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Properties

Label 21.21.3096148947...3329.1
Degree $21$
Signature $[21, 0]$
Discriminant $31^{14}\cdot 577^{10}$
Root discriminant $203.74$
Ramified primes $31, 577$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $D_{21}$ (as 21T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![43025225, -3015176110, -20539678739, -17641871062, 64981505313, 38245288476, -49423979231, -29542123459, 10809094658, 7604045846, -919804797, -902360226, 25506215, 56351019, 583246, -1932617, -43578, 35703, 664, -322, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 2*x^20 - 322*x^19 + 664*x^18 + 35703*x^17 - 43578*x^16 - 1932617*x^15 + 583246*x^14 + 56351019*x^13 + 25506215*x^12 - 902360226*x^11 - 919804797*x^10 + 7604045846*x^9 + 10809094658*x^8 - 29542123459*x^7 - 49423979231*x^6 + 38245288476*x^5 + 64981505313*x^4 - 17641871062*x^3 - 20539678739*x^2 - 3015176110*x + 43025225)
 
gp: K = bnfinit(x^21 - 2*x^20 - 322*x^19 + 664*x^18 + 35703*x^17 - 43578*x^16 - 1932617*x^15 + 583246*x^14 + 56351019*x^13 + 25506215*x^12 - 902360226*x^11 - 919804797*x^10 + 7604045846*x^9 + 10809094658*x^8 - 29542123459*x^7 - 49423979231*x^6 + 38245288476*x^5 + 64981505313*x^4 - 17641871062*x^3 - 20539678739*x^2 - 3015176110*x + 43025225, 1)
 

Normalized defining polynomial

\( x^{21} - 2 x^{20} - 322 x^{19} + 664 x^{18} + 35703 x^{17} - 43578 x^{16} - 1932617 x^{15} + 583246 x^{14} + 56351019 x^{13} + 25506215 x^{12} - 902360226 x^{11} - 919804797 x^{10} + 7604045846 x^{9} + 10809094658 x^{8} - 29542123459 x^{7} - 49423979231 x^{6} + 38245288476 x^{5} + 64981505313 x^{4} - 17641871062 x^{3} - 20539678739 x^{2} - 3015176110 x + 43025225 \)

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:  \(3096148947286301526552289503870245765743349723329=31^{14}\cdot 577^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $203.74$
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:  $31, 577$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{5} a^{15} - \frac{2}{5} a^{14} + \frac{2}{5} a^{13} - \frac{2}{5} a^{12} - \frac{2}{5} a^{11} - \frac{2}{5} a^{10} - \frac{1}{5} a^{9} + \frac{1}{5} a^{8} + \frac{1}{5} a^{6} + \frac{1}{5} a^{5} + \frac{1}{5} a^{4} - \frac{1}{5} a^{3} + \frac{2}{5} a$, $\frac{1}{5} a^{16} - \frac{2}{5} a^{14} + \frac{2}{5} a^{13} - \frac{1}{5} a^{12} - \frac{1}{5} a^{11} - \frac{1}{5} a^{9} + \frac{2}{5} a^{8} + \frac{1}{5} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{1}{5} a^{4} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{455} a^{17} + \frac{2}{35} a^{16} + \frac{8}{91} a^{15} - \frac{4}{455} a^{14} + \frac{8}{91} a^{13} + \frac{79}{455} a^{12} + \frac{4}{13} a^{11} - \frac{5}{91} a^{10} + \frac{134}{455} a^{9} - \frac{44}{91} a^{8} + \frac{27}{65} a^{7} - \frac{142}{455} a^{6} - \frac{144}{455} a^{5} + \frac{211}{455} a^{4} + \frac{1}{35} a^{3} - \frac{9}{455} a^{2} - \frac{92}{455} a - \frac{38}{91}$, $\frac{1}{455} a^{18} + \frac{1}{455} a^{16} - \frac{43}{455} a^{15} + \frac{53}{455} a^{14} + \frac{8}{91} a^{13} - \frac{3}{455} a^{12} + \frac{66}{455} a^{11} + \frac{21}{65} a^{10} + \frac{118}{455} a^{9} - \frac{6}{455} a^{8} + \frac{131}{455} a^{7} + \frac{18}{91} a^{6} + \frac{6}{65} a^{5} - \frac{3}{7} a^{4} + \frac{108}{455} a^{3} + \frac{51}{455} a^{2} - \frac{73}{455} a - \frac{1}{7}$, $\frac{1}{5915} a^{19} + \frac{1}{5915} a^{18} - \frac{6}{5915} a^{17} - \frac{19}{845} a^{16} - \frac{88}{5915} a^{15} - \frac{176}{1183} a^{14} - \frac{1972}{5915} a^{13} - \frac{14}{169} a^{12} + \frac{81}{455} a^{11} + \frac{2806}{5915} a^{10} - \frac{27}{845} a^{9} - \frac{1611}{5915} a^{8} + \frac{809}{5915} a^{7} - \frac{2514}{5915} a^{6} + \frac{535}{1183} a^{5} - \frac{2656}{5915} a^{4} - \frac{1661}{5915} a^{3} + \frac{2043}{5915} a^{2} - \frac{586}{5915} a - \frac{202}{1183}$, $\frac{1}{844817547370913925246080673738269517480249167233151569226202766241659681643308531523615640338817115} a^{20} - \frac{26212627416585013480462112376812362813591593586865306644376957498082011260677722032713583225058}{844817547370913925246080673738269517480249167233151569226202766241659681643308531523615640338817115} a^{19} + \frac{127357019347852372687170876760939467549556696284364181111886692918332674244626177081095531198046}{844817547370913925246080673738269517480249167233151569226202766241659681643308531523615640338817115} a^{18} - \frac{342590935747247178034838285579188283631724580525201514287037457502785842850896110698131562968577}{844817547370913925246080673738269517480249167233151569226202766241659681643308531523615640338817115} a^{17} - \frac{1635232371729456666375959129686251826712504652015734284814272170217430847633420695800697991880195}{24137644210597540721316590678236271928007119063804330549320079035475990904094529472103304009680489} a^{16} + \frac{77646549107093558711033801017562631360460162571335158848931315162969994079046322319427978076805047}{844817547370913925246080673738269517480249167233151569226202766241659681643308531523615640338817115} a^{15} - \frac{330207846331427810779807596390139457288312841457976093380430062067205122516505114228775844347302979}{844817547370913925246080673738269517480249167233151569226202766241659681643308531523615640338817115} a^{14} + \frac{12616089027548989918066407053959617054874489790273891252299214478163488009904012885326152884228600}{168963509474182785049216134747653903496049833446630313845240553248331936328661706304723128067763423} a^{13} - \frac{208215773034899269516658449405965492199338925034578085726905438223142878649171594228031039651270186}{844817547370913925246080673738269517480249167233151569226202766241659681643308531523615640338817115} a^{12} - \frac{30737194193073779824414489878047863544895894285244888965841465398170664110671984867722204643419364}{844817547370913925246080673738269517480249167233151569226202766241659681643308531523615640338817115} a^{11} + \frac{149702852219709915097425770125593036105283259584886655718939215270536575070462637073990498890832283}{844817547370913925246080673738269517480249167233151569226202766241659681643308531523615640338817115} a^{10} + \frac{11837291728546523129177253485731934497496601424025868239077511450510997748039117724320544101921970}{24137644210597540721316590678236271928007119063804330549320079035475990904094529472103304009680489} a^{9} - \frac{103797795620197242690618180842754615855554198817488611784532194464694262715464645617429739659393019}{844817547370913925246080673738269517480249167233151569226202766241659681643308531523615640338817115} a^{8} - \frac{123952250337832134740560389516388452980809040085362390238641798510673273872645411667489646138472928}{844817547370913925246080673738269517480249167233151569226202766241659681643308531523615640338817115} a^{7} + \frac{74648685304449146921666688904736608205141066851465612566551866550389942073095828322425971119845559}{844817547370913925246080673738269517480249167233151569226202766241659681643308531523615640338817115} a^{6} + \frac{17977674613404689358552509353781754469372100314056976740654210231568463574446106613718902228889599}{168963509474182785049216134747653903496049833446630313845240553248331936328661706304723128067763423} a^{5} + \frac{20489316574877737444289929974812419280286343340098754534987989691331964485537619297478640734013256}{844817547370913925246080673738269517480249167233151569226202766241659681643308531523615640338817115} a^{4} - \frac{165651296254613030159677656156528807101908481297752957000188746834722477569101433637606564690819368}{844817547370913925246080673738269517480249167233151569226202766241659681643308531523615640338817115} a^{3} + \frac{32833402253614168472477672366570873759840445047776039640308324118952641271089157227449883933022416}{844817547370913925246080673738269517480249167233151569226202766241659681643308531523615640338817115} a^{2} - \frac{63580143720726825937421459178417655631007151099245394696738255576490972743492569397855492252246701}{168963509474182785049216134747653903496049833446630313845240553248331936328661706304723128067763423} a - \frac{29592352686146011523040431512126224491131255134970724827842906605597409322460178316895699182044742}{168963509474182785049216134747653903496049833446630313845240553248331936328661706304723128067763423}$

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:  \( 238900599465000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{21}$ (as 21T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 42
The 12 conjugacy class representatives for $D_{21}$
Character table for $D_{21}$

Intermediate fields

3.3.554497.1, 7.7.192100033.1

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

Sibling fields

Galois closure: 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 $21$ $21$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ $21$ $21$ $21$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ R ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{3}$

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
31Data not computed
577Data not computed