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Properties

Label 22.0.19225541155...6343.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,7^{11}\cdot 89^{20}$
Root discriminant $156.57$
Ramified primes $7, 89$
Class number $1084931$ (GRH)
Class group $[1084931]$ (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![639704333, -867599298, 963528183, -437293737, 150503099, -21663766, 59561975, -56103742, 19675195, 3187790, -3715861, -923591, 1619754, -268055, -199899, 54452, 19825, -8101, -333, 427, -23, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 9*x^21 - 23*x^20 + 427*x^19 - 333*x^18 - 8101*x^17 + 19825*x^16 + 54452*x^15 - 199899*x^14 - 268055*x^13 + 1619754*x^12 - 923591*x^11 - 3715861*x^10 + 3187790*x^9 + 19675195*x^8 - 56103742*x^7 + 59561975*x^6 - 21663766*x^5 + 150503099*x^4 - 437293737*x^3 + 963528183*x^2 - 867599298*x + 639704333)
 
gp: K = bnfinit(x^22 - 9*x^21 - 23*x^20 + 427*x^19 - 333*x^18 - 8101*x^17 + 19825*x^16 + 54452*x^15 - 199899*x^14 - 268055*x^13 + 1619754*x^12 - 923591*x^11 - 3715861*x^10 + 3187790*x^9 + 19675195*x^8 - 56103742*x^7 + 59561975*x^6 - 21663766*x^5 + 150503099*x^4 - 437293737*x^3 + 963528183*x^2 - 867599298*x + 639704333, 1)
 

Normalized defining polynomial

\( x^{22} - 9 x^{21} - 23 x^{20} + 427 x^{19} - 333 x^{18} - 8101 x^{17} + 19825 x^{16} + 54452 x^{15} - 199899 x^{14} - 268055 x^{13} + 1619754 x^{12} - 923591 x^{11} - 3715861 x^{10} + 3187790 x^{9} + 19675195 x^{8} - 56103742 x^{7} + 59561975 x^{6} - 21663766 x^{5} + 150503099 x^{4} - 437293737 x^{3} + 963528183 x^{2} - 867599298 x + 639704333 \)

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:  \(-1922554115559596594815766124679345986918150636343=-\,7^{11}\cdot 89^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $156.57$
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:  $7, 89$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(623=7\cdot 89\)
Dirichlet character group:    $\lbrace$$\chi_{623}(512,·)$, $\chi_{623}(1,·)$, $\chi_{623}(134,·)$, $\chi_{623}(449,·)$, $\chi_{623}(8,·)$, $\chi_{623}(461,·)$, $\chi_{623}(78,·)$, $\chi_{623}(601,·)$, $\chi_{623}(153,·)$, $\chi_{623}(538,·)$, $\chi_{623}(90,·)$, $\chi_{623}(223,·)$, $\chi_{623}(97,·)$, $\chi_{623}(484,·)$, $\chi_{623}(358,·)$, $\chi_{623}(167,·)$, $\chi_{623}(64,·)$, $\chi_{623}(477,·)$, $\chi_{623}(372,·)$, $\chi_{623}(566,·)$, $\chi_{623}(573,·)$, $\chi_{623}(447,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{37} a^{15} - \frac{8}{37} a^{14} + \frac{13}{37} a^{13} + \frac{8}{37} a^{12} - \frac{5}{37} a^{11} + \frac{3}{37} a^{10} + \frac{9}{37} a^{9} + \frac{9}{37} a^{8} - \frac{6}{37} a^{7} - \frac{18}{37} a^{6} + \frac{17}{37} a^{5} + \frac{3}{37} a^{4} + \frac{12}{37} a^{3} + \frac{4}{37} a^{2} - \frac{4}{37} a - \frac{1}{37}$, $\frac{1}{37} a^{16} - \frac{14}{37} a^{14} + \frac{1}{37} a^{13} - \frac{15}{37} a^{12} - \frac{4}{37} a^{10} + \frac{7}{37} a^{9} - \frac{8}{37} a^{8} + \frac{8}{37} a^{7} - \frac{16}{37} a^{6} - \frac{9}{37} a^{5} - \frac{1}{37} a^{4} - \frac{11}{37} a^{3} - \frac{9}{37} a^{2} + \frac{4}{37} a - \frac{8}{37}$, $\frac{1}{37} a^{17} - \frac{18}{37} a^{13} + \frac{1}{37} a^{12} + \frac{12}{37} a^{10} + \frac{7}{37} a^{9} - \frac{14}{37} a^{8} + \frac{11}{37} a^{7} - \frac{2}{37} a^{6} + \frac{15}{37} a^{5} - \frac{6}{37} a^{4} + \frac{11}{37} a^{3} - \frac{14}{37} a^{2} + \frac{10}{37} a - \frac{14}{37}$, $\frac{1}{37} a^{18} - \frac{18}{37} a^{14} + \frac{1}{37} a^{13} + \frac{12}{37} a^{11} + \frac{7}{37} a^{10} - \frac{14}{37} a^{9} + \frac{11}{37} a^{8} - \frac{2}{37} a^{7} + \frac{15}{37} a^{6} - \frac{6}{37} a^{5} + \frac{11}{37} a^{4} - \frac{14}{37} a^{3} + \frac{10}{37} a^{2} - \frac{14}{37} a$, $\frac{1}{37} a^{19} + \frac{5}{37} a^{14} + \frac{12}{37} a^{13} + \frac{8}{37} a^{12} - \frac{9}{37} a^{11} + \frac{3}{37} a^{10} - \frac{12}{37} a^{9} + \frac{12}{37} a^{8} + \frac{18}{37} a^{7} + \frac{3}{37} a^{6} - \frac{16}{37} a^{5} + \frac{3}{37} a^{4} + \frac{4}{37} a^{3} - \frac{16}{37} a^{2} + \frac{2}{37} a - \frac{18}{37}$, $\frac{1}{677618999} a^{20} + \frac{863887}{677618999} a^{19} + \frac{2420329}{677618999} a^{18} - \frac{467112}{677618999} a^{17} - \frac{3118574}{677618999} a^{16} + \frac{6239916}{677618999} a^{15} - \frac{273271215}{677618999} a^{14} + \frac{174789670}{677618999} a^{13} + \frac{68525117}{677618999} a^{12} - \frac{90058046}{677618999} a^{11} + \frac{326778181}{677618999} a^{10} + \frac{168887638}{677618999} a^{9} - \frac{182055027}{677618999} a^{8} + \frac{298447882}{677618999} a^{7} + \frac{319570531}{677618999} a^{6} - \frac{7597661}{18314027} a^{5} - \frac{103630759}{677618999} a^{4} - \frac{3476195}{18314027} a^{3} - \frac{153955802}{677618999} a^{2} + \frac{258323764}{677618999} a + \frac{241861221}{677618999}$, $\frac{1}{9797086316190442999698655289138875728092695103472572135700539142874288229} a^{21} - \frac{6228866783687849050929157258465154442830173981794323313525034996}{9797086316190442999698655289138875728092695103472572135700539142874288229} a^{20} - \frac{352648170410877024945532144219440828246887035685278040975546092825959}{264786116653795756748612305111861506164667435228988436100014571429034817} a^{19} - \frac{39169527273315063725462736737869786442413330226519556030430171729473135}{9797086316190442999698655289138875728092695103472572135700539142874288229} a^{18} - \frac{24160047639235545437226615931306614271839642301137534950668452269655944}{9797086316190442999698655289138875728092695103472572135700539142874288229} a^{17} - \frac{107766826503662001440806346574750123816493857071943767010946795206148123}{9797086316190442999698655289138875728092695103472572135700539142874288229} a^{16} - \frac{79701952102910894132029581994859200385791623414165795509800709643792165}{9797086316190442999698655289138875728092695103472572135700539142874288229} a^{15} + \frac{2582332747010673776219329425641830031801802337037382565249514959616684254}{9797086316190442999698655289138875728092695103472572135700539142874288229} a^{14} + \frac{2715376026697680315085814214175775283706483902311066334222856552251953869}{9797086316190442999698655289138875728092695103472572135700539142874288229} a^{13} - \frac{3482582700696762844703420939643663271749106491810870472382796495478991097}{9797086316190442999698655289138875728092695103472572135700539142874288229} a^{12} - \frac{2497881661746084458202915985410425708409583963071735357352159271477385796}{9797086316190442999698655289138875728092695103472572135700539142874288229} a^{11} + \frac{933341879009140469724029407279963644584491576099918307640969558025194224}{9797086316190442999698655289138875728092695103472572135700539142874288229} a^{10} - \frac{2436829702400119792159510143097104068291986579888452181174271747830200420}{9797086316190442999698655289138875728092695103472572135700539142874288229} a^{9} - \frac{2770115633473236576104658345558540237571090912134557295038990633192397278}{9797086316190442999698655289138875728092695103472572135700539142874288229} a^{8} - \frac{4832297079331134201325407277294687204035669902901404133170306026052574120}{9797086316190442999698655289138875728092695103472572135700539142874288229} a^{7} - \frac{3634941116737292806501457919270083660756247987429634590256148957452490057}{9797086316190442999698655289138875728092695103472572135700539142874288229} a^{6} + \frac{70044348357567172766305508984541993723775414990657758077945600331800332}{264786116653795756748612305111861506164667435228988436100014571429034817} a^{5} + \frac{544490011638021399432487509047232438753391376973692484845154903992835594}{9797086316190442999698655289138875728092695103472572135700539142874288229} a^{4} - \frac{104999769204290647935481728585759542218985538634301583516840036983667484}{264786116653795756748612305111861506164667435228988436100014571429034817} a^{3} - \frac{855054570260313142100509126718041639534820142689374543424473582954102864}{9797086316190442999698655289138875728092695103472572135700539142874288229} a^{2} + \frac{1265457731946216137668498192068295975538497896036574844639149557541251044}{9797086316190442999698655289138875728092695103472572135700539142874288229} a - \frac{1849661033316241457007901986560621703498331774402963444642698036174675302}{9797086316190442999698655289138875728092695103472572135700539142874288229}$

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

Class group and class number

$C_{1084931}$, which has order $1084931$ (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:  \( 866679281.3791491 \) (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

\(\Q(\sqrt{-7}) \), 11.11.31181719929966183601.1

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 ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ $22$ $22$ R ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ $22$ $22$ $22$ ${\href{/LocalNumberField/23.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/29.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{22}$ $22$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ $22$

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
7Data not computed
89Data not computed