136310294198933}{6783291829100195288461329793511215120639275316030004060989126} a^{8} + \frac{874201445075714956409496788934743308430930701595666897095529}{6783291829100195288461329793511215120639275316030004060989126} a^{7} - \frac{2661488941221478640345855830035258530199146518694364022436007}{13566583658200390576922659587022430241278550632060008121978252} a^{6} + \frac{3195058857831471706595185150435504035223429516056422856266283}{6783291829100195288461329793511215120639275316030004060989126} a^{5} - \frac{659714243610419303449926191483617967811549511881287123416401}{3391645914550097644230664896755607560319637658015002030494563} a^{4} + \frac{135339959212250958740474929497844981222067787807069754167421}{399017166417658546380078223147718536508192665648823768293478} a^{3} + \frac{267775563179453464812037508743276789523568789068173013922897}{798034332835317092760156446295437073016385331297647536586956} a^{2} + \frac{23765061053055456653054816960686610714656105425322657184452}{199508583208829273190039111573859268254096332824411884146739} a - \frac{60440224615185121950988045880256180214808237538490146637076}{199508583208829273190039111573859268254096332824411884146739}$

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:  \( 24991394430400000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

21T24:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 882
The 26 conjugacy class representatives for t21n24
Character table for t21n24 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\)

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
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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ R ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ $21$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{7}$

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
$2$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
$3$3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$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$$[2]^{3}$
7Data not computed
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.6.0.1$x^{6} - x + 12$$1$$6$$0$$C_6$$[\ ]^{6}$
17.7.6.1$x^{7} - 17$$7$$1$$6$$F_7$$[\ ]_{7}^{6}$
17.7.6.1$x^{7} - 17$$7$$1$$6$$F_7$$[\ ]_{7}^{6}$