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Properties

Label 21.7.18787846783...0099.1
Degree $21$
Signature $[7, 7]$
Discriminant $-\,3^{9}\cdot 43^{3}\cdot 127^{12}\cdot 173539^{3}\cdot 22027339^{12}$
Root discriminant $3{,}841{,}230.19$
Ramified primes $3, 43, 127, 173539, 22027339$
Class number Not computed
Class group Not computed
Galois group 21T138

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-479285774976425895157444496145072595909425826941886537129, 0, 0, 28783133010994748402746840721476260664822283722824, 0, 0, -730517554800618820192741239594008021785368, 0, 0, 10123880900858011705357861577882618, 0, 0, -82344046413632782590316922, 0, 0, 390201702244539146, 0, 0, -985345269, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 985345269*x^18 + 390201702244539146*x^15 - 82344046413632782590316922*x^12 + 10123880900858011705357861577882618*x^9 - 730517554800618820192741239594008021785368*x^6 + 28783133010994748402746840721476260664822283722824*x^3 - 479285774976425895157444496145072595909425826941886537129)
 
gp: K = bnfinit(x^21 - 985345269*x^18 + 390201702244539146*x^15 - 82344046413632782590316922*x^12 + 10123880900858011705357861577882618*x^9 - 730517554800618820192741239594008021785368*x^6 + 28783133010994748402746840721476260664822283722824*x^3 - 479285774976425895157444496145072595909425826941886537129, 1)
 

Normalized defining polynomial

\( x^{21} - 985345269 x^{18} + 390201702244539146 x^{15} - 82344046413632782590316922 x^{12} + 10123880900858011705357861577882618 x^{9} - 730517554800618820192741239594008021785368 x^{6} + 28783133010994748402746840721476260664822283722824 x^{3} - 479285774976425895157444496145072595909425826941886537129 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[7, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1878784678390609885463278613336761712345663997297459976819101301866168285944837147062927218547397519053883556305417013657324906923305650099=-\,3^{9}\cdot 43^{3}\cdot 127^{12}\cdot 173539^{3}\cdot 22027339^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $3{,}841{,}230.19$
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:  $3, 43, 127, 173539, 22027339$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2797472053} a^{6} - \frac{985345269}{2797472053} a^{3}$, $\frac{1}{2797472053} a^{7} - \frac{985345269}{2797472053} a^{4}$, $\frac{1}{2797472053} a^{8} - \frac{985345269}{2797472053} a^{5}$, $\frac{1}{7825849887316034809} a^{9} - \frac{985345269}{7825849887316034809} a^{6} + \frac{139483682}{2797472053} a^{3}$, $\frac{1}{23477549661948104427} a^{10} + \frac{1}{23477549661948104427} a^{9} - \frac{3782817322}{23477549661948104427} a^{7} - \frac{3782817322}{23477549661948104427} a^{6} + \frac{1}{3} a^{5} - \frac{1672643102}{8392416159} a^{4} + \frac{1307433668}{2797472053} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{23477549661948104427} a^{11} - \frac{1}{23477549661948104427} a^{9} - \frac{3782817322}{23477549661948104427} a^{8} - \frac{1812126784}{23477549661948104427} a^{6} + \frac{1307433668}{2797472053} a^{5} - \frac{1}{3} a^{4} + \frac{3643333640}{8392416159} a^{3} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{65677789052219419670858078631} a^{12} - \frac{328448423}{21892596350739806556952692877} a^{9} + \frac{1}{8392416159} a^{7} + \frac{139483682}{23477549661948104427} a^{6} - \frac{328448423}{2797472053} a^{4} - \frac{935998037}{2797472053} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{65677789052219419670858078631} a^{13} - \frac{328448423}{21892596350739806556952692877} a^{10} + \frac{1}{8392416159} a^{8} + \frac{139483682}{23477549661948104427} a^{7} - \frac{328448423}{2797472053} a^{5} - \frac{935998037}{2797472053} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{65677789052219419670858078631} a^{14} - \frac{328448423}{21892596350739806556952692877} a^{11} + \frac{1}{23477549661948104427} a^{9} + \frac{139483682}{23477549661948104427} a^{8} - \frac{328448423}{7825849887316034809} a^{6} - \frac{935998037}{2797472053} a^{5} + \frac{1}{3} a^{4} - \frac{2657988371}{8392416159} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{1653586014387718657377935401495490994987} a^{15} - \frac{12175233481}{1653586014387718657377935401495490994987} a^{12} + \frac{26460641182}{591100101469974777037722707679} a^{9} - \frac{1}{8392416159} a^{7} + \frac{16726029539}{211297946957532939843} a^{6} + \frac{328448423}{2797472053} a^{4} - \frac{10507159352}{75531745431} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{10}{27}$, $\frac{1}{1653586014387718657377935401495490994987} a^{16} - \frac{12175233481}{1653586014387718657377935401495490994987} a^{13} + \frac{1283392705}{591100101469974777037722707679} a^{10} - \frac{1}{23477549661948104427} a^{9} - \frac{1}{8392416159} a^{8} - \frac{24760359994}{211297946957532939843} a^{7} + \frac{3782817322}{23477549661948104427} a^{6} - \frac{1812126784}{8392416159} a^{5} + \frac{31150950829}{75531745431} a^{4} + \frac{1672643102}{8392416159} a^{3} - \frac{8}{27} a - \frac{1}{3}$, $\frac{1}{4960758043163155972133806204486472984961} a^{17} + \frac{1}{4960758043163155972133806204486472984961} a^{16} - \frac{1}{4960758043163155972133806204486472984961} a^{15} + \frac{13002014996}{4960758043163155972133806204486472984961} a^{14} + \frac{13002014996}{4960758043163155972133806204486472984961} a^{13} - \frac{13002014996}{4960758043163155972133806204486472984961} a^{12} + \frac{17592533761}{1773300304409924331113168123037} a^{11} + \frac{17592533761}{1773300304409924331113168123037} a^{10} + \frac{57939211670}{1773300304409924331113168123037} a^{9} - \frac{57550362754}{633893840872598819529} a^{8} - \frac{57550362754}{633893840872598819529} a^{7} + \frac{30946040491}{633893840872598819529} a^{6} - \frac{84706529519}{226595236293} a^{5} + \frac{66356961343}{226595236293} a^{4} - \frac{62590901929}{226595236293} a^{3} + \frac{19}{81} a^{2} - \frac{8}{81} a + \frac{35}{81}$, $\frac{1}{35885664566617511106862801677465741607882175330073781509} a^{18} - \frac{1}{4960758043163155972133806204486472984961} a^{16} - \frac{297657663356798}{3987296062957501234095866853051749067542463925563753501} a^{15} - \frac{13002014996}{4960758043163155972133806204486472984961} a^{13} - \frac{157162866477614}{4275963427782813207709853950471349785680018651} a^{12} - \frac{17592533761}{1773300304409924331113168123037} a^{10} + \frac{7797023995914160}{169834421409829787402540792363779863} a^{9} + \frac{57550362754}{633893840872598819529} a^{7} - \frac{172533372481550}{4302280755849736816934157} a^{6} - \frac{1}{3} a^{5} + \frac{84706529519}{226595236293} a^{4} - \frac{281783357621563015}{585946503458688789} a^{3} + \frac{1}{3} a^{2} - \frac{19}{81} a + \frac{26988904}{209455713}$, $\frac{1}{35885664566617511106862801677465741607882175330073781509} a^{19} + \frac{4554988479896087}{35885664566617511106862801677465741607882175330073781509} a^{16} - \frac{1}{4960758043163155972133806204486472984961} a^{15} + \frac{33150070924318666}{12827890283348439623129561851414049357040055953} a^{13} - \frac{13002014996}{4960758043163155972133806204486472984961} a^{12} + \frac{60696204790944410}{4585529378065404259868601393822056301} a^{10} - \frac{17592533761}{1773300304409924331113168123037} a^{9} + \frac{49557755674842070}{1639168967978749727251913817} a^{7} + \frac{57550362754}{633893840872598819529} a^{6} - \frac{1}{3} a^{5} + \frac{6588868251991666}{585946503458688789} a^{4} - \frac{66356961343}{226595236293} a^{3} + \frac{1}{3} a^{2} + \frac{25373497}{69818571} a + \frac{8}{81}$, $\frac{1}{35885664566617511106862801677465741607882175330073781509} a^{20} - \frac{297657663356798}{3987296062957501234095866853051749067542463925563753501} a^{17} + \frac{1}{4960758043163155972133806204486472984961} a^{16} + \frac{1}{4960758043163155972133806204486472984961} a^{15} - \frac{157162866477614}{4275963427782813207709853950471349785680018651} a^{14} + \frac{13002014996}{4960758043163155972133806204486472984961} a^{13} + \frac{13002014996}{4960758043163155972133806204486472984961} a^{12} + \frac{563116545806891}{169834421409829787402540792363779863} a^{11} + \frac{17592533761}{1773300304409924331113168123037} a^{10} - \frac{57939211670}{1773300304409924331113168123037} a^{9} + \frac{66125228286872104}{546389655992916575750637939} a^{8} - \frac{57550362754}{633893840872598819529} a^{7} - \frac{106477785922}{633893840872598819529} a^{6} + \frac{225629195858876753}{585946503458688789} a^{5} + \frac{66356961343}{226595236293} a^{4} - \frac{61868266670}{226595236293} a^{3} + \frac{26988904}{209455713} a^{2} - \frac{8}{81} a - \frac{35}{81}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Not computed

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $13$
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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

21T138:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7348320
The 118 conjugacy class representatives for t21n138 are not computed
Character table for t21n138 is not computed

Intermediate fields

7.7.67159593.1

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 ${\href{/LocalNumberField/2.14.0.1}{14} }{,}\,{\href{/LocalNumberField/2.7.0.1}{7} }$ R ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ $18{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.14.0.1}{14} }{,}\,{\href{/LocalNumberField/29.7.0.1}{7} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{3}$ $15{,}\,{\href{/LocalNumberField/37.6.0.1}{6} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }$ R ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.14.0.1}{14} }{,}\,{\href{/LocalNumberField/53.7.0.1}{7} }$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
$3$3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
43Data not computed
$127$127.3.0.1$x^{3} - x + 9$$1$$3$$0$$C_3$$[\ ]^{3}$
127.6.4.3$x^{6} + 635 x^{3} + 145161$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
127.12.8.3$x^{12} + 96774 x^{6} - 2048383 x^{3} + 2341301769$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
173539Data not computed
22027339Data not computed