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Properties

Label 22.22.5881591469...9233.1
Degree $22$
Signature $[22, 0]$
Discriminant $17^{11}\cdot 23^{20}$
Root discriminant $71.31$
Ramified primes $17, 23$
Class number $1$ (GRH)
Class group Trivial (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![-6439, 43730, 143967, -1207355, 1151278, 3693202, -6210705, -2261923, 8093877, -974567, -4447882, 1345876, 1217089, -503122, -171900, 91788, 11069, -8947, -45, 447, -29, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 9*x^21 - 29*x^20 + 447*x^19 - 45*x^18 - 8947*x^17 + 11069*x^16 + 91788*x^15 - 171900*x^14 - 503122*x^13 + 1217089*x^12 + 1345876*x^11 - 4447882*x^10 - 974567*x^9 + 8093877*x^8 - 2261923*x^7 - 6210705*x^6 + 3693202*x^5 + 1151278*x^4 - 1207355*x^3 + 143967*x^2 + 43730*x - 6439)
 
gp: K = bnfinit(x^22 - 9*x^21 - 29*x^20 + 447*x^19 - 45*x^18 - 8947*x^17 + 11069*x^16 + 91788*x^15 - 171900*x^14 - 503122*x^13 + 1217089*x^12 + 1345876*x^11 - 4447882*x^10 - 974567*x^9 + 8093877*x^8 - 2261923*x^7 - 6210705*x^6 + 3693202*x^5 + 1151278*x^4 - 1207355*x^3 + 143967*x^2 + 43730*x - 6439, 1)
 

Normalized defining polynomial

\( x^{22} - 9 x^{21} - 29 x^{20} + 447 x^{19} - 45 x^{18} - 8947 x^{17} + 11069 x^{16} + 91788 x^{15} - 171900 x^{14} - 503122 x^{13} + 1217089 x^{12} + 1345876 x^{11} - 4447882 x^{10} - 974567 x^{9} + 8093877 x^{8} - 2261923 x^{7} - 6210705 x^{6} + 3693202 x^{5} + 1151278 x^{4} - 1207355 x^{3} + 143967 x^{2} + 43730 x - 6439 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[22, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(58815914699238651208660872676277748369233=17^{11}\cdot 23^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $71.31$
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:  $17, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(391=17\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{391}(256,·)$, $\chi_{391}(1,·)$, $\chi_{391}(324,·)$, $\chi_{391}(271,·)$, $\chi_{391}(16,·)$, $\chi_{391}(18,·)$, $\chi_{391}(154,·)$, $\chi_{391}(220,·)$, $\chi_{391}(288,·)$, $\chi_{391}(35,·)$, $\chi_{391}(101,·)$, $\chi_{391}(358,·)$, $\chi_{391}(169,·)$, $\chi_{391}(239,·)$, $\chi_{391}(305,·)$, $\chi_{391}(50,·)$, $\chi_{391}(307,·)$, $\chi_{391}(52,·)$, $\chi_{391}(118,·)$, $\chi_{391}(186,·)$, $\chi_{391}(188,·)$, $\chi_{391}(254,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{47} a^{20} - \frac{15}{47} a^{19} + \frac{11}{47} a^{18} + \frac{3}{47} a^{17} - \frac{2}{47} a^{16} - \frac{14}{47} a^{15} + \frac{20}{47} a^{14} + \frac{13}{47} a^{13} - \frac{18}{47} a^{12} - \frac{12}{47} a^{11} + \frac{9}{47} a^{10} + \frac{13}{47} a^{9} - \frac{1}{47} a^{8} - \frac{8}{47} a^{7} + \frac{11}{47} a^{6} + \frac{4}{47} a^{5} + \frac{6}{47} a^{4} - \frac{12}{47} a^{3} + \frac{20}{47} a^{2} - \frac{9}{47} a$, $\frac{1}{1096882583607897978241161423944468782359578936014115295323} a^{21} + \frac{1816345314998901452225616817923887197357250498468443452}{1096882583607897978241161423944468782359578936014115295323} a^{20} - \frac{86040047911695148436607269350434594441733558247322686153}{1096882583607897978241161423944468782359578936014115295323} a^{19} - \frac{209694481869847852956926610439648579879969103948922074227}{1096882583607897978241161423944468782359578936014115295323} a^{18} + \frac{411122704658749410441992170475544619977745391318093721905}{1096882583607897978241161423944468782359578936014115295323} a^{17} - \frac{230820325581086565888118504689690443362265086565851410856}{1096882583607897978241161423944468782359578936014115295323} a^{16} + \frac{247518495405641945175400904929655069095443446017158958688}{1096882583607897978241161423944468782359578936014115295323} a^{15} - \frac{209833726433122126804504617693884585032244654715812154893}{1096882583607897978241161423944468782359578936014115295323} a^{14} - \frac{503665192936623323248042410341175398322943257946793079310}{1096882583607897978241161423944468782359578936014115295323} a^{13} - \frac{382430920148906940244422813899507147648058250476006942315}{1096882583607897978241161423944468782359578936014115295323} a^{12} - \frac{295148758811230687882130882730856000785453438284342011527}{1096882583607897978241161423944468782359578936014115295323} a^{11} + \frac{397994956994386334141902249927772429242960298444016963781}{1096882583607897978241161423944468782359578936014115295323} a^{10} - \frac{252378659989973395428910879624463610411577093266212371931}{1096882583607897978241161423944468782359578936014115295323} a^{9} + \frac{101464932554470992451461478130522121800050380896621338650}{1096882583607897978241161423944468782359578936014115295323} a^{8} + \frac{321731883347663629004733652212452227640328874848658035070}{1096882583607897978241161423944468782359578936014115295323} a^{7} + \frac{287300286444461748433027325022547881605742776122191176154}{1096882583607897978241161423944468782359578936014115295323} a^{6} + \frac{238134263520402587171442320892828298857247489401924433015}{1096882583607897978241161423944468782359578936014115295323} a^{5} + \frac{209788124351642606219133796840475204411586411541095546449}{1096882583607897978241161423944468782359578936014115295323} a^{4} + \frac{452481572463537102030837150342157555436191371962340251392}{1096882583607897978241161423944468782359578936014115295323} a^{3} + \frac{174970686443127132279290265332071292177361970554201591639}{1096882583607897978241161423944468782359578936014115295323} a^{2} - \frac{421971385200719108651958470640576329929870343428723423933}{1096882583607897978241161423944468782359578936014115295323} a + \frac{40326583495504690638108662647349356971951675296474041}{170349834385447736953123376913258080813725568568739757}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $21$
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:  \( 22589470687200 \) (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

\(\Q(\sqrt{17}) \), \(\Q(\zeta_{23})^+\)

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 ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ $22$ $22$ $22$ $22$ ${\href{/LocalNumberField/13.11.0.1}{11} }^{2}$ R ${\href{/LocalNumberField/19.11.0.1}{11} }^{2}$ R $22$ $22$ $22$ $22$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{22}$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$

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
17Data not computed
23Data not computed