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Properties

Label 22.0.10870284342...4375.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,5^{11}\cdot 67^{21}$
Root discriminant $123.75$
Ramified primes $5, 67$
Class number $306522$ (GRH)
Class group $[306522]$ (GRH)
Galois group $C_{22}$ (as 22T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![48482809, 41416437, 42185124, 4152387, -25724027, -4039339, 45275429, 54976444, 46733335, 34080648, 19147508, 8660900, 3834110, 1343170, 407666, 135984, 28072, 7027, 1709, 107, 69, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 + 69*x^20 + 107*x^19 + 1709*x^18 + 7027*x^17 + 28072*x^16 + 135984*x^15 + 407666*x^14 + 1343170*x^13 + 3834110*x^12 + 8660900*x^11 + 19147508*x^10 + 34080648*x^9 + 46733335*x^8 + 54976444*x^7 + 45275429*x^6 - 4039339*x^5 - 25724027*x^4 + 4152387*x^3 + 42185124*x^2 + 41416437*x + 48482809)
 
gp: K = bnfinit(x^22 - x^21 + 69*x^20 + 107*x^19 + 1709*x^18 + 7027*x^17 + 28072*x^16 + 135984*x^15 + 407666*x^14 + 1343170*x^13 + 3834110*x^12 + 8660900*x^11 + 19147508*x^10 + 34080648*x^9 + 46733335*x^8 + 54976444*x^7 + 45275429*x^6 - 4039339*x^5 - 25724027*x^4 + 4152387*x^3 + 42185124*x^2 + 41416437*x + 48482809, 1)
 

Normalized defining polynomial

\( x^{22} - x^{21} + 69 x^{20} + 107 x^{19} + 1709 x^{18} + 7027 x^{17} + 28072 x^{16} + 135984 x^{15} + 407666 x^{14} + 1343170 x^{13} + 3834110 x^{12} + 8660900 x^{11} + 19147508 x^{10} + 34080648 x^{9} + 46733335 x^{8} + 54976444 x^{7} + 45275429 x^{6} - 4039339 x^{5} - 25724027 x^{4} + 4152387 x^{3} + 42185124 x^{2} + 41416437 x + 48482809 \)

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:  \(-10870284342485680666407125885565079749365234375=-\,5^{11}\cdot 67^{21}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $123.75$
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:  $5, 67$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(335=5\cdot 67\)
Dirichlet character group:    $\lbrace$$\chi_{335}(1,·)$, $\chi_{335}(131,·)$, $\chi_{335}(196,·)$, $\chi_{335}(204,·)$, $\chi_{335}(139,·)$, $\chi_{335}(76,·)$, $\chi_{335}(334,·)$, $\chi_{335}(81,·)$, $\chi_{335}(259,·)$, $\chi_{335}(216,·)$, $\chi_{335}(91,·)$, $\chi_{335}(156,·)$, $\chi_{335}(94,·)$, $\chi_{335}(226,·)$, $\chi_{335}(209,·)$, $\chi_{335}(109,·)$, $\chi_{335}(241,·)$, $\chi_{335}(179,·)$, $\chi_{335}(244,·)$, $\chi_{335}(126,·)$, $\chi_{335}(119,·)$, $\chi_{335}(254,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{29} a^{14} - \frac{5}{29} a^{13} - \frac{8}{29} a^{12} + \frac{13}{29} a^{11} - \frac{14}{29} a^{10} + \frac{11}{29} a^{9} + \frac{12}{29} a^{8} + \frac{7}{29} a^{7} - \frac{10}{29} a^{6} + \frac{1}{29} a^{5} - \frac{9}{29} a^{4} + \frac{14}{29} a^{3} + \frac{7}{29} a^{2} - \frac{4}{29} a$, $\frac{1}{29} a^{15} - \frac{4}{29} a^{13} + \frac{2}{29} a^{12} - \frac{7}{29} a^{11} - \frac{1}{29} a^{10} + \frac{9}{29} a^{9} + \frac{9}{29} a^{8} - \frac{4}{29} a^{7} + \frac{9}{29} a^{6} - \frac{4}{29} a^{5} - \frac{2}{29} a^{4} - \frac{10}{29} a^{3} + \frac{2}{29} a^{2} + \frac{9}{29} a$, $\frac{1}{29} a^{16} + \frac{11}{29} a^{13} - \frac{10}{29} a^{12} - \frac{7}{29} a^{11} + \frac{11}{29} a^{10} - \frac{5}{29} a^{9} - \frac{14}{29} a^{8} + \frac{8}{29} a^{7} + \frac{14}{29} a^{6} + \frac{2}{29} a^{5} + \frac{12}{29} a^{4} + \frac{8}{29} a^{2} + \frac{13}{29} a$, $\frac{1}{29} a^{17} - \frac{13}{29} a^{13} - \frac{6}{29} a^{12} + \frac{13}{29} a^{11} + \frac{4}{29} a^{10} + \frac{10}{29} a^{9} - \frac{8}{29} a^{8} - \frac{5}{29} a^{7} - \frac{4}{29} a^{6} + \frac{1}{29} a^{5} + \frac{12}{29} a^{4} - \frac{1}{29} a^{3} - \frac{6}{29} a^{2} - \frac{14}{29} a$, $\frac{1}{29} a^{18} - \frac{13}{29} a^{13} - \frac{4}{29} a^{12} - \frac{1}{29} a^{11} + \frac{2}{29} a^{10} - \frac{10}{29} a^{9} + \frac{6}{29} a^{8} - \frac{13}{29} a^{6} - \frac{4}{29} a^{5} - \frac{2}{29} a^{4} + \frac{2}{29} a^{3} - \frac{10}{29} a^{2} + \frac{6}{29} a$, $\frac{1}{29} a^{19} - \frac{11}{29} a^{13} + \frac{11}{29} a^{12} - \frac{3}{29} a^{11} + \frac{11}{29} a^{10} + \frac{4}{29} a^{9} + \frac{11}{29} a^{8} - \frac{9}{29} a^{7} + \frac{11}{29} a^{6} + \frac{11}{29} a^{5} + \frac{1}{29} a^{4} - \frac{2}{29} a^{3} + \frac{10}{29} a^{2} + \frac{6}{29} a$, $\frac{1}{841} a^{20} - \frac{7}{841} a^{19} + \frac{10}{841} a^{18} + \frac{3}{841} a^{16} - \frac{9}{841} a^{15} + \frac{12}{841} a^{14} + \frac{173}{841} a^{13} - \frac{265}{841} a^{12} + \frac{334}{841} a^{11} - \frac{246}{841} a^{10} - \frac{47}{841} a^{9} - \frac{366}{841} a^{8} + \frac{237}{841} a^{7} + \frac{28}{841} a^{6} + \frac{152}{841} a^{5} - \frac{385}{841} a^{4} - \frac{8}{841} a^{3} + \frac{32}{841} a^{2} + \frac{5}{29} a$, $\frac{1}{9979337181346683634440965893350919756678686080476354932858763098544279494754609} a^{21} - \frac{175598113808716509250190197545648032358865287908251948864543751914372804175}{9979337181346683634440965893350919756678686080476354932858763098544279494754609} a^{20} - \frac{40040480828282422847179879665261793298662469945006930882503297071922846452406}{9979337181346683634440965893350919756678686080476354932858763098544279494754609} a^{19} + \frac{109942926537638182495618119809666818609684953095781092004767298914129728294703}{9979337181346683634440965893350919756678686080476354932858763098544279494754609} a^{18} + \frac{142656479266406910616769112944838730730526169946944958053493112642695674208260}{9979337181346683634440965893350919756678686080476354932858763098544279494754609} a^{17} - \frac{53642689088452243363675267832366222065712091229130954311730220915496111030030}{9979337181346683634440965893350919756678686080476354932858763098544279494754609} a^{16} + \frac{93067227708459841757029748819812993170225071747924723790868569317297852717267}{9979337181346683634440965893350919756678686080476354932858763098544279494754609} a^{15} + \frac{126511798831182913579175632706225154486152488374252433134513096804980426449254}{9979337181346683634440965893350919756678686080476354932858763098544279494754609} a^{14} + \frac{3433065573192042466362363283096009852236308176042188295377156310310475954669324}{9979337181346683634440965893350919756678686080476354932858763098544279494754609} a^{13} - \frac{3605800884408882699086465424905852625924882452426872398480301799274175230465825}{9979337181346683634440965893350919756678686080476354932858763098544279494754609} a^{12} - \frac{154571431217561048959783364097180900937329155488847309109645657595775062464059}{344115075218851159808309168736238612299265037257805342512371141329113086026021} a^{11} + \frac{334483686937474256538514782404624102751201129825361975905036386951010124563199}{9979337181346683634440965893350919756678686080476354932858763098544279494754609} a^{10} - \frac{131114859291401217515040275595886882442013104559906163429538207610205551284072}{344115075218851159808309168736238612299265037257805342512371141329113086026021} a^{9} + \frac{4334466379971109724950026659757192618542216943076762007944886035310677642774058}{9979337181346683634440965893350919756678686080476354932858763098544279494754609} a^{8} + \frac{3125416491483470862283297728235898547104744072278252725331682659422352590718955}{9979337181346683634440965893350919756678686080476354932858763098544279494754609} a^{7} + \frac{1635981395262235521365045125176645879023002842474142051636277235457623946522642}{9979337181346683634440965893350919756678686080476354932858763098544279494754609} a^{6} - \frac{716708245274408456973392326836387102798744758232735716978979773075680080693315}{9979337181346683634440965893350919756678686080476354932858763098544279494754609} a^{5} - \frac{19938366200893396664733019265429754108726047613929223649067811205585822113949}{344115075218851159808309168736238612299265037257805342512371141329113086026021} a^{4} + \frac{3903891086780914810056961230150683515034856897654355278515490728602876640488252}{9979337181346683634440965893350919756678686080476354932858763098544279494754609} a^{3} + \frac{260031937211573439196274027853155174536854038387359514216211252722446904991287}{9979337181346683634440965893350919756678686080476354932858763098544279494754609} a^{2} + \frac{113070000245462819025738197312704309661739048898427175833894198983595706928538}{344115075218851159808309168736238612299265037257805342512371141329113086026021} a - \frac{2457583092432538922356450150744943431539444084585175405941217276456005071990}{11866037076512108958907212715042710768940173698545011810771418666521140897449}$

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

Class group and class number

$C_{306522}$, which has order $306522$ (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:  \( 338444542.042557 \) (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

\(\Q(\sqrt{-335}) \), 11.11.1822837804551761449.1

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 ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ R ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/13.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/19.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{22}$ $22$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{11}$ $22$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$

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
5Data not computed
67Data not computed