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Properties

Label 21.3.16474122069...0000.1
Degree $21$
Signature $[3, 9]$
Discriminant $-\,2^{26}\cdot 3^{19}\cdot 5^{19}\cdot 7^{21}\cdot 2129^{7}$
Root discriminant $2461.93$
Ramified primes $2, 3, 5, 7, 2129$
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![1281974857676400, 2011887049200, -7218977715600, -231401632434000, 302507514834000, -223602818989200, 79566366868800, 1697893096120, -8783229793240, 889660452520, 364571599000, -47160362200, -8161548500, 1138001760, 108271479, -15509193, -856121, 123515, 3745, -539, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 - 539*x^19 + 3745*x^18 + 123515*x^17 - 856121*x^16 - 15509193*x^15 + 108271479*x^14 + 1138001760*x^13 - 8161548500*x^12 - 47160362200*x^11 + 364571599000*x^10 + 889660452520*x^9 - 8783229793240*x^8 + 1697893096120*x^7 + 79566366868800*x^6 - 223602818989200*x^5 + 302507514834000*x^4 - 231401632434000*x^3 - 7218977715600*x^2 + 2011887049200*x + 1281974857676400)
 
gp: K = bnfinit(x^21 - 7*x^20 - 539*x^19 + 3745*x^18 + 123515*x^17 - 856121*x^16 - 15509193*x^15 + 108271479*x^14 + 1138001760*x^13 - 8161548500*x^12 - 47160362200*x^11 + 364571599000*x^10 + 889660452520*x^9 - 8783229793240*x^8 + 1697893096120*x^7 + 79566366868800*x^6 - 223602818989200*x^5 + 302507514834000*x^4 - 231401632434000*x^3 - 7218977715600*x^2 + 2011887049200*x + 1281974857676400, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} - 539 x^{19} + 3745 x^{18} + 123515 x^{17} - 856121 x^{16} - 15509193 x^{15} + 108271479 x^{14} + 1138001760 x^{13} - 8161548500 x^{12} - 47160362200 x^{11} + 364571599000 x^{10} + 889660452520 x^{9} - 8783229793240 x^{8} + 1697893096120 x^{7} + 79566366868800 x^{6} - 223602818989200 x^{5} + 302507514834000 x^{4} - 231401632434000 x^{3} - 7218977715600 x^{2} + 2011887049200 x + 1281974857676400 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[3, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-164741220697322129579481287495063613302478454386458880000000000000000000=-\,2^{26}\cdot 3^{19}\cdot 5^{19}\cdot 7^{21}\cdot 2129^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2461.93$
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:  $2, 3, 5, 7, 2129$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{240} a^{14} - \frac{17}{240} a^{13} + \frac{17}{80} a^{12} - \frac{5}{48} a^{11} - \frac{11}{48} a^{10} + \frac{3}{80} a^{9} - \frac{23}{240} a^{8} - \frac{11}{240} a^{7} + \frac{1}{8} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{240} a^{15} + \frac{1}{120} a^{13} + \frac{1}{120} a^{12} + \frac{17}{120} a^{10} + \frac{1}{24} a^{9} - \frac{7}{40} a^{8} + \frac{83}{240} a^{7} - \frac{3}{8} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{240} a^{16} + \frac{3}{20} a^{13} + \frac{3}{40} a^{12} - \frac{3}{20} a^{11} - \frac{1}{4} a^{9} + \frac{3}{80} a^{8} + \frac{13}{60} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{480} a^{17} - \frac{1}{480} a^{16} - \frac{1}{480} a^{15} - \frac{1}{480} a^{14} + \frac{3}{160} a^{13} + \frac{97}{480} a^{12} + \frac{1}{480} a^{11} + \frac{7}{160} a^{10} - \frac{17}{240} a^{9} + \frac{1}{5} a^{8} + \frac{19}{60} a^{7} - \frac{1}{8} a^{6} - \frac{3}{8} a^{5} + \frac{1}{8} a^{4} + \frac{1}{8} a^{3} - \frac{1}{2}$, $\frac{1}{480} a^{18} + \frac{13}{80} a^{13} - \frac{1}{16} a^{12} - \frac{3}{16} a^{11} + \frac{1}{32} a^{10} - \frac{11}{48} a^{9} - \frac{19}{80} a^{8} + \frac{7}{16} a^{7} + \frac{1}{8} a^{6} + \frac{1}{4} a^{4} - \frac{3}{8} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{7200} a^{19} + \frac{1}{2400} a^{18} + \frac{1}{1200} a^{17} + \frac{1}{720} a^{16} - \frac{1}{1200} a^{14} + \frac{41}{600} a^{13} + \frac{67}{600} a^{12} - \frac{17}{96} a^{11} + \frac{19}{288} a^{10} + \frac{37}{240} a^{9} - \frac{49}{240} a^{8} + \frac{47}{144} a^{7} - \frac{47}{120} a^{6} + \frac{1}{20} a^{5} + \frac{1}{8} a^{4} - \frac{5}{24} a^{3} - \frac{1}{4} a - \frac{1}{12}$, $\frac{1}{510284681602364090960415207917133982930905875996502886517798259129984795046228363078177604533225078294343142261571134205989600} a^{20} - \frac{13821850651397179985161095922337316463380041859727252006440796739318467249237753756388324550485782211325920412936203918437}{255142340801182045480207603958566991465452937998251443258899129564992397523114181539088802266612539147171571130785567102994800} a^{19} - \frac{34143075298657910541095175733645712255293424149066040757414221947814353097116093918146175607106671226258069668429429910137}{34018978773490939397361013861142265528727058399766859101186550608665653003081890871878506968881671886289542817438075613732640} a^{18} - \frac{92057402928338408933243850415875323249393916445752976751342168001978870519591748317527227346031476448287571368484240174497}{510284681602364090960415207917133982930905875996502886517798259129984795046228363078177604533225078294343142261571134205989600} a^{17} - \frac{14698554446862761164151870420605825933881494903983031524118894486080505341439412851371838654169106929079848105158097062657}{20411387264094563638416608316685359317236235039860115460711930365199391801849134523127104181329003131773725690462845368239584} a^{16} + \frac{187640065249411326572012929005820069617496160401355022606908340547075877012584303601227484033116753471482095288269911827683}{170094893867454696986805069305711327643635291998834295505932753043328265015409454359392534844408359431447714087190378068663200} a^{15} - \frac{146787001577307319125744530103868277372223932334443971305854382592672352073082571816531614794487943452871935796832119664827}{170094893867454696986805069305711327643635291998834295505932753043328265015409454359392534844408359431447714087190378068663200} a^{14} + \frac{6024883510263158339169983995416797653228076896294300016261286232913900992226554138055672320706220824941772423857620009614357}{34018978773490939397361013861142265528727058399766859101186550608665653003081890871878506968881671886289542817438075613732640} a^{13} - \frac{4566024506313564898549676922144673469527630596805031158918786366295116321225025966468738830953922978483639741654975670714059}{42523723466863674246701267326427831910908822999708573876483188260832066253852363589848133711102089857861928521797594517165800} a^{12} - \frac{4667067689428777977491430164464437506303662270433308699822180542508644651992865662326224462303418002065221726994152551928339}{102056936320472818192083041583426796586181175199300577303559651825996959009245672615635520906645015658868628452314226841197920} a^{11} + \frac{607645029886462654796341205479774859316845112250801287115796218624709867656789797596842067220932074407762351102183733754677}{12757117040059102274010380197928349573272646899912572162944956478249619876155709076954440113330626957358578556539278355149740} a^{10} - \frac{2944447409480137533153624980066218205647135794204917671660725831106191994121522878562519279401060738410857276304781750141351}{17009489386745469698680506930571132764363529199883429550593275304332826501540945435939253484440835943144771408719037806866320} a^{9} - \frac{1235823878788312435818319646235351173400256331529052479199263379531728984570600076513877549104401546312798924872226534715033}{6378558520029551137005190098964174786636323449956286081472478239124809938077854538477220056665313478679289278269639177574870} a^{8} + \frac{110557582058991961632618692048360149024663711091114571200542124250127400806706002739659467514524700773425871120706873713247}{1275711704005910227401038019792834957327264689991257216294495647824961987615570907695444011333062695735857855653927835514974} a^{7} - \frac{681305325197650795451846106400367357885483315671034976433446711001873626452546554601228292469405313133171313420380395627395}{1700948938674546969868050693057113276436352919988342955059327530433282650154094543593925348444083594314477140871903780686632} a^{6} - \frac{521110912629854098258661845756367239986346803432423690842831129076147107105050584152012409599961341114043928472443100502127}{1417457448895455808223375577547594397030294099990285795882772942027735541795078786328271123703402995262064284059919817238860} a^{5} - \frac{414488171547318545215785620211522362411269514202191130976839984211603597841196362648672938026089385791982004771985659556585}{1700948938674546969868050693057113276436352919988342955059327530433282650154094543593925348444083594314477140871903780686632} a^{4} + \frac{49078321814702078150603186125817878891059139444579649125025678845474546376882324759899124350817750742770504613238254395567}{850474469337273484934025346528556638218176459994171477529663765216641325077047271796962674222041797157238570435951890343316} a^{3} - \frac{77848827265254564321965892408878643414940668893513941566517909230116114254466815603487259394716046223639943949465566066775}{283491489779091161644675115509518879406058819998057159176554588405547108359015757265654224740680599052412856811983963447772} a^{2} - \frac{91015730581088631298040046235311029596763468107821988390310442659740364183601990847427489646379305916861480548773113403740}{212618617334318371233506336632139159554544114998542869382415941304160331269261817949240668555510449289309642608987972585829} a + \frac{128989677706075131032872887192040986663162592813842950815290931857583955756324515639146420566074154888554723320488555691679}{425237234668636742467012673264278319109088229997085738764831882608320662538523635898481337111020898578619285217975945171658}$

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:  $11$
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

3.3.127740.1, 7.1.600362847000000.40

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 ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ $21$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.14.0.1}{14} }{,}\,{\href{/LocalNumberField/43.7.0.1}{7} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
$2$2.7.6.1$x^{7} - 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
2.14.20.8$x^{14} + 2 x^{13} + 2 x^{11} + 2 x^{8} + 2 x^{7} + 2 x^{6} + 2$$14$$1$$20$$(C_7:C_3) \times C_2$$[2]_{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
2129Data not computed