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Properties

Label 21.21.2118943095...6944.1
Degree $21$
Signature $[21, 0]$
Discriminant $2^{44}\cdot 809^{6}\cdot 207278461946293^{2}$
Root discriminant $668.43$
Ramified primes $2, 809, 207278461946293$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group 21T145

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-481522673032, -2561280441560, -5928205956328, -7721036828824, -6030549227920, -2659740900832, -388239708418, 212495615538, 112623606960, 8590831880, -6779995633, -1525715221, 168613567, 75996139, -1014192, -2041324, -33086, 32242, 621, -279, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 3*x^20 - 279*x^19 + 621*x^18 + 32242*x^17 - 33086*x^16 - 2041324*x^15 - 1014192*x^14 + 75996139*x^13 + 168613567*x^12 - 1525715221*x^11 - 6779995633*x^10 + 8590831880*x^9 + 112623606960*x^8 + 212495615538*x^7 - 388239708418*x^6 - 2659740900832*x^5 - 6030549227920*x^4 - 7721036828824*x^3 - 5928205956328*x^2 - 2561280441560*x - 481522673032)
 
gp: K = bnfinit(x^21 - 3*x^20 - 279*x^19 + 621*x^18 + 32242*x^17 - 33086*x^16 - 2041324*x^15 - 1014192*x^14 + 75996139*x^13 + 168613567*x^12 - 1525715221*x^11 - 6779995633*x^10 + 8590831880*x^9 + 112623606960*x^8 + 212495615538*x^7 - 388239708418*x^6 - 2659740900832*x^5 - 6030549227920*x^4 - 7721036828824*x^3 - 5928205956328*x^2 - 2561280441560*x - 481522673032, 1)
 

Normalized defining polynomial

\( x^{21} - 3 x^{20} - 279 x^{19} + 621 x^{18} + 32242 x^{17} - 33086 x^{16} - 2041324 x^{15} - 1014192 x^{14} + 75996139 x^{13} + 168613567 x^{12} - 1525715221 x^{11} - 6779995633 x^{10} + 8590831880 x^{9} + 112623606960 x^{8} + 212495615538 x^{7} - 388239708418 x^{6} - 2659740900832 x^{5} - 6030549227920 x^{4} - 7721036828824 x^{3} - 5928205956328 x^{2} - 2561280441560 x - 481522673032 \)

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:  \(211894309505097231632930771477455617607260995268779331026944=2^{44}\cdot 809^{6}\cdot 207278461946293^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $668.43$
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:  $2, 809, 207278461946293$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{8}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{15} - \frac{1}{4} a^{9} + \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{596} a^{18} + \frac{23}{596} a^{17} + \frac{79}{596} a^{16} - \frac{19}{596} a^{15} + \frac{2}{149} a^{14} + \frac{11}{149} a^{13} + \frac{10}{149} a^{12} + \frac{44}{149} a^{11} - \frac{297}{596} a^{10} - \frac{295}{596} a^{9} + \frac{229}{596} a^{8} - \frac{45}{596} a^{7} - \frac{1}{298} a^{6} + \frac{105}{298} a^{5} - \frac{67}{298} a^{4} + \frac{109}{298} a^{3} + \frac{5}{149} a^{2} - \frac{11}{149} a - \frac{31}{149}$, $\frac{1}{596} a^{19} - \frac{3}{596} a^{17} - \frac{12}{149} a^{16} - \frac{1}{298} a^{15} - \frac{35}{149} a^{14} - \frac{39}{298} a^{13} - \frac{37}{149} a^{12} - \frac{173}{596} a^{11} - \frac{5}{149} a^{10} + \frac{11}{596} a^{9} + \frac{13}{149} a^{8} + \frac{72}{149} a^{7} + \frac{64}{149} a^{6} - \frac{49}{149} a^{5} - \frac{69}{149} a^{4} + \frac{18}{149} a^{3} + \frac{23}{149} a^{2} + \frac{73}{149} a - \frac{32}{149}$, $\frac{1}{129647401721115148671320948007158524736156} a^{20} - \frac{86844474876770692842212557077262977735}{129647401721115148671320948007158524736156} a^{19} - \frac{17450548498357230601957546271790479380}{32411850430278787167830237001789631184039} a^{18} - \frac{5629089321417981877558125163259614501095}{129647401721115148671320948007158524736156} a^{17} - \frac{18437269263204464372803573272713402662247}{129647401721115148671320948007158524736156} a^{16} - \frac{4415881310602565289774435672000729861908}{32411850430278787167830237001789631184039} a^{15} + \frac{5141204318840503581300719413154567390609}{64823700860557574335660474003579262368078} a^{14} + \frac{2979557366278210946793851004302645631793}{64823700860557574335660474003579262368078} a^{13} - \frac{16213440750263534220214906020583741480421}{129647401721115148671320948007158524736156} a^{12} + \frac{44694723482791825090290072593579408048147}{129647401721115148671320948007158524736156} a^{11} - \frac{132516857632776973070802954280305835360}{1409210888272990746427401608773462225393} a^{10} - \frac{15320225240949231362244757349926837504789}{129647401721115148671320948007158524736156} a^{9} - \frac{15748218847070853390573490946275116810495}{129647401721115148671320948007158524736156} a^{8} - \frac{19322211353408675359449650239614622644941}{64823700860557574335660474003579262368078} a^{7} - \frac{18538130636942756444710805836221743066465}{64823700860557574335660474003579262368078} a^{6} - \frac{61321482455303519095503367181042251481}{1409210888272990746427401608773462225393} a^{5} - \frac{31196556632362340514981941791352492452669}{64823700860557574335660474003579262368078} a^{4} - \frac{6513802718726352368278449462885794109705}{32411850430278787167830237001789631184039} a^{3} - \frac{4521860539440227363879184503004151190912}{32411850430278787167830237001789631184039} a^{2} - \frac{11787949920869243903543572674924144356775}{32411850430278787167830237001789631184039} a + \frac{205577453396972897044024513218427530317}{531341810332439133898856344291633298099}$

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:  \( 426259268490000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

21T145:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 23514624
The 132 conjugacy class representatives for t21n145 are not computed
Character table for t21n145 is not computed

Intermediate fields

7.7.670188544.2

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

Sibling fields

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 $21$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{3}$ $21$ ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ $21$ $21$ ${\href{/LocalNumberField/19.9.0.1}{9} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/43.9.0.1}{9} }{,}\,{\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.4.8.8$x^{4} + 4 x + 2$$4$$1$$8$$S_4$$[8/3, 8/3]_{3}^{2}$
2.6.10.5$x^{6} + 2 x^{5} + 6$$6$$1$$10$$S_4\times C_2$$[2, 8/3, 8/3]_{3}^{2}$
2.8.24.132$x^{8} + 12 x^{2} + 8 x + 10$$8$$1$$24$$C_2 \wr S_4$$[2, 8/3, 8/3, 10/3, 10/3, 4]_{3}^{2}$
809Data not computed
207278461946293Data not computed