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Properties

Label 22.0.38877404501...7392.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,2^{33}\cdot 11^{40}$
Root discriminant $221.30$
Ramified primes $2, 11$
Class number $795765707$ (GRH)
Class group $[23, 34598509]$ (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![8288294787, -793159290, 1197228681, -275105160, -35720190, 90770922, 349976088, -18728094, -96420511, 10122948, 28614850, -2188980, -4596438, 483450, 620873, -49896, -52415, 3762, 3135, -66, -88, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 88*x^20 - 66*x^19 + 3135*x^18 + 3762*x^17 - 52415*x^16 - 49896*x^15 + 620873*x^14 + 483450*x^13 - 4596438*x^12 - 2188980*x^11 + 28614850*x^10 + 10122948*x^9 - 96420511*x^8 - 18728094*x^7 + 349976088*x^6 + 90770922*x^5 - 35720190*x^4 - 275105160*x^3 + 1197228681*x^2 - 793159290*x + 8288294787)
 
gp: K = bnfinit(x^22 - 88*x^20 - 66*x^19 + 3135*x^18 + 3762*x^17 - 52415*x^16 - 49896*x^15 + 620873*x^14 + 483450*x^13 - 4596438*x^12 - 2188980*x^11 + 28614850*x^10 + 10122948*x^9 - 96420511*x^8 - 18728094*x^7 + 349976088*x^6 + 90770922*x^5 - 35720190*x^4 - 275105160*x^3 + 1197228681*x^2 - 793159290*x + 8288294787, 1)
 

Normalized defining polynomial

\( x^{22} - 88 x^{20} - 66 x^{19} + 3135 x^{18} + 3762 x^{17} - 52415 x^{16} - 49896 x^{15} + 620873 x^{14} + 483450 x^{13} - 4596438 x^{12} - 2188980 x^{11} + 28614850 x^{10} + 10122948 x^{9} - 96420511 x^{8} - 18728094 x^{7} + 349976088 x^{6} + 90770922 x^{5} - 35720190 x^{4} - 275105160 x^{3} + 1197228681 x^{2} - 793159290 x + 8288294787 \)

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:  \(-3887740450132432227599338487287650162637926293307392=-\,2^{33}\cdot 11^{40}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $221.30$
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:  $2, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(968=2^{3}\cdot 11^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{968}(1,·)$, $\chi_{968}(771,·)$, $\chi_{968}(705,·)$, $\chi_{968}(265,·)$, $\chi_{968}(859,·)$, $\chi_{968}(331,·)$, $\chi_{968}(529,·)$, $\chi_{968}(595,·)$, $\chi_{968}(67,·)$, $\chi_{968}(89,·)$, $\chi_{968}(793,·)$, $\chi_{968}(155,·)$, $\chi_{968}(353,·)$, $\chi_{968}(419,·)$, $\chi_{968}(243,·)$, $\chi_{968}(881,·)$, $\chi_{968}(617,·)$, $\chi_{968}(683,·)$, $\chi_{968}(177,·)$, $\chi_{968}(947,·)$, $\chi_{968}(441,·)$, $\chi_{968}(507,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{5} - \frac{1}{3} a$, $\frac{1}{9} a^{6} + \frac{1}{9} a^{4} - \frac{2}{9} a^{2}$, $\frac{1}{9} a^{7} + \frac{1}{9} a^{5} + \frac{1}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{9} a^{8} - \frac{1}{9} a^{2}$, $\frac{1}{81} a^{9} - \frac{1}{27} a^{7} + \frac{1}{27} a^{5} - \frac{10}{81} a^{3} + \frac{1}{9} a$, $\frac{1}{81} a^{10} - \frac{1}{27} a^{8} + \frac{1}{27} a^{6} - \frac{10}{81} a^{4} + \frac{1}{9} a^{2}$, $\frac{1}{81} a^{11} + \frac{1}{27} a^{7} + \frac{8}{81} a^{5} - \frac{4}{27} a^{3}$, $\frac{1}{243} a^{12} - \frac{1}{243} a^{10} - \frac{1}{81} a^{8} - \frac{4}{243} a^{6} - \frac{11}{243} a^{4} + \frac{2}{27} a^{2}$, $\frac{1}{243} a^{13} - \frac{1}{243} a^{11} - \frac{13}{243} a^{7} - \frac{2}{243} a^{5} - \frac{4}{81} a^{3} + \frac{1}{9} a$, $\frac{1}{729} a^{14} + \frac{1}{729} a^{12} + \frac{1}{243} a^{11} + \frac{4}{729} a^{10} - \frac{1}{243} a^{9} - \frac{37}{729} a^{8} - \frac{1}{81} a^{7} + \frac{8}{729} a^{6} + \frac{23}{243} a^{5} - \frac{94}{729} a^{4} + \frac{16}{243} a^{3} + \frac{40}{81} a^{2} - \frac{4}{27} a - \frac{1}{3}$, $\frac{1}{2187} a^{15} + \frac{1}{2187} a^{13} + \frac{1}{729} a^{12} - \frac{5}{2187} a^{11} - \frac{1}{729} a^{10} - \frac{10}{2187} a^{9} - \frac{10}{243} a^{8} - \frac{19}{2187} a^{7} - \frac{4}{729} a^{6} - \frac{247}{2187} a^{5} + \frac{70}{729} a^{4} - \frac{23}{243} a^{3} - \frac{4}{81} a^{2} + \frac{2}{9} a$, $\frac{1}{6561} a^{16} + \frac{1}{6561} a^{15} + \frac{1}{6561} a^{14} - \frac{5}{6561} a^{13} - \frac{11}{6561} a^{12} + \frac{28}{6561} a^{11} + \frac{23}{6561} a^{10} + \frac{8}{6561} a^{9} + \frac{323}{6561} a^{8} - \frac{157}{6561} a^{7} + \frac{101}{6561} a^{6} - \frac{208}{6561} a^{5} - \frac{218}{2187} a^{4} - \frac{17}{729} a^{3} + \frac{89}{243} a^{2} + \frac{11}{27} a$, $\frac{1}{6561} a^{17} + \frac{1}{2187} a^{14} - \frac{2}{2187} a^{13} - \frac{2}{2187} a^{12} + \frac{22}{6561} a^{11} - \frac{2}{2187} a^{10} - \frac{4}{729} a^{9} + \frac{107}{2187} a^{8} - \frac{103}{2187} a^{7} - \frac{88}{2187} a^{6} - \frac{68}{6561} a^{5} + \frac{353}{2187} a^{4} + \frac{44}{729} a^{3} + \frac{40}{243} a^{2} - \frac{1}{3}$, $\frac{1}{177147} a^{18} - \frac{1}{19683} a^{17} - \frac{1}{59049} a^{15} + \frac{34}{59049} a^{14} + \frac{77}{59049} a^{13} + \frac{31}{177147} a^{12} + \frac{149}{59049} a^{11} + \frac{19}{19683} a^{10} + \frac{19}{59049} a^{9} + \frac{1823}{59049} a^{8} + \frac{1273}{59049} a^{7} + \frac{1678}{177147} a^{6} + \frac{5839}{59049} a^{5} - \frac{256}{2187} a^{4} + \frac{857}{6561} a^{3} - \frac{79}{2187} a^{2} + \frac{118}{243} a - \frac{13}{27}$, $\frac{1}{42692427} a^{19} + \frac{17}{14230809} a^{18} + \frac{56}{1581201} a^{17} - \frac{1045}{14230809} a^{16} + \frac{2170}{14230809} a^{15} + \frac{4502}{14230809} a^{14} + \frac{57577}{42692427} a^{13} + \frac{5395}{14230809} a^{12} - \frac{3322}{4743603} a^{11} + \frac{19954}{14230809} a^{10} + \frac{59573}{14230809} a^{9} - \frac{727895}{14230809} a^{8} + \frac{1640272}{42692427} a^{7} - \frac{252361}{4743603} a^{6} - \frac{453262}{4743603} a^{5} - \frac{87262}{1581201} a^{4} + \frac{13349}{175689} a^{3} + \frac{9016}{175689} a^{2} - \frac{2557}{19521} a + \frac{721}{2169}$, $\frac{1}{44074082015001} a^{20} - \frac{179795}{44074082015001} a^{19} + \frac{20036947}{14691360671667} a^{18} - \frac{807677956}{14691360671667} a^{17} - \frac{8965259}{181374823107} a^{16} + \frac{224273510}{4897120223889} a^{15} + \frac{10596637792}{44074082015001} a^{14} + \frac{58639488010}{44074082015001} a^{13} - \frac{6540000251}{4897120223889} a^{12} + \frac{16401170881}{14691360671667} a^{11} - \frac{4036337939}{4897120223889} a^{10} - \frac{15411584866}{4897120223889} a^{9} - \frac{1334688311504}{44074082015001} a^{8} - \frac{1745020255406}{44074082015001} a^{7} - \frac{453531439723}{14691360671667} a^{6} - \frac{600082166762}{4897120223889} a^{5} + \frac{224288372063}{1632373407963} a^{4} + \frac{50903258446}{544124469321} a^{3} + \frac{85764648518}{181374823107} a^{2} - \frac{8156440265}{20152758123} a - \frac{567297010}{2239195347}$, $\frac{1}{1350887708062586761640368227741893219999440353} a^{21} - \frac{487286519636316524384466774463}{450295902687528920546789409247297739999813451} a^{20} + \frac{487775655305556778839716208673025909}{1350887708062586761640368227741893219999440353} a^{19} - \frac{1104393737948674022589993450077050130696}{450295902687528920546789409247297739999813451} a^{18} - \frac{30356339201948556963679809596034277539971}{450295902687528920546789409247297739999813451} a^{17} + \frac{4279166937214747136600193782216606722937}{150098634229176306848929803082432579999937817} a^{16} + \frac{280203928320069077511054311683693630488523}{1350887708062586761640368227741893219999440353} a^{15} - \frac{178105838975159937621963370631801950768798}{450295902687528920546789409247297739999813451} a^{14} - \frac{2263130750728026436943343579384264363434590}{1350887708062586761640368227741893219999440353} a^{13} + \frac{314081423140111694738168137233649328703553}{450295902687528920546789409247297739999813451} a^{12} - \frac{2169518373013286369573306589127688797229224}{450295902687528920546789409247297739999813451} a^{11} + \frac{458347267631822376999747216774531063406681}{150098634229176306848929803082432579999937817} a^{10} - \frac{1642669159960001340145942750625093412462488}{1350887708062586761640368227741893219999440353} a^{9} + \frac{5290032570141580471394948121410805669585530}{450295902687528920546789409247297739999813451} a^{8} - \frac{9655513479974500162996589618251867788258754}{1350887708062586761640368227741893219999440353} a^{7} - \frac{7553875645154483165744763264228823203124883}{450295902687528920546789409247297739999813451} a^{6} - \frac{17204688328341683149302118584995827195970425}{150098634229176306848929803082432579999937817} a^{5} + \frac{7147126454309885514420234006316150743348364}{50032878076392102282976601027477526666645939} a^{4} - \frac{2312456842058817929696634723465389421531307}{16677626025464034094325533675825842222215313} a^{3} + \frac{2398789093390602209957670725309911280221}{23067255913504888097269064558541967112331} a^{2} - \frac{39888310534628874278730194694262765202914}{617689852794964225715760506512068230452419} a + \frac{28972731100175102055917785013503049505925}{68632205866107136190640056279118692272491}$

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

Class group and class number

$C_{23}\times C_{34598509}$, which has order $795765707$ (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:  \( 285114946276.13544 \) (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

\(\Q(\sqrt{-2}) \), 11.11.672749994932560009201.1

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 R ${\href{/LocalNumberField/3.1.0.1}{1} }^{22}$ $22$ $22$ R $22$ ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/19.11.0.1}{11} }^{2}$ $22$ $22$ $22$ $22$ ${\href{/LocalNumberField/41.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ $22$ $22$ ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$

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
2Data not computed
$11$11.11.20.9$x^{11} - 11 x^{10} + 11$$11$$1$$20$$C_{11}$$[2]$
11.11.20.9$x^{11} - 11 x^{10} + 11$$11$$1$$20$$C_{11}$$[2]$