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Properties

Label 21.1.51594841943...0000.1
Degree $21$
Signature $[1, 10]$
Discriminant $2^{33}\cdot 3^{19}\cdot 5^{19}\cdot 7^{15}\cdot 13^{7}\cdot 71^{7}$
Root discriminant $1346.39$
Ramified primes $2, 3, 5, 7, 13, 71$
Class number Not computed
Class group Not computed
Galois group $S_3\times F_7$ (as 21T15)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![900920827165696, -632942656095232, 173123472026624, 17265195121664, -42702941725696, 11837589381632, 574658074368, -544613923904, 59488601536, 15950698176, -7552096384, 739819136, 144182136, -39409664, 1327111, 735063, -101773, -1981, 1589, -91, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 - 91*x^19 + 1589*x^18 - 1981*x^17 - 101773*x^16 + 735063*x^15 + 1327111*x^14 - 39409664*x^13 + 144182136*x^12 + 739819136*x^11 - 7552096384*x^10 + 15950698176*x^9 + 59488601536*x^8 - 544613923904*x^7 + 574658074368*x^6 + 11837589381632*x^5 - 42702941725696*x^4 + 17265195121664*x^3 + 173123472026624*x^2 - 632942656095232*x + 900920827165696)
 
gp: K = bnfinit(x^21 - 7*x^20 - 91*x^19 + 1589*x^18 - 1981*x^17 - 101773*x^16 + 735063*x^15 + 1327111*x^14 - 39409664*x^13 + 144182136*x^12 + 739819136*x^11 - 7552096384*x^10 + 15950698176*x^9 + 59488601536*x^8 - 544613923904*x^7 + 574658074368*x^6 + 11837589381632*x^5 - 42702941725696*x^4 + 17265195121664*x^3 + 173123472026624*x^2 - 632942656095232*x + 900920827165696, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} - 91 x^{19} + 1589 x^{18} - 1981 x^{17} - 101773 x^{16} + 735063 x^{15} + 1327111 x^{14} - 39409664 x^{13} + 144182136 x^{12} + 739819136 x^{11} - 7552096384 x^{10} + 15950698176 x^{9} + 59488601536 x^{8} - 544613923904 x^{7} + 574658074368 x^{6} + 11837589381632 x^{5} - 42702941725696 x^{4} + 17265195121664 x^{3} + 173123472026624 x^{2} - 632942656095232 x + 900920827165696 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(515948419432548652795384693815714046409770106880000000000000000000=2^{33}\cdot 3^{19}\cdot 5^{19}\cdot 7^{15}\cdot 13^{7}\cdot 71^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1346.39$
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:  $2, 3, 5, 7, 13, 71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{3}$, $\frac{1}{20} a^{8} + \frac{1}{20} a^{6} - \frac{1}{4} a^{5} - \frac{1}{5} a^{4} + \frac{1}{4} a^{3} + \frac{3}{10} a^{2} - \frac{1}{5}$, $\frac{1}{20} a^{9} + \frac{1}{20} a^{7} + \frac{1}{20} a^{5} + \frac{1}{20} a^{3} - \frac{1}{5} a$, $\frac{1}{40} a^{10} - \frac{1}{40} a^{9} + \frac{1}{10} a^{7} - \frac{1}{8} a^{6} + \frac{9}{40} a^{5} + \frac{1}{10} a^{3} - \frac{1}{2} a^{2} - \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{40} a^{11} - \frac{1}{40} a^{9} - \frac{1}{40} a^{7} + \frac{9}{40} a^{5} - \frac{2}{5} a^{3} + \frac{1}{5} a$, $\frac{1}{80} a^{12} - \frac{1}{80} a^{10} - \frac{1}{40} a^{9} - \frac{1}{80} a^{8} + \frac{1}{10} a^{7} + \frac{9}{80} a^{6} - \frac{1}{40} a^{5} - \frac{1}{5} a^{4} + \frac{7}{20} a^{3} + \frac{1}{10} a^{2} - \frac{2}{5} a$, $\frac{1}{80} a^{13} - \frac{1}{80} a^{11} + \frac{1}{80} a^{9} + \frac{1}{80} a^{7} - \frac{7}{40} a^{5} - \frac{1}{4} a^{3} + \frac{2}{5} a$, $\frac{1}{2400} a^{14} - \frac{1}{160} a^{13} - \frac{7}{2400} a^{12} - \frac{1}{160} a^{11} - \frac{3}{800} a^{10} + \frac{1}{160} a^{9} - \frac{1}{480} a^{8} - \frac{19}{160} a^{7} + \frac{7}{120} a^{6} - \frac{1}{40} a^{5} - \frac{9}{100} a^{4} - \frac{1}{20} a^{3} - \frac{34}{75} a^{2} + \frac{1}{5} a + \frac{37}{75}$, $\frac{1}{4800} a^{15} - \frac{1}{4800} a^{14} + \frac{23}{4800} a^{13} + \frac{7}{4800} a^{12} + \frac{7}{1600} a^{11} + \frac{3}{1600} a^{10} + \frac{17}{960} a^{9} - \frac{23}{960} a^{8} + \frac{23}{480} a^{7} + \frac{17}{240} a^{6} - \frac{49}{200} a^{5} - \frac{23}{100} a^{4} - \frac{83}{300} a^{3} - \frac{13}{75} a^{2} + \frac{67}{150} a - \frac{11}{75}$, $\frac{1}{9600} a^{16} - \frac{1}{9600} a^{15} - \frac{1}{9600} a^{14} + \frac{7}{9600} a^{13} - \frac{17}{3200} a^{12} + \frac{3}{3200} a^{11} + \frac{61}{9600} a^{10} + \frac{5}{384} a^{9} + \frac{11}{960} a^{8} + \frac{29}{480} a^{7} - \frac{49}{400} a^{6} - \frac{43}{200} a^{5} + \frac{19}{150} a^{4} + \frac{19}{300} a^{3} - \frac{17}{300} a^{2} + \frac{49}{150} a + \frac{1}{25}$, $\frac{1}{48000} a^{17} + \frac{1}{24000} a^{16} - \frac{1}{12000} a^{15} + \frac{1}{320} a^{13} + \frac{61}{12000} a^{12} + \frac{7}{12000} a^{11} + \frac{7}{1500} a^{10} - \frac{179}{9600} a^{9} - \frac{29}{1600} a^{8} + \frac{27}{250} a^{7} + \frac{253}{3000} a^{6} - \frac{491}{3000} a^{5} + \frac{8}{75} a^{4} + \frac{77}{300} a^{3} + \frac{151}{500} a^{2} + \frac{41}{250} a + \frac{62}{375}$, $\frac{1}{672000} a^{18} + \frac{1}{134400} a^{17} - \frac{23}{672000} a^{16} - \frac{9}{224000} a^{15} + \frac{1}{8960} a^{14} - \frac{443}{96000} a^{13} + \frac{187}{134400} a^{12} - \frac{8257}{672000} a^{11} + \frac{719}{84000} a^{10} + \frac{97}{11200} a^{9} + \frac{177}{56000} a^{8} + \frac{299}{3360} a^{7} - \frac{629}{5250} a^{6} - \frac{11}{5250} a^{5} + \frac{73}{4200} a^{4} + \frac{39}{1750} a^{3} - \frac{67}{350} a^{2} + \frac{389}{5250} a + \frac{793}{2625}$, $\frac{1}{4032000} a^{19} + \frac{1}{4032000} a^{18} - \frac{1}{268800} a^{17} + \frac{121}{4032000} a^{16} - \frac{209}{4032000} a^{15} - \frac{107}{1344000} a^{14} - \frac{5701}{4032000} a^{13} + \frac{55}{32256} a^{12} + \frac{1}{48000} a^{11} + \frac{11341}{1008000} a^{10} - \frac{1663}{126000} a^{9} + \frac{481}{168000} a^{8} - \frac{5927}{50400} a^{7} + \frac{1739}{15750} a^{6} - \frac{2881}{42000} a^{5} + \frac{2237}{31500} a^{4} + \frac{8327}{31500} a^{3} + \frac{31}{70} a^{2} - \frac{3569}{15750} a - \frac{592}{7875}$, $\frac{1}{1388804904468542003781052260626066128262932339251834954549073022156033955339841110105648744439467969408000} a^{20} - \frac{2085688095365116132304975466453390272320464490329649249403134752875695944410130447354114990343577}{1388804904468542003781052260626066128262932339251834954549073022156033955339841110105648744439467969408000} a^{19} - \frac{293695345119415192436411473496594047874098677012748427121428323457643765675147075030931348946250307}{462934968156180667927017420208688709420977446417278318183024340718677985113280370035216248146489323136000} a^{18} - \frac{3082740798695827321243557191800831281113190043042441251469304135907535779690402738968875178356570193}{1388804904468542003781052260626066128262932339251834954549073022156033955339841110105648744439467969408000} a^{17} + \frac{8887991031855281487433850956720783133118053004785238808908414895242242624288298375770568573495411907}{198400700638363143397293180089438018323276048464547850649867574593719136477120158586521249205638281344000} a^{16} - \frac{37561492192965655516171274950593465029034367581833532456034487410563996477833103541323215034196418269}{462934968156180667927017420208688709420977446417278318183024340718677985113280370035216248146489323136000} a^{15} + \frac{242793619915407418503124423978437931265606742867318705701721625374440184742529307382653426105396959717}{1388804904468542003781052260626066128262932339251834954549073022156033955339841110105648744439467969408000} a^{14} + \frac{145446162663573423061706016820582908524771709724430953674830470641122576603680278358351662807567133421}{1388804904468542003781052260626066128262932339251834954549073022156033955339841110105648744439467969408000} a^{13} - \frac{1230798691884839423725874589611007591906300615912480616077778686995386916673108717996558753468381280127}{231467484078090333963508710104344354710488723208639159091512170359338992556640185017608124073244661568000} a^{12} - \frac{2883680876761053040075405653224324879274723355814530021706038936184348679785349807975044770497950553761}{347201226117135500945263065156516532065733084812958738637268255539008488834960277526412186109866992352000} a^{11} - \frac{18638572184713864820723270904408925940904999425795588768726450942039084836443977472714995822528055119}{34720122611713550094526306515651653206573308481295873863726825553900848883496027752641218610986699235200} a^{10} - \frac{171477773582668937925203956271863091417882228555008050551683737827355314706861074324088242571882519629}{19288957006507527830292392508695362892540726934053263257626014196611582713053348751467343672770388464000} a^{9} + \frac{35240337117632997369102681666297188674325670300214351229795873895563722529364641592159343732270812869}{21700076632320968809078941572282283254108317800809921164829265971188030552185017345400761631866687022000} a^{8} - \frac{3316703616863719437695136444398461116992519041629517219055776247059554924563848601583000640867824512501}{43400153264641937618157883144564566508216635601619842329658531942376061104370034690801523263733374044000} a^{7} - \frac{338634255175757127233126229775295240615004849755489751530147719793944849947690855086830642821395679089}{14466717754880645872719294381521522169405545200539947443219510647458687034790011563600507754577791348000} a^{6} + \frac{2180132142860765484476745903267690600773915376778904392768543765470452026613876077995823113902408597323}{21700076632320968809078941572282283254108317800809921164829265971188030552185017345400761631866687022000} a^{5} + \frac{898840488950094904181874856520031791898163206167489100646399897714294946754777007318824029571105075191}{10850038316160484404539470786141141627054158900404960582414632985594015276092508672700380815933343511000} a^{4} - \frac{135933459905837594662453170880275279859252276173561404136086976026457499532681729095395439651823196181}{904169859680040367044955898845095135587846575033746715201219415466167939674375722725031734661111959250} a^{3} - \frac{357944131702732218869221143036013754738433558173688889493146598140337121383971994576709765316280848889}{775002736868606028895676484724367259075297064314640041601045213256715376863750619478598629709524536500} a^{2} + \frac{29375677662278043287171513712817526113627239689567052301959674883229588262537551140474042281462557948}{193750684217151507223919121181091814768824266078660010400261303314178844215937654869649657427381134125} a - \frac{59874832875415894584507006702093235965050940099407685737072699766565890263722259094045167192535074766}{150694976613340061174159316474182522597974429172291119200203235911027989945729287120838622443518659875}$

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:  $10$
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

$S_3\times F_7$ (as 21T15):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 252
The 21 conjugacy class representatives for $S_3\times F_7$
Character table for $S_3\times F_7$ is not computed

Intermediate fields

3.1.110760.1, 7.1.12252303000000.10

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 R R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ $21$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.14.0.1}{14} }{,}\,{\href{/LocalNumberField/43.7.0.1}{7} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.7.6.1$x^{7} - 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
2.14.27.21$x^{14} + 4 x^{12} + 4 x^{11} - 4 x^{10} + 8 x^{9} + 2 x^{8} + 8 x^{6} + 4 x^{5} - 4 x^{4} + 6 x^{2} - 4 x - 6$$14$$1$$27$$(C_7:C_3) \times C_2$$[3]_{7}^{3}$
$3$3.7.6.1$x^{7} - 3$$7$$1$$6$$F_7$$[\ ]_{7}^{6}$
3.14.13.1$x^{14} - 3$$14$$1$$13$$F_7 \times C_2$$[\ ]_{14}^{6}$
$5$5.7.6.1$x^{7} - 5$$7$$1$$6$$F_7$$[\ ]_{7}^{6}$
5.14.13.2$x^{14} + 10$$14$$1$$13$$F_7 \times C_2$$[\ ]_{14}^{6}$
7Data not computed
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$71$$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
71.2.1.1$x^{2} - 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.1$x^{2} - 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.1$x^{2} - 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.1$x^{2} - 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.1$x^{2} - 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.1$x^{2} - 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.1$x^{2} - 71$$2$$1$$1$$C_2$$[\ ]_{2}$