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Properties

Label 22.0.578...471.1
Degree $22$
Signature $(0, 11)$
Discriminant $-5.790\times 10^{26}$
Root discriminant \(16.46\)
Ramified prime $271$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $D_{11}$ (as 22T2)

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Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^22 - 11*x^21 + 67*x^20 - 285*x^19 + 929*x^18 - 2433*x^17 + 5271*x^16 - 9630*x^15 + 15091*x^14 - 20633*x^13 + 25057*x^12 - 27453*x^11 + 27302*x^10 - 24426*x^9 + 19254*x^8 - 13050*x^7 + 7570*x^6 - 3890*x^5 + 1921*x^4 - 927*x^3 + 362*x^2 - 87*x + 9)
 
Copy content gp:K = bnfinit(y^22 - 11*y^21 + 67*y^20 - 285*y^19 + 929*y^18 - 2433*y^17 + 5271*y^16 - 9630*y^15 + 15091*y^14 - 20633*y^13 + 25057*y^12 - 27453*y^11 + 27302*y^10 - 24426*y^9 + 19254*y^8 - 13050*y^7 + 7570*y^6 - 3890*y^5 + 1921*y^4 - 927*y^3 + 362*y^2 - 87*y + 9, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 11*x^21 + 67*x^20 - 285*x^19 + 929*x^18 - 2433*x^17 + 5271*x^16 - 9630*x^15 + 15091*x^14 - 20633*x^13 + 25057*x^12 - 27453*x^11 + 27302*x^10 - 24426*x^9 + 19254*x^8 - 13050*x^7 + 7570*x^6 - 3890*x^5 + 1921*x^4 - 927*x^3 + 362*x^2 - 87*x + 9);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^22 - 11*x^21 + 67*x^20 - 285*x^19 + 929*x^18 - 2433*x^17 + 5271*x^16 - 9630*x^15 + 15091*x^14 - 20633*x^13 + 25057*x^12 - 27453*x^11 + 27302*x^10 - 24426*x^9 + 19254*x^8 - 13050*x^7 + 7570*x^6 - 3890*x^5 + 1921*x^4 - 927*x^3 + 362*x^2 - 87*x + 9)
 

\( x^{22} - 11 x^{21} + 67 x^{20} - 285 x^{19} + 929 x^{18} - 2433 x^{17} + 5271 x^{16} - 9630 x^{15} + \cdots + 9 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $22$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $(0, 11)$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(-578978183833808423828407471\) \(\medspace = -\,271^{11}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(16.46\)
Copy content comment:Root discriminant
 
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  $271^{1/2}\approx 16.46207763315433$
Ramified primes:   \(271\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant(OK))
 
Discriminant root field:  \(\Q(\sqrt{-271}) \)
$\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$:   $D_{11}$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(K)
 
This field is Galois over $\Q$.
This is not a CM field.
Maximal CM subfield:  \(\Q(\sqrt{-271}) \)

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{4}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{5}$, $\frac{1}{3}a^{14}-\frac{1}{3}a^{6}$, $\frac{1}{3}a^{15}-\frac{1}{3}a^{7}$, $\frac{1}{3}a^{16}-\frac{1}{3}a^{8}$, $\frac{1}{3}a^{17}-\frac{1}{3}a$, $\frac{1}{27}a^{18}-\frac{1}{9}a^{16}+\frac{1}{9}a^{15}+\frac{1}{9}a^{13}-\frac{2}{27}a^{10}+\frac{1}{9}a^{9}-\frac{2}{9}a^{8}-\frac{1}{9}a^{7}-\frac{1}{9}a^{5}+\frac{10}{27}a^{2}-\frac{1}{9}a-\frac{1}{3}$, $\frac{1}{81}a^{19}+\frac{1}{81}a^{18}+\frac{2}{27}a^{17}-\frac{1}{9}a^{16}-\frac{2}{27}a^{15}+\frac{1}{27}a^{14}+\frac{4}{27}a^{13}+\frac{7}{81}a^{11}+\frac{1}{81}a^{10}-\frac{4}{27}a^{9}-\frac{7}{27}a^{7}+\frac{8}{27}a^{6}+\frac{5}{27}a^{5}+\frac{1}{3}a^{4}-\frac{26}{81}a^{3}+\frac{34}{81}a^{2}-\frac{4}{27}a-\frac{4}{9}$, $\frac{1}{148473}a^{20}-\frac{10}{148473}a^{19}+\frac{2560}{148473}a^{18}-\frac{7585}{49491}a^{17}+\frac{4171}{49491}a^{16}+\frac{6152}{49491}a^{15}-\frac{5677}{49491}a^{14}-\frac{587}{3807}a^{13}-\frac{24644}{148473}a^{12}-\frac{2884}{148473}a^{11}+\frac{4027}{148473}a^{10}-\frac{5464}{49491}a^{9}-\frac{9223}{49491}a^{8}+\frac{9751}{49491}a^{7}+\frac{21301}{49491}a^{6}-\frac{19351}{49491}a^{5}-\frac{515}{11421}a^{4}+\frac{4829}{148473}a^{3}-\frac{22877}{148473}a^{2}-\frac{20704}{49491}a-\frac{5644}{16497}$, $\frac{1}{148473}a^{21}+\frac{209}{49491}a^{19}+\frac{1012}{148473}a^{18}+\frac{2380}{16497}a^{17}+\frac{430}{5499}a^{16}-\frac{6479}{49491}a^{15}-\frac{82}{16497}a^{14}+\frac{21376}{148473}a^{13}-\frac{623}{49491}a^{12}+\frac{3949}{49491}a^{11}+\frac{22045}{148473}a^{10}-\frac{7040}{49491}a^{9}+\frac{5501}{16497}a^{8}-\frac{334}{49491}a^{7}+\frac{4675}{16497}a^{6}-\frac{20828}{148473}a^{5}-\frac{20707}{49491}a^{4}+\frac{2620}{16497}a^{3}-\frac{56258}{148473}a^{2}-\frac{2179}{49491}a+\frac{383}{16497}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $3$

Class group and class number

Ideal class group:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $10$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $\frac{695605}{49491}a^{21}-\frac{817052}{5499}a^{20}+\frac{4828385}{5499}a^{19}-\frac{179646464}{49491}a^{18}+\frac{21088270}{1833}a^{17}-\frac{161011082}{5499}a^{16}+\frac{1015964059}{16497}a^{15}-\frac{199954799}{1833}a^{14}+\frac{8197221595}{49491}a^{13}-\frac{3621182783}{16497}a^{12}+\frac{4270128809}{16497}a^{11}-\frac{13653529436}{49491}a^{10}+\frac{338312881}{1269}a^{9}-\frac{1265856827}{5499}a^{8}+\frac{2854663187}{16497}a^{7}-\frac{67307030}{611}a^{6}+\frac{2960457967}{49491}a^{5}-\frac{485022211}{16497}a^{4}+\frac{240466393}{16497}a^{3}-\frac{339984797}{49491}a^{2}+\frac{36140630}{16497}a-\frac{1657276}{5499}$, $\frac{987238}{148473}a^{21}-\frac{10616428}{148473}a^{20}+\frac{63466003}{148473}a^{19}-\frac{88361210}{49491}a^{18}+\frac{21747550}{3807}a^{17}-\frac{726083383}{49491}a^{16}+\frac{1540616651}{49491}a^{15}-\frac{2752404185}{49491}a^{14}+\frac{12638647951}{148473}a^{13}-\frac{16872199033}{148473}a^{12}+\frac{20021639923}{148473}a^{11}-\frac{7153237508}{49491}a^{10}+\frac{147952879}{1053}a^{9}-\frac{6049184477}{49491}a^{8}+\frac{4589771410}{49491}a^{7}-\frac{2954170348}{49491}a^{6}+\frac{4851949267}{148473}a^{5}-\frac{2393298787}{148473}a^{4}+\frac{1181839885}{148473}a^{3}-\frac{187988756}{49491}a^{2}+\frac{6904601}{5499}a-\frac{987905}{5499}$, $\frac{1307986}{49491}a^{21}-\frac{4583665}{16497}a^{20}+\frac{80954372}{49491}a^{19}-\frac{111168563}{16497}a^{18}+\frac{27014380}{1269}a^{17}-\frac{296957152}{5499}a^{16}+\frac{69172609}{611}a^{15}-\frac{70154648}{351}a^{14}+\frac{14970526717}{49491}a^{13}-\frac{2197699951}{5499}a^{12}+\frac{23260042046}{49491}a^{11}-\frac{8243165807}{16497}a^{10}+\frac{7945030928}{16497}a^{9}-\frac{2279156447}{5499}a^{8}+\frac{189533336}{611}a^{7}-\frac{3240176960}{16497}a^{6}+\frac{5255733004}{49491}a^{5}-\frac{859641359}{16497}a^{4}+\frac{1281839342}{49491}a^{3}-\frac{200139800}{16497}a^{2}+\frac{20822071}{5499}a-\frac{310185}{611}$, $\frac{987833}{49491}a^{21}-\frac{31462118}{148473}a^{20}+\frac{186488921}{148473}a^{19}-\frac{59467139}{11421}a^{18}+\frac{818854685}{49491}a^{17}-\frac{160713890}{3807}a^{16}+\frac{4405332749}{49491}a^{15}-\frac{7822607692}{49491}a^{14}+\frac{3968590802}{16497}a^{13}-\frac{47439689600}{148473}a^{12}+\frac{4311589952}{11421}a^{11}-\frac{59845497863}{148473}a^{10}+\frac{19313532335}{49491}a^{9}-\frac{16715879449}{49491}a^{8}+\frac{12603503470}{49491}a^{7}-\frac{8053232504}{49491}a^{6}+\frac{4386460132}{49491}a^{5}-\frac{6476442677}{148473}a^{4}+\frac{3206959451}{148473}a^{3}-\frac{116647355}{11421}a^{2}+\frac{163649417}{49491}a-\frac{7705027}{16497}$, $\frac{243190}{148473}a^{21}-\frac{2091038}{148473}a^{20}+\frac{10400150}{148473}a^{19}-\frac{35622727}{148473}a^{18}+\frac{30316835}{49491}a^{17}-\frac{58872899}{49491}a^{16}+\frac{28577749}{16497}a^{15}-\frac{84180361}{49491}a^{14}+\frac{85427923}{148473}a^{13}+\frac{254809444}{148473}a^{12}-\frac{684943330}{148473}a^{11}+\frac{1099522703}{148473}a^{10}-\frac{159919778}{16497}a^{9}+\frac{42403481}{3807}a^{8}-\frac{182470372}{16497}a^{7}+\frac{451992991}{49491}a^{6}-\frac{868671833}{148473}a^{5}+\frac{451903303}{148473}a^{4}-\frac{205677853}{148473}a^{3}+\frac{117792068}{148473}a^{2}-\frac{20578022}{49491}a+\frac{1448092}{16497}$, $\frac{987238}{148473}a^{21}-\frac{10115570}{148473}a^{20}+\frac{58457423}{148473}a^{19}-\frac{18205819}{11421}a^{18}+\frac{245075711}{49491}a^{17}-\frac{47021780}{3807}a^{16}+\frac{419913493}{16497}a^{15}-\frac{2185201768}{49491}a^{14}+\frac{207447626}{3159}a^{13}-\frac{269573572}{3159}a^{12}+\frac{1129063004}{11421}a^{11}-\frac{15391506583}{148473}a^{10}+\frac{1624036862}{16497}a^{9}-\frac{4113184921}{49491}a^{8}+\frac{1000297115}{16497}a^{7}-\frac{1839631640}{49491}a^{6}+\frac{2906012101}{148473}a^{5}-\frac{1411279970}{148473}a^{4}+\frac{711347060}{148473}a^{3}-\frac{24693853}{11421}a^{2}+\frac{28523236}{49491}a-\frac{950198}{16497}$, $\frac{1512778}{148473}a^{21}-\frac{397792}{3807}a^{20}+\frac{9969230}{16497}a^{19}-\frac{363522758}{148473}a^{18}+\frac{125563523}{16497}a^{17}-\frac{313406059}{16497}a^{16}+\frac{1938935743}{49491}a^{15}-\frac{124652066}{1833}a^{14}+\frac{319705526}{3159}a^{13}-\frac{5131832}{39}a^{12}+\frac{7547963051}{49491}a^{11}-\frac{23755880390}{148473}a^{10}+\frac{7523849128}{49491}a^{9}-\frac{235418024}{1833}a^{8}+\frac{4641357125}{49491}a^{7}-\frac{316485653}{5499}a^{6}+\frac{4504818655}{148473}a^{5}-\frac{729970565}{49491}a^{4}+\frac{368215969}{49491}a^{3}-\frac{498362381}{148473}a^{2}+\frac{44661860}{49491}a-\frac{1559542}{16497}$, $\frac{3209632}{148473}a^{21}-\frac{3797698}{16497}a^{20}+\frac{67672595}{49491}a^{19}-\frac{64860086}{11421}a^{18}+\frac{298226368}{16497}a^{17}-\frac{19543669}{423}a^{16}+\frac{4829326522}{49491}a^{15}-\frac{2863085842}{16497}a^{14}+\frac{39276470104}{148473}a^{13}-\frac{17412162776}{49491}a^{12}+\frac{1584285640}{3807}a^{11}-\frac{66034244636}{148473}a^{10}+\frac{21332456896}{49491}a^{9}-\frac{6162310159}{16497}a^{8}+\frac{13960652141}{49491}a^{7}-\frac{2978498975}{16497}a^{6}+\frac{14613941326}{148473}a^{5}-\frac{2397451660}{49491}a^{4}+\frac{131781757}{5499}a^{3}-\frac{129807059}{11421}a^{2}+\frac{182895455}{49491}a-\frac{8585191}{16497}$, $\frac{222094}{148473}a^{21}-\frac{2829142}{148473}a^{20}+\frac{18706591}{148473}a^{19}-\frac{2175782}{3807}a^{18}+\frac{97149121}{49491}a^{17}-\frac{20464633}{3807}a^{16}+\frac{598791392}{49491}a^{15}-\frac{1130384477}{49491}a^{14}+\frac{5455968370}{148473}a^{13}-\frac{7608869122}{148473}a^{12}+\frac{720473947}{11421}a^{11}-\frac{3453284651}{49491}a^{10}+\frac{3465373082}{49491}a^{9}-\frac{3132455810}{49491}a^{8}+\frac{2490590665}{49491}a^{7}-\frac{1690339228}{49491}a^{6}+\frac{2885900968}{148473}a^{5}-\frac{1442303383}{148473}a^{4}+\frac{697855975}{148473}a^{3}-\frac{9001142}{3807}a^{2}+\frac{5043008}{5499}a-\frac{857846}{5499}$, $\frac{711668}{49491}a^{21}-\frac{22686592}{148473}a^{20}+\frac{134530654}{148473}a^{19}-\frac{557854111}{148473}a^{18}+\frac{45461392}{3807}a^{17}-\frac{1508037289}{49491}a^{16}+\frac{3179680780}{49491}a^{15}-\frac{5645450132}{49491}a^{14}+\frac{8589994891}{49491}a^{13}-\frac{34215338854}{148473}a^{12}+\frac{40408725661}{148473}a^{11}-\frac{43125942988}{148473}a^{10}+\frac{13911570586}{49491}a^{9}-\frac{12033538046}{49491}a^{8}+\frac{9064721885}{49491}a^{7}-\frac{5783755645}{49491}a^{6}+\frac{116463631}{1833}a^{5}-\frac{4638579394}{148473}a^{4}+\frac{2297503807}{148473}a^{3}-\frac{1087045504}{148473}a^{2}+\frac{116257519}{49491}a-\frac{5351744}{16497}$ Copy content Toggle raw display (assuming GRH)
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 33482.7947011 \) (assuming GRH)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{11}\cdot 33482.7947011 \cdot 1}{2\cdot\sqrt{578978183833808423828407471}}\cr\approx \mathstrut & 0.419217261313 \end{aligned}\] (assuming GRH)

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^22 - 11*x^21 + 67*x^20 - 285*x^19 + 929*x^18 - 2433*x^17 + 5271*x^16 - 9630*x^15 + 15091*x^14 - 20633*x^13 + 25057*x^12 - 27453*x^11 + 27302*x^10 - 24426*x^9 + 19254*x^8 - 13050*x^7 + 7570*x^6 - 3890*x^5 + 1921*x^4 - 927*x^3 + 362*x^2 - 87*x + 9) DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent() hK = K.class_number(); wK = K.unit_group().torsion_generator().order(); 2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^22 - 11*x^21 + 67*x^20 - 285*x^19 + 929*x^18 - 2433*x^17 + 5271*x^16 - 9630*x^15 + 15091*x^14 - 20633*x^13 + 25057*x^12 - 27453*x^11 + 27302*x^10 - 24426*x^9 + 19254*x^8 - 13050*x^7 + 7570*x^6 - 3890*x^5 + 1921*x^4 - 927*x^3 + 362*x^2 - 87*x + 9, 1); [polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 11*x^21 + 67*x^20 - 285*x^19 + 929*x^18 - 2433*x^17 + 5271*x^16 - 9630*x^15 + 15091*x^14 - 20633*x^13 + 25057*x^12 - 27453*x^11 + 27302*x^10 - 24426*x^9 + 19254*x^8 - 13050*x^7 + 7570*x^6 - 3890*x^5 + 1921*x^4 - 927*x^3 + 362*x^2 - 87*x + 9); OK := Integers(K); DK := Discriminant(OK); UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK); r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK); hK := #clK; wK := #TorsionSubgroup(UK); 2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = polynomial_ring(QQ); K, a = number_field(x^22 - 11*x^21 + 67*x^20 - 285*x^19 + 929*x^18 - 2433*x^17 + 5271*x^16 - 9630*x^15 + 15091*x^14 - 20633*x^13 + 25057*x^12 - 27453*x^11 + 27302*x^10 - 24426*x^9 + 19254*x^8 - 13050*x^7 + 7570*x^6 - 3890*x^5 + 1921*x^4 - 927*x^3 + 362*x^2 - 87*x + 9); OK = ring_of_integers(K); DK = discriminant(OK); UK, fUK = unit_group(OK); clK, fclK = class_group(OK); r1,r2 = signature(K); RK = regulator(K); RR = parent(RK); hK = order(clK); wK = torsion_units_order(K); 2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$D_{11}$ (as 22T2):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A solvable group of order 22
The 7 conjugacy class representatives for $D_{11}$
Character table for $D_{11}$

Intermediate fields

\(\Q(\sqrt{-271}) \), 11.1.1461660310351.1 x11

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 11 sibling: 11.1.1461660310351.1
Minimal sibling: 11.1.1461660310351.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.11.0.1}{11} }^{2}$ ${\href{/padicField/3.2.0.1}{2} }^{11}$ ${\href{/padicField/5.11.0.1}{11} }^{2}$ ${\href{/padicField/7.11.0.1}{11} }^{2}$ ${\href{/padicField/11.11.0.1}{11} }^{2}$ ${\href{/padicField/13.2.0.1}{2} }^{11}$ ${\href{/padicField/17.11.0.1}{11} }^{2}$ ${\href{/padicField/19.2.0.1}{2} }^{11}$ ${\href{/padicField/23.2.0.1}{2} }^{11}$ ${\href{/padicField/29.2.0.1}{2} }^{11}$ ${\href{/padicField/31.11.0.1}{11} }^{2}$ ${\href{/padicField/37.11.0.1}{11} }^{2}$ ${\href{/padicField/41.11.0.1}{11} }^{2}$ ${\href{/padicField/43.2.0.1}{2} }^{11}$ ${\href{/padicField/47.2.0.1}{2} }^{11}$ ${\href{/padicField/53.11.0.1}{11} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{11}$

Cycle lengths which are repeated in a cycle type are indicated by exponents.

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Oscar: p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(271\) Copy content Toggle raw display Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)