↑

Properties

Label 22.0.82487448322...5931.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,331^{21}$
Root discriminant $254.27$
Ramified prime $331$
Class number $4623$ (GRH)
Class group $[4623]$ (GRH)
Galois group $C_{22}$ (as 22T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9703203584, -18990975168, 10949586736, 2958199440, -2509852460, -1492486348, 1779169827, -322947527, -106359704, 58800883, 19719085, -13427544, 2439822, 1322066, -437088, 46590, 30942, -5596, 399, 469, 8, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 + 8*x^20 + 469*x^19 + 399*x^18 - 5596*x^17 + 30942*x^16 + 46590*x^15 - 437088*x^14 + 1322066*x^13 + 2439822*x^12 - 13427544*x^11 + 19719085*x^10 + 58800883*x^9 - 106359704*x^8 - 322947527*x^7 + 1779169827*x^6 - 1492486348*x^5 - 2509852460*x^4 + 2958199440*x^3 + 10949586736*x^2 - 18990975168*x + 9703203584)
 
gp: K = bnfinit(x^22 - x^21 + 8*x^20 + 469*x^19 + 399*x^18 - 5596*x^17 + 30942*x^16 + 46590*x^15 - 437088*x^14 + 1322066*x^13 + 2439822*x^12 - 13427544*x^11 + 19719085*x^10 + 58800883*x^9 - 106359704*x^8 - 322947527*x^7 + 1779169827*x^6 - 1492486348*x^5 - 2509852460*x^4 + 2958199440*x^3 + 10949586736*x^2 - 18990975168*x + 9703203584, 1)
 

Normalized defining polynomial

\( x^{22} - x^{21} + 8 x^{20} + 469 x^{19} + 399 x^{18} - 5596 x^{17} + 30942 x^{16} + 46590 x^{15} - 437088 x^{14} + 1322066 x^{13} + 2439822 x^{12} - 13427544 x^{11} + 19719085 x^{10} + 58800883 x^{9} - 106359704 x^{8} - 322947527 x^{7} + 1779169827 x^{6} - 1492486348 x^{5} - 2509852460 x^{4} + 2958199440 x^{3} + 10949586736 x^{2} - 18990975168 x + 9703203584 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 11]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-82487448322624491126367023211989174496912600822635931=-\,331^{21}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $254.27$
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:  $331$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(331\)
Dirichlet character group:    $\lbrace$$\chi_{331}(1,·)$, $\chi_{331}(74,·)$, $\chi_{331}(257,·)$, $\chi_{331}(330,·)$, $\chi_{331}(270,·)$, $\chi_{331}(80,·)$, $\chi_{331}(274,·)$, $\chi_{331}(211,·)$, $\chi_{331}(85,·)$, $\chi_{331}(151,·)$, $\chi_{331}(220,·)$, $\chi_{331}(164,·)$, $\chi_{331}(293,·)$, $\chi_{331}(38,·)$, $\chi_{331}(167,·)$, $\chi_{331}(111,·)$, $\chi_{331}(180,·)$, $\chi_{331}(246,·)$, $\chi_{331}(120,·)$, $\chi_{331}(57,·)$, $\chi_{331}(251,·)$, $\chi_{331}(61,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{5}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{9} - \frac{1}{8} a^{6} + \frac{1}{8} a^{3}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{10} - \frac{1}{8} a^{7} + \frac{1}{8} a^{4}$, $\frac{1}{16} a^{14} - \frac{1}{16} a^{13} - \frac{1}{16} a^{12} + \frac{1}{16} a^{11} + \frac{1}{16} a^{10} + \frac{1}{16} a^{9} + \frac{1}{16} a^{8} + \frac{1}{16} a^{7} - \frac{3}{16} a^{6} + \frac{3}{16} a^{5} + \frac{3}{16} a^{4} + \frac{3}{16} a^{3} - \frac{3}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{16} a^{15} - \frac{1}{8} a^{11} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} + \frac{1}{8} a^{5} + \frac{1}{16} a^{3} + \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{32} a^{16} + \frac{1}{16} a^{10} - \frac{1}{4} a^{7} - \frac{3}{32} a^{4} + \frac{1}{4} a$, $\frac{1}{992} a^{17} + \frac{1}{248} a^{16} - \frac{9}{496} a^{15} + \frac{13}{496} a^{14} + \frac{5}{496} a^{13} - \frac{23}{496} a^{12} + \frac{3}{62} a^{11} + \frac{23}{496} a^{10} + \frac{41}{496} a^{9} + \frac{43}{496} a^{8} + \frac{87}{496} a^{7} + \frac{75}{496} a^{6} - \frac{169}{992} a^{5} + \frac{59}{496} a^{4} - \frac{37}{124} a^{3} - \frac{15}{124} a^{2} + \frac{13}{31} a + \frac{15}{31}$, $\frac{1}{1984} a^{18} - \frac{1}{1984} a^{17} - \frac{7}{1984} a^{16} - \frac{1}{248} a^{15} - \frac{29}{992} a^{14} - \frac{17}{992} a^{13} + \frac{23}{496} a^{12} + \frac{29}{496} a^{11} - \frac{37}{496} a^{10} + \frac{55}{992} a^{9} - \frac{97}{992} a^{8} - \frac{19}{992} a^{7} - \frac{485}{1984} a^{6} + \frac{405}{1984} a^{5} - \frac{173}{1984} a^{4} + \frac{401}{992} a^{3} - \frac{211}{496} a^{2} - \frac{7}{124} a + \frac{9}{31}$, $\frac{1}{3968} a^{19} - \frac{1}{3968} a^{18} - \frac{1}{3968} a^{17} + \frac{1}{248} a^{16} + \frac{41}{1984} a^{15} + \frac{61}{1984} a^{14} + \frac{19}{496} a^{13} - \frac{5}{124} a^{12} + \frac{35}{992} a^{11} - \frac{55}{1984} a^{10} - \frac{99}{1984} a^{9} - \frac{9}{1984} a^{8} + \frac{559}{3968} a^{7} - \frac{679}{3968} a^{6} + \frac{797}{3968} a^{5} + \frac{11}{1984} a^{4} + \frac{399}{992} a^{3} - \frac{21}{248} a^{2} - \frac{7}{31} a + \frac{7}{31}$, $\frac{1}{10209664} a^{20} - \frac{361}{10209664} a^{19} + \frac{11}{10209664} a^{18} - \frac{637}{1276208} a^{17} - \frac{21135}{5104832} a^{16} + \frac{147449}{5104832} a^{15} + \frac{15213}{1276208} a^{14} - \frac{6481}{319052} a^{13} + \frac{80801}{2552416} a^{12} + \frac{186881}{5104832} a^{11} + \frac{67461}{5104832} a^{10} - \frac{564865}{5104832} a^{9} + \frac{1094767}{10209664} a^{8} - \frac{58591}{10209664} a^{7} - \frac{964903}{10209664} a^{6} + \frac{1065519}{5104832} a^{5} + \frac{489463}{2552416} a^{4} - \frac{489027}{1276208} a^{3} - \frac{143893}{638104} a^{2} - \frac{24623}{319052} a - \frac{22377}{79763}$, $\frac{1}{597395540169586401279185523994232909989321563878622522925658577149975573582336491277824} a^{21} + \frac{915976562250805276275696919628271379245959214184250667355929200393688910415887}{37337221260599150079949095249639556874332597742413907682853661071873473348896030704864} a^{20} - \frac{2545891390224425166928712663356108252831409767146687258487507272289589612336805529}{37337221260599150079949095249639556874332597742413907682853661071873473348896030704864} a^{19} + \frac{133799131791376721136866215324617824569747969961971615891911319085273501432326937749}{597395540169586401279185523994232909989321563878622522925658577149975573582336491277824} a^{18} + \frac{11276459199906758818292788547203883828188584623328021670707267856136797610066914037}{149348885042396600319796380998558227497330390969655630731414644287493893395584122819456} a^{17} - \frac{64359746437951877733106660174934990555304764918440204588662526365033024219337670139}{37337221260599150079949095249639556874332597742413907682853661071873473348896030704864} a^{16} - \frac{8068347794382606876432202997780520857051554402120748312610412251648659401876148831465}{298697770084793200639592761997116454994660781939311261462829288574987786791168245638912} a^{15} - \frac{2578717273433782987516417783838330166400483550745395538338411055801806595346144721237}{149348885042396600319796380998558227497330390969655630731414644287493893395584122819456} a^{14} - \frac{2057338198174739876262220510369966029251007139079546441894194270801640448070562476145}{149348885042396600319796380998558227497330390969655630731414644287493893395584122819456} a^{13} + \frac{17916773445856374055815994537844968838475563421354912583745133223376390742375591285399}{298697770084793200639592761997116454994660781939311261462829288574987786791168245638912} a^{12} + \frac{4348922683181889262601644543540255280576670200456123293971691706633041423378542795575}{149348885042396600319796380998558227497330390969655630731414644287493893395584122819456} a^{11} - \frac{15230943783901895839095503233021080817482835924199689883155506862933788891156106508243}{149348885042396600319796380998558227497330390969655630731414644287493893395584122819456} a^{10} - \frac{47079185798089208654262368926865476596950544930803064516631733400538095472050041882015}{597395540169586401279185523994232909989321563878622522925658577149975573582336491277824} a^{9} + \frac{10618015021529039286315140334571200494585050179256005803141119105904508935478105833857}{149348885042396600319796380998558227497330390969655630731414644287493893395584122819456} a^{8} - \frac{6786983580790723457005370062460114733796108438416953038229692335638288208136129620479}{149348885042396600319796380998558227497330390969655630731414644287493893395584122819456} a^{7} + \frac{141694844699709034630759187073729513457391649005207372210333933876375317898107956885965}{597395540169586401279185523994232909989321563878622522925658577149975573582336491277824} a^{6} - \frac{1236368571193786074584981734296333716008906788569736594812429137099627776465809755999}{18668610630299575039974547624819778437166298871206953841426830535936736674448015352432} a^{5} - \frac{17338341348423459025571778719672369424749128872292065236717493870256922315115267904909}{149348885042396600319796380998558227497330390969655630731414644287493893395584122819456} a^{4} - \frac{1603416229360667137213430180064670294072568111835822887308806407080867008592010808635}{18668610630299575039974547624819778437166298871206953841426830535936736674448015352432} a^{3} + \frac{1470053189360901588531739008858340639793430042695045353120051192684818195909917706177}{37337221260599150079949095249639556874332597742413907682853661071873473348896030704864} a^{2} - \frac{2180747854400062764533385799051440780616423070627607492386760860406400853174467473435}{9334305315149787519987273812409889218583149435603476920713415267968368337224007676216} a - \frac{80572217275606709747464637753877049193388571873798813361545798633926504843068574533}{2333576328787446879996818453102472304645787358900869230178353816992092084306001919054}$

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

Class group and class number

$C_{4623}$, which has order $4623$ (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:  $10$
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:  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:  \( 1372307132022027.0 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{22}$ (as 22T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 22
The 22 conjugacy class representatives for $C_{22}$
Character table for $C_{22}$ is not computed

Intermediate fields

\(\Q(\sqrt{-331}) \), 11.11.15786284949774657045043801.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.2.0.1}{2} }^{11}$ $22$ ${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$ $22$ $22$ $22$ ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/19.11.0.1}{11} }^{2}$ $22$ $22$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{22}$ $22$ $22$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ $22$

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
331Data not computed