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Properties

Label 22.0.48157344753...2467.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,23^{21}\cdot 29^{11}$
Root discriminant $107.41$
Ramified primes $23, 29$
Class number Not computed
Class group Not computed
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![393724105097, -348245590008, 348245590008, -205313114014, 205313114014, -72590100591, 72590100591, -15708809124, 15708809124, -2165644489, 2165644489, -195729633, 195729633, -11726597, 11726597, -461105, 461105, -11432, 11432, -162, 162, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 + 162*x^20 - 162*x^19 + 11432*x^18 - 11432*x^17 + 461105*x^16 - 461105*x^15 + 11726597*x^14 - 11726597*x^13 + 195729633*x^12 - 195729633*x^11 + 2165644489*x^10 - 2165644489*x^9 + 15708809124*x^8 - 15708809124*x^7 + 72590100591*x^6 - 72590100591*x^5 + 205313114014*x^4 - 205313114014*x^3 + 348245590008*x^2 - 348245590008*x + 393724105097)
 
gp: K = bnfinit(x^22 - x^21 + 162*x^20 - 162*x^19 + 11432*x^18 - 11432*x^17 + 461105*x^16 - 461105*x^15 + 11726597*x^14 - 11726597*x^13 + 195729633*x^12 - 195729633*x^11 + 2165644489*x^10 - 2165644489*x^9 + 15708809124*x^8 - 15708809124*x^7 + 72590100591*x^6 - 72590100591*x^5 + 205313114014*x^4 - 205313114014*x^3 + 348245590008*x^2 - 348245590008*x + 393724105097, 1)
 

Normalized defining polynomial

\( x^{22} - x^{21} + 162 x^{20} - 162 x^{19} + 11432 x^{18} - 11432 x^{17} + 461105 x^{16} - 461105 x^{15} + 11726597 x^{14} - 11726597 x^{13} + 195729633 x^{12} - 195729633 x^{11} + 2165644489 x^{10} - 2165644489 x^{9} + 15708809124 x^{8} - 15708809124 x^{7} + 72590100591 x^{6} - 72590100591 x^{5} + 205313114014 x^{4} - 205313114014 x^{3} + 348245590008 x^{2} - 348245590008 x + 393724105097 \)

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:  \(-481573447532424402907134681295965659189012467=-\,23^{21}\cdot 29^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $107.41$
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:  $23, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(667=23\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{667}(1,·)$, $\chi_{667}(260,·)$, $\chi_{667}(581,·)$, $\chi_{667}(262,·)$, $\chi_{667}(521,·)$, $\chi_{667}(146,·)$, $\chi_{667}(405,·)$, $\chi_{667}(86,·)$, $\chi_{667}(407,·)$, $\chi_{667}(666,·)$, $\chi_{667}(28,·)$, $\chi_{667}(349,·)$, $\chi_{667}(608,·)$, $\chi_{667}(610,·)$, $\chi_{667}(550,·)$, $\chi_{667}(233,·)$, $\chi_{667}(434,·)$, $\chi_{667}(117,·)$, $\chi_{667}(57,·)$, $\chi_{667}(59,·)$, $\chi_{667}(318,·)$, $\chi_{667}(639,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{73138542583} a^{12} + \frac{30287213200}{73138542583} a^{11} + \frac{84}{73138542583} a^{10} - \frac{8317946256}{73138542583} a^{9} + \frac{2646}{73138542583} a^{8} - \frac{13486867419}{73138542583} a^{7} + \frac{38416}{73138542583} a^{6} - \frac{652405071}{73138542583} a^{5} + \frac{252105}{73138542583} a^{4} - \frac{3262025355}{73138542583} a^{3} + \frac{605052}{73138542583} a^{2} - \frac{4566835497}{73138542583} a + \frac{235298}{73138542583}$, $\frac{1}{73138542583} a^{13} + \frac{91}{73138542583} a^{11} + \frac{7405135349}{73138542583} a^{10} + \frac{3185}{73138542583} a^{9} + \frac{6389676349}{73138542583} a^{8} + \frac{53508}{73138542583} a^{7} - \frac{26299285907}{73138542583} a^{6} + \frac{436982}{73138542583} a^{5} + \frac{29561311262}{73138542583} a^{4} + \frac{1529437}{73138542583} a^{3} + \frac{30326046834}{73138542583} a^{2} + \frac{1529437}{73138542583} a + \frac{25759211337}{73138542583}$, $\frac{1}{73138542583} a^{14} + \frac{30533352303}{73138542583} a^{11} - \frac{4459}{73138542583} a^{10} + \frac{31937359815}{73138542583} a^{9} - \frac{187278}{73138542583} a^{8} + \frac{30788967894}{73138542583} a^{7} - \frac{3058874}{73138542583} a^{6} + \frac{15791630140}{73138542583} a^{5} - \frac{21412118}{73138542583} a^{4} + \frac{34616183807}{73138542583} a^{3} - \frac{53530295}{73138542583} a^{2} + \frac{2509986066}{73138542583} a - \frac{21412118}{73138542583}$, $\frac{1}{73138542583} a^{15} - \frac{5145}{73138542583} a^{11} + \frac{26984756768}{73138542583} a^{10} - \frac{240100}{73138542583} a^{9} - \frac{15510214212}{73138542583} a^{8} - \frac{4537890}{73138542583} a^{7} - \frac{30663038337}{73138542583} a^{6} - \frac{39530064}{73138542583} a^{5} + \frac{36025068993}{73138542583} a^{4} - \frac{144120025}{73138542583} a^{3} + \frac{20519019029}{73138542583} a^{2} - \frac{148237740}{73138542583} a + \frac{35446279379}{73138542583}$, $\frac{1}{73138542583} a^{16} - \frac{3537573605}{73138542583} a^{11} + \frac{192080}{73138542583} a^{10} - \frac{25296290277}{73138542583} a^{9} + \frac{9075780}{73138542583} a^{8} - \frac{12118997825}{73138542583} a^{7} + \frac{158120256}{73138542583} a^{6} - \frac{29364605067}{73138542583} a^{5} + \frac{1152960200}{73138542583} a^{4} - \frac{13875180939}{73138542583} a^{3} + \frac{2964754800}{73138542583} a^{2} + \frac{16549816457}{73138542583} a + \frac{1210608210}{73138542583}$, $\frac{1}{73138542583} a^{17} + \frac{233240}{73138542583} a^{11} - \frac{20694277789}{73138542583} a^{10} + \frac{12245100}{73138542583} a^{9} - \frac{13432689619}{73138542583} a^{8} + \frac{246861216}{73138542583} a^{7} - \frac{21349114601}{73138542583} a^{6} + \frac{2240036960}{73138542583} a^{5} - \frac{25269749516}{73138542583} a^{4} + \frac{8400138600}{73138542583} a^{3} + \frac{33085977422}{73138542583} a^{2} + \frac{8820145530}{73138542583} a - \frac{5759027833}{73138542583}$, $\frac{1}{73138542583} a^{18} + \frac{22111418432}{73138542583} a^{11} - \frac{7347060}{73138542583} a^{10} - \frac{8628496837}{73138542583} a^{9} - \frac{370291824}{73138542583} a^{8} - \frac{33108801871}{73138542583} a^{7} - \frac{6720110880}{73138542583} a^{6} + \frac{13520437884}{73138542583} a^{5} + \frac{22737710983}{73138542583} a^{4} + \frac{7621286673}{73138542583} a^{3} + \frac{13974902216}{73138542583} a^{2} - \frac{26781886365}{73138542583} a + \frac{18257637063}{73138542583}$, $\frac{1}{73138542583} a^{19} - \frac{9306276}{73138542583} a^{11} + \frac{35614462033}{73138542583} a^{10} - \frac{521151456}{73138542583} a^{9} - \frac{29087906543}{73138542583} a^{8} - \frac{10944180576}{73138542583} a^{7} + \frac{12303513134}{73138542583} a^{6} - \frac{29007142793}{73138542583} a^{5} + \frac{8777535824}{73138542583} a^{4} - \frac{25333738915}{73138542583} a^{3} - \frac{9379179886}{73138542583} a^{2} + \frac{21736373546}{73138542583} a + \frac{10830971552}{73138542583}$, $\frac{1}{73138542583} a^{20} + \frac{31795224999}{73138542583} a^{11} + \frac{260575728}{73138542583} a^{10} - \frac{30614987829}{73138542583} a^{9} + \frac{13680225720}{73138542583} a^{8} - \frac{30454762874}{73138542583} a^{7} + \frac{35948585691}{73138542583} a^{6} - \frac{3041547193}{73138542583} a^{5} - \frac{19608390591}{73138542583} a^{4} + \frac{4663947612}{73138542583} a^{3} + \frac{20849501007}{73138542583} a^{2} + \frac{1235933016}{73138542583} a - \frac{4408147242}{73138542583}$, $\frac{1}{73138542583} a^{21} + \frac{342005643}{73138542583} a^{11} + \frac{4712187826}{73138542583} a^{10} + \frac{19950329175}{73138542583} a^{9} + \frac{21842402805}{73138542583} a^{8} - \frac{7904145318}{73138542583} a^{7} - \frac{34743972677}{73138542583} a^{6} + \frac{10019359567}{73138542583} a^{5} + \frac{34317043768}{73138542583} a^{4} + \frac{22711166631}{73138542583} a^{3} + \frac{13892529724}{73138542583} a^{2} - \frac{16429523885}{73138542583} a - \frac{11330999632}{73138542583}$

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:  $10$
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:  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

$C_{22}$ (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

\(\Q(\sqrt{-667}) \), \(\Q(\zeta_{23})^+\)

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 $22$ $22$ $22$ $22$ ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/13.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/19.11.0.1}{11} }^{2}$ R R $22$ ${\href{/LocalNumberField/37.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{11}$ $22$ ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$

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
23Data not computed
29Data not computed