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Properties

Label 21.3.39066061807...3063.1
Degree $21$
Signature $[3, 9]$
Discriminant $-\,7^{17}\cdot 617^{18}$
Root discriminant $1190.69$
Ramified primes $7, 617$
Class number Not computed
Class group Not computed
Galois group $F_7$ (as 21T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![293698546559488, -428208522920192, 157072709296608, -17385833768968, -11010699170856, 10689969421306, -307347689851, -1367906097928, -44219027574, 18233746589, -3215159260, -574361697, -106972970, -20945477, 1715360, -387753, 53056, 29, -34, 21, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 + 21*x^19 - 34*x^18 + 29*x^17 + 53056*x^16 - 387753*x^15 + 1715360*x^14 - 20945477*x^13 - 106972970*x^12 - 574361697*x^11 - 3215159260*x^10 + 18233746589*x^9 - 44219027574*x^8 - 1367906097928*x^7 - 307347689851*x^6 + 10689969421306*x^5 - 11010699170856*x^4 - 17385833768968*x^3 + 157072709296608*x^2 - 428208522920192*x + 293698546559488)
 
gp: K = bnfinit(x^21 - 7*x^20 + 21*x^19 - 34*x^18 + 29*x^17 + 53056*x^16 - 387753*x^15 + 1715360*x^14 - 20945477*x^13 - 106972970*x^12 - 574361697*x^11 - 3215159260*x^10 + 18233746589*x^9 - 44219027574*x^8 - 1367906097928*x^7 - 307347689851*x^6 + 10689969421306*x^5 - 11010699170856*x^4 - 17385833768968*x^3 + 157072709296608*x^2 - 428208522920192*x + 293698546559488, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} + 21 x^{19} - 34 x^{18} + 29 x^{17} + 53056 x^{16} - 387753 x^{15} + 1715360 x^{14} - 20945477 x^{13} - 106972970 x^{12} - 574361697 x^{11} - 3215159260 x^{10} + 18233746589 x^{9} - 44219027574 x^{8} - 1367906097928 x^{7} - 307347689851 x^{6} + 10689969421306 x^{5} - 11010699170856 x^{4} - 17385833768968 x^{3} + 157072709296608 x^{2} - 428208522920192 x + 293698546559488 \)

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:  \(-39066061807525750900462785203767943376542499971141582990405683063=-\,7^{17}\cdot 617^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1190.69$
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:  $7, 617$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{13} - \frac{1}{8} a^{11} - \frac{1}{4} a^{10} - \frac{1}{8} a^{9} + \frac{1}{8} a^{7} - \frac{1}{4} a^{6} - \frac{3}{8} a^{5} + \frac{1}{4} a^{4} - \frac{1}{8} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{16} - \frac{1}{8} a^{14} - \frac{1}{8} a^{12} - \frac{1}{8} a^{10} - \frac{1}{8} a^{8} + \frac{3}{8} a^{6} + \frac{3}{8} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{17} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{3}{8} a^{3}$, $\frac{1}{172760} a^{18} + \frac{1217}{21595} a^{17} + \frac{10167}{172760} a^{16} + \frac{1247}{43190} a^{15} - \frac{19073}{172760} a^{14} - \frac{10389}{86380} a^{13} - \frac{20047}{172760} a^{12} - \frac{1411}{12340} a^{11} + \frac{1563}{24680} a^{10} - \frac{765}{8638} a^{9} - \frac{6351}{172760} a^{8} - \frac{2837}{17276} a^{7} + \frac{21391}{172760} a^{6} - \frac{334}{3085} a^{5} + \frac{1977}{12340} a^{4} - \frac{2437}{6170} a^{3} + \frac{445}{2468} a^{2} - \frac{451}{1234} a - \frac{601}{3085}$, $\frac{1}{4677649760} a^{19} - \frac{8039}{4677649760} a^{18} + \frac{24892537}{4677649760} a^{17} + \frac{71616579}{2338824880} a^{16} + \frac{82394817}{4677649760} a^{15} + \frac{15794479}{584706220} a^{14} + \frac{477024763}{4677649760} a^{13} + \frac{19455631}{292353110} a^{12} + \frac{12957513}{668235680} a^{11} + \frac{27677611}{467764976} a^{10} - \frac{976542181}{4677649760} a^{9} - \frac{13206163}{233882488} a^{8} + \frac{1003315401}{4677649760} a^{7} - \frac{109777667}{2338824880} a^{6} - \frac{50446939}{167058920} a^{5} + \frac{321012387}{668235680} a^{4} - \frac{4243947}{9546224} a^{3} - \frac{6946241}{16705892} a^{2} + \frac{17176289}{83529460} a - \frac{1901006}{4176473}$, $\frac{1}{880569794965544197594186496764237089187174239086024116474500343269386599084056930097310691914557061982439244226211432877027127680} a^{20} + \frac{5846393464850321178513754496909764567588004017004114848745800587296218187594864485984128899193210986293276554629343491}{125795684995077742513455213823462441312453462726574873782071477609912371297722418585330098844936723140348463460887347553861018240} a^{19} + \frac{1788387941949680996261634489509754776577940793215226405308767418711202025491972789865958359301031318239097450394298749533041}{880569794965544197594186496764237089187174239086024116474500343269386599084056930097310691914557061982439244226211432877027127680} a^{18} + \frac{20056939193090776069881877776534266012939771637778772592148685072768197969325283109991674036397696542597693368782014312645058589}{440284897482772098797093248382118544593587119543012058237250171634693299542028465048655345957278530991219622113105716438513563840} a^{17} + \frac{50167655082220172958222087177831654805045681791297174528830652351061749699174900582745634500241850816676217141961165885864226741}{880569794965544197594186496764237089187174239086024116474500343269386599084056930097310691914557061982439244226211432877027127680} a^{16} - \frac{9827993500878723754976381380877243482314529535653661513047033974387948415572571909846001483301451516455039149474194563757461153}{220142448741386049398546624191059272296793559771506029118625085817346649771014232524327672978639265495609811056552858219256781920} a^{15} - \frac{92892704285050145177300213842524859280112825849680203924023218583385446973093380344858284984486046302351961703801918876669654169}{880569794965544197594186496764237089187174239086024116474500343269386599084056930097310691914557061982439244226211432877027127680} a^{14} - \frac{1003236667566104942210006395284911716389899260366267968480875502007744093770596747081756349230847179342872798190280499283721199}{44028489748277209879709324838211854459358711954301205823725017163469329954202846504865534595727853099121962211310571643851356384} a^{13} - \frac{51823728960701412813942767017219416982933054756466233709262128683657567796195414048333437885384238399981086623636372278510087989}{880569794965544197594186496764237089187174239086024116474500343269386599084056930097310691914557061982439244226211432877027127680} a^{12} + \frac{3671593884815266973165862510086482518206418479862919489879055465223915601150368446622404403684691835843113137152938943752938717}{440284897482772098797093248382118544593587119543012058237250171634693299542028465048655345957278530991219622113105716438513563840} a^{11} + \frac{3894434695672339347702886553027329640497159893429014616801197683463437659180203808172908730420384326286173419103020907456236881}{125795684995077742513455213823462441312453462726574873782071477609912371297722418585330098844936723140348463460887347553861018240} a^{10} + \frac{19473980179582232958082376770039127536327408022855552492230835700216933113128152750561111108409354438468531219489744948701399823}{110071224370693024699273312095529636148396779885753014559312542908673324885507116262163836489319632747804905528276429109628390960} a^{9} + \frac{150212045437731662215549160115196420992272708198346496757306251247728286925722806965565705197855294416956230484483483186185986013}{880569794965544197594186496764237089187174239086024116474500343269386599084056930097310691914557061982439244226211432877027127680} a^{8} - \frac{8470438962912761742703205225400385769765129523699959983678734868480179716471714557220640737414608955487898563842951883494045947}{62897842497538871256727606911731220656226731363287436891035738804956185648861209292665049422468361570174231730443673776930509120} a^{7} - \frac{1840399972437194344938987484452591586651103241559277902274929502511660466724844732203191912135789878991032273193846310520714027}{13758903046336628087409164011941204518549597485719126819914067863584165610688389532770479561164954093475613191034553638703548870} a^{6} + \frac{123165480011649289509597596979999643930088615068081136545847542848717882597843981983717425217294216601312655975439444386918855}{25159136999015548502691042764692488262490692545314974756414295521982474259544483717066019768987344628069692692177469510772203648} a^{5} + \frac{28228602150668678567202931598932692096583616654339119906583497182795649544878782086336393205150655420996828815388059026245366397}{62897842497538871256727606911731220656226731363287436891035738804956185648861209292665049422468361570174231730443673776930509120} a^{4} + \frac{1637543348477527115711287786149226210773803433103185442365065706048726998550378346399451622746966596872570042431107808323677}{16981058989616325933241794522605621127491018186632677346391938122288387054228188253959246604338110575101034484461035036968280} a^{3} - \frac{7826911064242180518429484489117966770903960111397889004256416834184278671047398439010528356816212487567807508569417021160857951}{15724460624384717814181901727932805164056682840821859222758934701239046412215302323166262355617090392543557932610918444232627280} a^{2} + \frac{44852944910370426204080205582573122873434989675868484314149659705151587146285313993982391305793637365919208231406911189688719}{280793939721155675253248245141657235072440765014676057549266691093554400218130398627968970636019471295420677368052115075582630} a - \frac{427761871475109745339879838984653491363839847535801611696009983358738665557365503836318247315869252398510991977998395242300408}{982778789024044863386368857995800322753542677551366201422433418827440400763456395197891397226068149533972370788182402764539205}$

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

$F_7$ (as 21T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 42
The 7 conjugacy class representatives for $F_7$
Character table for $F_7$

Intermediate fields

\(\Q(\zeta_{7})^+\), Deg 7

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

Sibling fields

Galois closure: data not computed
Degree 7 sibling: data not computed
Degree 14 sibling: 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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
617Data not computed