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Properties

Label 22.0.83519904642...6992.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,2^{33}\cdot 89^{20}$
Root discriminant $167.39$
Ramified primes $2, 89$
Class number $1293259$ (GRH)
Class group $[1293259]$ (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![762593971, 263009438, 890303175, 188456410, 248446153, 4776110, 447202, -4737470, 10118056, 2955478, 1977573, -318952, -680544, -5440, 166656, 8722, -19345, -1192, 1486, 78, -57, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 2*x^21 - 57*x^20 + 78*x^19 + 1486*x^18 - 1192*x^17 - 19345*x^16 + 8722*x^15 + 166656*x^14 - 5440*x^13 - 680544*x^12 - 318952*x^11 + 1977573*x^10 + 2955478*x^9 + 10118056*x^8 - 4737470*x^7 + 447202*x^6 + 4776110*x^5 + 248446153*x^4 + 188456410*x^3 + 890303175*x^2 + 263009438*x + 762593971)
 
gp: K = bnfinit(x^22 - 2*x^21 - 57*x^20 + 78*x^19 + 1486*x^18 - 1192*x^17 - 19345*x^16 + 8722*x^15 + 166656*x^14 - 5440*x^13 - 680544*x^12 - 318952*x^11 + 1977573*x^10 + 2955478*x^9 + 10118056*x^8 - 4737470*x^7 + 447202*x^6 + 4776110*x^5 + 248446153*x^4 + 188456410*x^3 + 890303175*x^2 + 263009438*x + 762593971, 1)
 

Normalized defining polynomial

\( x^{22} - 2 x^{21} - 57 x^{20} + 78 x^{19} + 1486 x^{18} - 1192 x^{17} - 19345 x^{16} + 8722 x^{15} + 166656 x^{14} - 5440 x^{13} - 680544 x^{12} - 318952 x^{11} + 1977573 x^{10} + 2955478 x^{9} + 10118056 x^{8} - 4737470 x^{7} + 447202 x^{6} + 4776110 x^{5} + 248446153 x^{4} + 188456410 x^{3} + 890303175 x^{2} + 263009438 x + 762593971 \)

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:  \(-8351990464247387275322638622384170627163060436992=-\,2^{33}\cdot 89^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $167.39$
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:  $2, 89$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(712=2^{3}\cdot 89\)
Dirichlet character group:    $\lbrace$$\chi_{712}(1,·)$, $\chi_{712}(579,·)$, $\chi_{712}(601,·)$, $\chi_{712}(449,·)$, $\chi_{712}(275,·)$, $\chi_{712}(523,·)$, $\chi_{712}(401,·)$, $\chi_{712}(67,·)$, $\chi_{712}(153,·)$, $\chi_{712}(217,·)$, $\chi_{712}(331,·)$, $\chi_{712}(283,·)$, $\chi_{712}(395,·)$, $\chi_{712}(627,·)$, $\chi_{712}(97,·)$, $\chi_{712}(91,·)$, $\chi_{712}(105,·)$, $\chi_{712}(299,·)$, $\chi_{712}(625,·)$, $\chi_{712}(179,·)$, $\chi_{712}(121,·)$, $\chi_{712}(345,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{179} a^{17} + \frac{12}{179} a^{16} + \frac{17}{179} a^{15} + \frac{53}{179} a^{14} + \frac{78}{179} a^{13} - \frac{9}{179} a^{12} - \frac{21}{179} a^{11} - \frac{12}{179} a^{10} + \frac{41}{179} a^{9} + \frac{15}{179} a^{8} - \frac{53}{179} a^{7} - \frac{30}{179} a^{6} + \frac{27}{179} a^{5} + \frac{41}{179} a^{4} + \frac{50}{179} a^{3} + \frac{83}{179} a^{2} - \frac{19}{179} a + \frac{73}{179}$, $\frac{1}{6623} a^{18} - \frac{18}{6623} a^{17} + \frac{3237}{6623} a^{16} + \frac{2765}{6623} a^{15} + \frac{1710}{6623} a^{14} - \frac{380}{6623} a^{13} - \frac{1004}{6623} a^{12} - \frac{2067}{6623} a^{11} + \frac{580}{6623} a^{10} + \frac{217}{6623} a^{9} + \frac{1287}{6623} a^{8} + \frac{844}{6623} a^{7} + \frac{1464}{6623} a^{6} + \frac{1200}{6623} a^{5} + \frac{31}{179} a^{4} + \frac{15}{6623} a^{3} - \frac{361}{6623} a^{2} - \frac{1326}{6623} a + \frac{1748}{6623}$, $\frac{1}{6623} a^{19} - \frac{10}{6623} a^{17} - \frac{3}{37} a^{16} + \frac{1789}{6623} a^{15} + \frac{1318}{6623} a^{14} + \frac{70}{179} a^{13} - \frac{455}{6623} a^{12} - \frac{1735}{6623} a^{11} - \frac{628}{6623} a^{10} - \frac{2059}{6623} a^{9} + \frac{34}{6623} a^{8} - \frac{623}{6623} a^{7} + \frac{2651}{6623} a^{6} - \frac{3190}{6623} a^{5} + \frac{163}{6623} a^{4} - \frac{535}{6623} a^{3} + \frac{1241}{6623} a^{2} + \frac{302}{6623} a - \frac{3094}{6623}$, $\frac{1}{677618999} a^{20} - \frac{45393}{677618999} a^{19} + \frac{37910}{677618999} a^{18} + \frac{1401497}{677618999} a^{17} + \frac{152407027}{677618999} a^{16} - \frac{68665300}{677618999} a^{15} - \frac{17624556}{677618999} a^{14} - \frac{106118033}{677618999} a^{13} + \frac{1684376}{18314027} a^{12} + \frac{3113750}{18314027} a^{11} + \frac{305301398}{677618999} a^{10} - \frac{133730258}{677618999} a^{9} - \frac{73894263}{677618999} a^{8} - \frac{3139996}{6709099} a^{7} - \frac{65814422}{677618999} a^{6} - \frac{21967716}{677618999} a^{5} - \frac{248027392}{677618999} a^{4} - \frac{22554691}{677618999} a^{3} + \frac{303420431}{677618999} a^{2} - \frac{245391269}{677618999} a - \frac{6606681}{677618999}$, $\frac{1}{3229817576722022817368482683260787651150080023939091427827673078082623949} a^{21} + \frac{773239179975511251932654574015263536146115741107640506426519408}{3229817576722022817368482683260787651150080023939091427827673078082623949} a^{20} + \frac{104202095034356030356047741543842774633907521973637098774873706147023}{3229817576722022817368482683260787651150080023939091427827673078082623949} a^{19} + \frac{207418801944174492619671688670507370406721147883043241874439036253731}{3229817576722022817368482683260787651150080023939091427827673078082623949} a^{18} + \frac{2509347511155312857865084686157719483184987739133041653661464675259120}{3229817576722022817368482683260787651150080023939091427827673078082623949} a^{17} + \frac{47382383489420701748720073663975289413811282635282291600905092257535089}{3229817576722022817368482683260787651150080023939091427827673078082623949} a^{16} - \frac{61629682031171216877930411470961955616784531561164185798117032376193702}{3229817576722022817368482683260787651150080023939091427827673078082623949} a^{15} + \frac{520197120343559311564067660077970942660694007764117826979784683132291936}{3229817576722022817368482683260787651150080023939091427827673078082623949} a^{14} + \frac{585814573202356772351395910735639476522724417732777440091041124355392249}{3229817576722022817368482683260787651150080023939091427827673078082623949} a^{13} + \frac{273495591501433859248096314424952892530397023474081102328618866199526280}{3229817576722022817368482683260787651150080023939091427827673078082623949} a^{12} - \frac{1292034014455300129279310914074712496889000346904391960028459472397618583}{3229817576722022817368482683260787651150080023939091427827673078082623949} a^{11} + \frac{1085199196964908761983130416753124982081927336360423024282130312170946401}{3229817576722022817368482683260787651150080023939091427827673078082623949} a^{10} - \frac{913978802562784809250621017683334645333366661890465917319014634666589564}{3229817576722022817368482683260787651150080023939091427827673078082623949} a^{9} - \frac{657582331792864791132985432104881348570893844650006345131987637739402980}{3229817576722022817368482683260787651150080023939091427827673078082623949} a^{8} - \frac{1516230738370641793456662556370004792300229895290732712945187965987400616}{3229817576722022817368482683260787651150080023939091427827673078082623949} a^{7} - \frac{511381907458325501884150360234986041855192578050779458641808675793200819}{3229817576722022817368482683260787651150080023939091427827673078082623949} a^{6} + \frac{1237286060210860246796067155404825928965092351543534476079274624875531120}{3229817576722022817368482683260787651150080023939091427827673078082623949} a^{5} + \frac{926668357755814412846243750681807240254075297515069719172756900277326567}{3229817576722022817368482683260787651150080023939091427827673078082623949} a^{4} - \frac{6218338251478676689721248956406771652097168859276755682537306376874706}{18043673612972194510438450744473674028771396781782633675014933397109631} a^{3} - \frac{519434539297435585950181514484914673763552778007230361800461547370685892}{3229817576722022817368482683260787651150080023939091427827673078082623949} a^{2} - \frac{70951414858953133691720572785958216838448962058600631032056459834210209}{3229817576722022817368482683260787651150080023939091427827673078082623949} a - \frac{1448596035425276794534857236542118115798488220173648941594142657248232050}{3229817576722022817368482683260787651150080023939091427827673078082623949}$

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

Class group and class number

$C_{1293259}$, which has order $1293259$ (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

\(\Q(\sqrt{-2}) \), 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 R ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ $22$ $22$ ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/19.11.0.1}{11} }^{2}$ $22$ $22$ $22$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/41.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ $22$ $22$ ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$

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
$89$89.11.10.1$x^{11} - 89$$11$$1$$10$$C_{11}$$[\ ]_{11}$
89.11.10.1$x^{11} - 89$$11$$1$$10$$C_{11}$$[\ ]_{11}$