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Properties

Label 22.0.94801583326...6211.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,11^{11}\cdot 67^{20}$
Root discriminant $151.62$
Ramified primes $11, 67$
Class number $2694197$ (GRH)
Class group $[23, 117139]$ (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![589202287, 522560028, 683809164, 428625612, 349577039, 160917509, 94723994, 27209924, 11788951, 840645, 757511, -276013, 72521, 12225, 45513, 252, -1667, -1536, 341, 69, 8, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 9*x^21 + 8*x^20 + 69*x^19 + 341*x^18 - 1536*x^17 - 1667*x^16 + 252*x^15 + 45513*x^14 + 12225*x^13 + 72521*x^12 - 276013*x^11 + 757511*x^10 + 840645*x^9 + 11788951*x^8 + 27209924*x^7 + 94723994*x^6 + 160917509*x^5 + 349577039*x^4 + 428625612*x^3 + 683809164*x^2 + 522560028*x + 589202287)
 
gp: K = bnfinit(x^22 - 9*x^21 + 8*x^20 + 69*x^19 + 341*x^18 - 1536*x^17 - 1667*x^16 + 252*x^15 + 45513*x^14 + 12225*x^13 + 72521*x^12 - 276013*x^11 + 757511*x^10 + 840645*x^9 + 11788951*x^8 + 27209924*x^7 + 94723994*x^6 + 160917509*x^5 + 349577039*x^4 + 428625612*x^3 + 683809164*x^2 + 522560028*x + 589202287, 1)
 

Normalized defining polynomial

\( x^{22} - 9 x^{21} + 8 x^{20} + 69 x^{19} + 341 x^{18} - 1536 x^{17} - 1667 x^{16} + 252 x^{15} + 45513 x^{14} + 12225 x^{13} + 72521 x^{12} - 276013 x^{11} + 757511 x^{10} + 840645 x^{9} + 11788951 x^{8} + 27209924 x^{7} + 94723994 x^{6} + 160917509 x^{5} + 349577039 x^{4} + 428625612 x^{3} + 683809164 x^{2} + 522560028 x + 589202287 \)

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:  \(-948015833262595128634255168492301209005419806211=-\,11^{11}\cdot 67^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $151.62$
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:  $11, 67$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(737=11\cdot 67\)
Dirichlet character group:    $\lbrace$$\chi_{737}(1,·)$, $\chi_{737}(131,·)$, $\chi_{737}(263,·)$, $\chi_{737}(265,·)$, $\chi_{737}(76,·)$, $\chi_{737}(397,·)$, $\chi_{737}(461,·)$, $\chi_{737}(210,·)$, $\chi_{737}(595,·)$, $\chi_{737}(89,·)$, $\chi_{737}(604,·)$, $\chi_{737}(417,·)$, $\chi_{737}(483,·)$, $\chi_{737}(551,·)$, $\chi_{737}(617,·)$, $\chi_{737}(494,·)$, $\chi_{737}(560,·)$, $\chi_{737}(241,·)$, $\chi_{737}(692,·)$, $\chi_{737}(694,·)$, $\chi_{737}(375,·)$, $\chi_{737}(628,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{37} a^{17} + \frac{6}{37} a^{16} + \frac{1}{37} a^{15} + \frac{16}{37} a^{14} - \frac{4}{37} a^{13} - \frac{10}{37} a^{12} + \frac{3}{37} a^{11} + \frac{7}{37} a^{10} - \frac{4}{37} a^{9} - \frac{5}{37} a^{8} + \frac{18}{37} a^{7} - \frac{10}{37} a^{6} - \frac{5}{37} a^{5} + \frac{16}{37} a^{4} + \frac{16}{37} a^{3} - \frac{9}{37} a^{2} - \frac{11}{37} a - \frac{7}{37}$, $\frac{1}{37} a^{18} + \frac{2}{37} a^{16} + \frac{10}{37} a^{15} + \frac{11}{37} a^{14} + \frac{14}{37} a^{13} - \frac{11}{37} a^{12} - \frac{11}{37} a^{11} - \frac{9}{37} a^{10} - \frac{18}{37} a^{9} + \frac{11}{37} a^{8} - \frac{7}{37} a^{7} + \frac{18}{37} a^{6} + \frac{9}{37} a^{5} - \frac{6}{37} a^{4} + \frac{6}{37} a^{3} + \frac{6}{37} a^{2} - \frac{15}{37} a + \frac{5}{37}$, $\frac{1}{37} a^{19} - \frac{2}{37} a^{16} + \frac{9}{37} a^{15} - \frac{18}{37} a^{14} - \frac{3}{37} a^{13} + \frac{9}{37} a^{12} - \frac{15}{37} a^{11} + \frac{5}{37} a^{10} - \frac{18}{37} a^{9} + \frac{3}{37} a^{8} - \frac{18}{37} a^{7} - \frac{8}{37} a^{6} + \frac{4}{37} a^{5} + \frac{11}{37} a^{4} + \frac{11}{37} a^{3} + \frac{3}{37} a^{2} - \frac{10}{37} a + \frac{14}{37}$, $\frac{1}{256447} a^{20} - \frac{1587}{256447} a^{19} - \frac{1729}{256447} a^{18} - \frac{199}{256447} a^{17} - \frac{88518}{256447} a^{16} + \frac{47207}{256447} a^{15} + \frac{95858}{256447} a^{14} + \frac{2426}{6931} a^{13} + \frac{30186}{256447} a^{12} + \frac{53264}{256447} a^{11} - \frac{69362}{256447} a^{10} - \frac{103875}{256447} a^{9} - \frac{67213}{256447} a^{8} + \frac{116443}{256447} a^{7} + \frac{23286}{256447} a^{6} - \frac{101240}{256447} a^{5} + \frac{34731}{256447} a^{4} + \frac{96041}{256447} a^{3} + \frac{2319}{8843} a^{2} - \frac{76745}{256447} a + \frac{57910}{256447}$, $\frac{1}{854048722742953832717110273978698930680954226735986813772787601935939217} a^{21} - \frac{58238628526222943012702938159767059751892618490583577620849135729}{854048722742953832717110273978698930680954226735986813772787601935939217} a^{20} - \frac{452926252901941751824753300348600366690769947400379039844158594698744}{854048722742953832717110273978698930680954226735986813772787601935939217} a^{19} + \frac{215279620975017936079404189276015525329254816759474704484400190312358}{29449955956653580438521043930299963126929456094344372888716813859859973} a^{18} + \frac{3260445199982317554584415657281797968405952280448002118579379934758651}{854048722742953832717110273978698930680954226735986813772787601935939217} a^{17} - \frac{153632218259421166466186215492939905286207715164623371070953173622756341}{854048722742953832717110273978698930680954226735986813772787601935939217} a^{16} + \frac{170645905629739922569722092534396891242976818237898598042070686283485621}{854048722742953832717110273978698930680954226735986813772787601935939217} a^{15} + \frac{333541202203007888479040174490841686687022505591320375665400752803298128}{854048722742953832717110273978698930680954226735986813772787601935939217} a^{14} + \frac{111851250666698262458219412581163273196915054083207120982898348887658620}{854048722742953832717110273978698930680954226735986813772787601935939217} a^{13} + \frac{7550571227904857236604142116910279213893235210497265337721821727447837}{29449955956653580438521043930299963126929456094344372888716813859859973} a^{12} + \frac{337665142093970009311487365099741157943512616779042581615735913376629737}{854048722742953832717110273978698930680954226735986813772787601935939217} a^{11} + \frac{124316768267351412127682584820136322114657064856892341254213663414194765}{854048722742953832717110273978698930680954226735986813772787601935939217} a^{10} - \frac{8258480690479829437379746658651638001584994310575379095098615312991427}{29449955956653580438521043930299963126929456094344372888716813859859973} a^{9} - \frac{111420535469374701673189380187785456567831230387999580057971740165041737}{854048722742953832717110273978698930680954226735986813772787601935939217} a^{8} + \frac{8644845051519242976061107780387515018195478753642149563352056438899650}{23082397911971725208570547945370241369755519641513157128994259511782141} a^{7} - \frac{57506395091914665737558253086266707452017716087633867138792321233769757}{854048722742953832717110273978698930680954226735986813772787601935939217} a^{6} - \frac{427010359686603171898201487334041614775008028686832755764415359905919231}{854048722742953832717110273978698930680954226735986813772787601935939217} a^{5} + \frac{194665842031782246441580164256519117719417163472824139304559199358140940}{854048722742953832717110273978698930680954226735986813772787601935939217} a^{4} - \frac{320969079913490834848986562253617006396379688153035475507077936372160265}{854048722742953832717110273978698930680954226735986813772787601935939217} a^{3} - \frac{372208578694611886448141835106965376301951841396590417315131206643134046}{854048722742953832717110273978698930680954226735986813772787601935939217} a^{2} + \frac{140124994987077987222118224958179428829514386347300893951343853345981926}{854048722742953832717110273978698930680954226735986813772787601935939217} a + \frac{215222720844970172278711999041763628526552453482605810265187835408354344}{854048722742953832717110273978698930680954226735986813772787601935939217}$

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

Class group and class number

$C_{23}\times C_{117139}$, which has order $2694197$ (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

\(\Q(\sqrt{-11}) \), 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 $22$ ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$ $22$ R $22$ $22$ $22$ ${\href{/LocalNumberField/23.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{22}$ $22$ $22$ ${\href{/LocalNumberField/47.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$

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
11Data not computed
$67$67.11.10.1$x^{11} - 67$$11$$1$$10$$C_{11}$$[\ ]_{11}$
67.11.10.1$x^{11} - 67$$11$$1$$10$$C_{11}$$[\ ]_{11}$