Properties

Label 11.11.251...241.1
Degree $11$
Signature $(11, 0)$
Discriminant $2.513\times 10^{22}$
Root discriminant \(108.74\)
Ramified prime $631$
Class number $1$
Class group trivial
Galois group $C_{11}:C_5$ (as 11T3)

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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^11 - 3*x^10 - 49*x^9 + 245*x^8 + 164*x^7 - 2975*x^6 + 5783*x^5 - 1592*x^4 - 5667*x^3 + 5301*x^2 - 1319*x + 64)
 
Copy content gp:K = bnfinit(y^11 - 3*y^10 - 49*y^9 + 245*y^8 + 164*y^7 - 2975*y^6 + 5783*y^5 - 1592*y^4 - 5667*y^3 + 5301*y^2 - 1319*y + 64, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^11 - 3*x^10 - 49*x^9 + 245*x^8 + 164*x^7 - 2975*x^6 + 5783*x^5 - 1592*x^4 - 5667*x^3 + 5301*x^2 - 1319*x + 64);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^11 - 3*x^10 - 49*x^9 + 245*x^8 + 164*x^7 - 2975*x^6 + 5783*x^5 - 1592*x^4 - 5667*x^3 + 5301*x^2 - 1319*x + 64)
 

\( x^{11} - 3 x^{10} - 49 x^{9} + 245 x^{8} + 164 x^{7} - 2975 x^{6} + 5783 x^{5} - 1592 x^{4} - 5667 x^{3} + \cdots + 64 \) 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:  $11$
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:  $(11, 0)$
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:   \(25132452704633039250241\) \(\medspace = 631^{8}\) 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:  \(108.74\)
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:  not computed
Ramified primes:   \(631\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_1$
Copy content comment:Automorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{106632100889}a^{10}-\frac{1443129849}{106632100889}a^{9}-\frac{21574320898}{106632100889}a^{8}+\frac{11735221843}{106632100889}a^{7}-\frac{6142290314}{106632100889}a^{6}+\frac{21868640215}{106632100889}a^{5}+\frac{33928647585}{106632100889}a^{4}+\frac{30975362841}{106632100889}a^{3}-\frac{39945471027}{106632100889}a^{2}+\frac{42658875461}{106632100889}a+\frac{13357074299}{106632100889}$ 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:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Ideal class group:  Trivial group, which has order $1$
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 $1$
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:   $\frac{842778796}{106632100889}a^{10}+\frac{1569897967}{106632100889}a^{9}-\frac{43599181578}{106632100889}a^{8}+\frac{476617623}{106632100889}a^{7}+\frac{644287068088}{106632100889}a^{6}-\frac{616811200778}{106632100889}a^{5}-\frac{2863913360663}{106632100889}a^{4}+\frac{3858604005522}{106632100889}a^{3}+\frac{2604675898013}{106632100889}a^{2}-\frac{4208348754517}{106632100889}a+\frac{981501908323}{106632100889}$, $\frac{1725289889}{106632100889}a^{10}-\frac{7547598805}{106632100889}a^{9}-\frac{81101485992}{106632100889}a^{8}+\frac{545250096765}{106632100889}a^{7}-\frac{110836105674}{106632100889}a^{6}-\frac{6184888232428}{106632100889}a^{5}+\frac{15634975304087}{106632100889}a^{4}-\frac{7331857664946}{106632100889}a^{3}-\frac{16236969856335}{106632100889}a^{2}+\frac{16273628756779}{106632100889}a-\frac{1105928021109}{106632100889}$, $\frac{14085951292}{106632100889}a^{10}-\frac{36371114971}{106632100889}a^{9}-\frac{702972182706}{106632100889}a^{8}+\frac{3156079824228}{106632100889}a^{7}+\frac{3505579234755}{106632100889}a^{6}-\frac{40151486332347}{106632100889}a^{5}+\frac{65858478598904}{106632100889}a^{4}+\frac{657765296786}{106632100889}a^{3}-\frac{77153879873318}{106632100889}a^{2}+\frac{46681080618800}{106632100889}a-\frac{2751646458675}{106632100889}$, $\frac{2624776191}{106632100889}a^{10}+\frac{1871971936}{106632100889}a^{9}+\frac{134584532254}{106632100889}a^{8}-\frac{335438448740}{106632100889}a^{7}-\frac{1285858892031}{106632100889}a^{6}+\frac{5027698407176}{106632100889}a^{5}-\frac{2672734921820}{106632100889}a^{4}-\frac{4642125921035}{106632100889}a^{3}+\frac{3428254588839}{106632100889}a^{2}-\frac{1604718782938}{106632100889}a+\frac{273540314199}{106632100889}$, $\frac{991964742}{106632100889}a^{10}+\frac{89124151}{106632100889}a^{9}-\frac{50199287306}{106632100889}a^{8}+\frac{87926965166}{106632100889}a^{7}+\frac{532072265211}{106632100889}a^{6}-\frac{1484246650366}{106632100889}a^{5}+\frac{4305811238}{106632100889}a^{4}+\frac{1729256348689}{106632100889}a^{3}+\frac{705460160644}{106632100889}a^{2}-\frac{1335180788166}{106632100889}a+\frac{81642112009}{106632100889}$, $\frac{483710995}{106632100889}a^{10}+\frac{3696928598}{106632100889}a^{9}+\frac{28031240761}{106632100889}a^{8}-\frac{229592018227}{106632100889}a^{7}-\frac{66669431240}{106632100889}a^{6}+\frac{2945579605617}{106632100889}a^{5}-\frac{4724861269837}{106632100889}a^{4}-\frac{4069834820764}{106632100889}a^{3}+\frac{13356750428205}{106632100889}a^{2}-\frac{7550691088839}{106632100889}a+\frac{479492736181}{106632100889}$, $\frac{22567901361}{106632100889}a^{10}-\frac{47421676507}{106632100889}a^{9}-\frac{1149218588997}{106632100889}a^{8}+\frac{4495355676155}{106632100889}a^{7}+\frac{7777232958949}{106632100889}a^{6}-\frac{60182511106688}{106632100889}a^{5}+\frac{76059045868021}{106632100889}a^{4}+\frac{33209920255063}{106632100889}a^{3}-\frac{98202828878327}{106632100889}a^{2}+\frac{30726452227583}{106632100889}a-\frac{1454319045587}{106632100889}$, $\frac{24788771783}{106632100889}a^{10}-\frac{69193135601}{106632100889}a^{9}-\frac{1276525185596}{106632100889}a^{8}+\frac{5764316628539}{106632100889}a^{7}+\frac{7551636552218}{106632100889}a^{6}-\frac{74779614280098}{106632100889}a^{5}+\frac{103799500783605}{106632100889}a^{4}+\frac{36693443567906}{106632100889}a^{3}-\frac{131509957675458}{106632100889}a^{2}+\frac{42713076376597}{106632100889}a-\frac{2199437571327}{106632100889}$, $\frac{12568638330}{106632100889}a^{10}+\frac{45811939222}{106632100889}a^{9}+\frac{607410979505}{106632100889}a^{8}-\frac{3494700920307}{106632100889}a^{7}-\frac{881025370017}{106632100889}a^{6}+\frac{41122657003837}{106632100889}a^{5}-\frac{89995168851157}{106632100889}a^{4}+\frac{32428474393907}{106632100889}a^{3}+\frac{88862193709455}{106632100889}a^{2}-\frac{85404505149254}{106632100889}a+\frac{18084926168171}{106632100889}$, $\frac{148591170110}{106632100889}a^{10}+\frac{402357803115}{106632100889}a^{9}+\frac{7405914002106}{106632100889}a^{8}-\frac{34249691787129}{106632100889}a^{7}-\frac{34758736150762}{106632100889}a^{6}+\frac{433015958127054}{106632100889}a^{5}-\frac{729301393062930}{106632100889}a^{4}+\frac{7391817155684}{106632100889}a^{3}+\frac{855062632220116}{106632100889}a^{2}-\frac{522164632647929}{106632100889}a+\frac{29807564391369}{106632100889}$ Copy content Toggle raw display
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:  \( 165702859.05851 \)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 
Unit signature rank:  \( 11 \)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{11}\cdot(2\pi)^{0}\cdot 165702859.05851 \cdot 1}{2\cdot\sqrt{25132452704633039250241}}\cr\approx \mathstrut & 1.070317241710 \end{aligned}\]

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^11 - 3*x^10 - 49*x^9 + 245*x^8 + 164*x^7 - 2975*x^6 + 5783*x^5 - 1592*x^4 - 5667*x^3 + 5301*x^2 - 1319*x + 64) 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^11 - 3*x^10 - 49*x^9 + 245*x^8 + 164*x^7 - 2975*x^6 + 5783*x^5 - 1592*x^4 - 5667*x^3 + 5301*x^2 - 1319*x + 64, 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^11 - 3*x^10 - 49*x^9 + 245*x^8 + 164*x^7 - 2975*x^6 + 5783*x^5 - 1592*x^4 - 5667*x^3 + 5301*x^2 - 1319*x + 64); 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^11 - 3*x^10 - 49*x^9 + 245*x^8 + 164*x^7 - 2975*x^6 + 5783*x^5 - 1592*x^4 - 5667*x^3 + 5301*x^2 - 1319*x + 64); 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

$C_{11}:C_5$ (as 11T3):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
Copy content gp:polgalois(K.pol)
 
Copy content magma: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 55
The 7 conjugacy class representatives for $C_{11}:C_5$
Character table for $C_{11}:C_5$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.
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]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.5.0.1}{5} }^{2}{,}\,{\href{/padicField/2.1.0.1}{1} }$ ${\href{/padicField/3.5.0.1}{5} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }$ ${\href{/padicField/5.5.0.1}{5} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }$ ${\href{/padicField/7.5.0.1}{5} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ ${\href{/padicField/11.5.0.1}{5} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }$ ${\href{/padicField/13.5.0.1}{5} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.5.0.1}{5} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.5.0.1}{5} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.5.0.1}{5} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.5.0.1}{5} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.5.0.1}{5} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.11.0.1}{11} }$ ${\href{/padicField/41.11.0.1}{11} }$ ${\href{/padicField/43.11.0.1}{11} }$ ${\href{/padicField/47.5.0.1}{5} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.5.0.1}{5} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.5.0.1}{5} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$

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
\(631\) Copy content Toggle raw display $\Q_{631}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $5$$5$$1$$4$
Deg $5$$5$$1$$4$

Spectrum of ring of integers

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