Properties

Label 11.11.177...321.1
Degree $11$
Signature $(11, 0)$
Discriminant $1.771\times 10^{18}$
Root discriminant \(45.60\)
Ramified prime $191$
Class number $1$
Class group trivial
Galois group $C_{11}:C_5$ (as 11T3)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^11 - x^10 - 21*x^9 + 14*x^8 + 151*x^7 - 53*x^6 - 449*x^5 - 13*x^4 + 516*x^3 + 197*x^2 - 67*x - 19)
 
Copy content gp:K = bnfinit(y^11 - y^10 - 21*y^9 + 14*y^8 + 151*y^7 - 53*y^6 - 449*y^5 - 13*y^4 + 516*y^3 + 197*y^2 - 67*y - 19, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^11 - x^10 - 21*x^9 + 14*x^8 + 151*x^7 - 53*x^6 - 449*x^5 - 13*x^4 + 516*x^3 + 197*x^2 - 67*x - 19);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^11 - x^10 - 21*x^9 + 14*x^8 + 151*x^7 - 53*x^6 - 449*x^5 - 13*x^4 + 516*x^3 + 197*x^2 - 67*x - 19)
 

\( x^{11} - x^{10} - 21x^{9} + 14x^{8} + 151x^{7} - 53x^{6} - 449x^{5} - 13x^{4} + 516x^{3} + 197x^{2} - 67x - 19 \) 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:   \(1771197285652216321\) \(\medspace = 191^{8}\) 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:  \(45.60\)
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:   \(191\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{8627}a^{10}+\frac{1744}{8627}a^{9}-\frac{2072}{8627}a^{8}-\frac{913}{8627}a^{7}+\frac{2961}{8627}a^{6}-\frac{681}{8627}a^{5}+\frac{1732}{8627}a^{4}+\frac{2877}{8627}a^{3}-\frac{33}{8627}a^{2}+\frac{3001}{8627}a+\frac{89}{8627}$ 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:   $\frac{31499}{8627}a^{10}+\frac{71496}{8627}a^{9}+\frac{572055}{8627}a^{8}-\frac{1168476}{8627}a^{7}-\frac{3297556}{8627}a^{6}+\frac{5870457}{8627}a^{5}+\frac{6824837}{8627}a^{4}-\frac{8312416}{8627}a^{3}-\frac{5922522}{8627}a^{2}+\frac{1369233}{8627}a+\frac{431714}{8627}$, $\frac{277}{8627}a^{10}+\frac{24}{8627}a^{9}+\frac{4562}{8627}a^{8}+\frac{2718}{8627}a^{7}-\frac{17886}{8627}a^{6}-\frac{35665}{8627}a^{5}-\frac{13906}{8627}a^{4}+\frac{126160}{8627}a^{3}+\frac{104038}{8627}a^{2}-\frac{54847}{8627}a-\frac{16026}{8627}$, $\frac{10967}{8627}a^{10}+\frac{25492}{8627}a^{9}+\frac{198527}{8627}a^{8}-\frac{417172}{8627}a^{7}-\frac{1140023}{8627}a^{6}+\frac{2093906}{8627}a^{5}+\frac{2348354}{8627}a^{4}-\frac{2944927}{8627}a^{3}-\frac{2062276}{8627}a^{2}+\frac{431388}{8627}a+\frac{145447}{8627}$, $\frac{8498}{8627}a^{10}-\frac{17928}{8627}a^{9}-\frac{155435}{8627}a^{8}+\frac{290317}{8627}a^{7}+\frac{903454}{8627}a^{6}-\frac{1447757}{8627}a^{5}-\frac{1897066}{8627}a^{4}+\frac{2018546}{8627}a^{3}+\frac{1652014}{8627}a^{2}-\frac{309486}{8627}a-\frac{158140}{8627}$, $\frac{24538}{8627}a^{10}+\frac{56037}{8627}a^{9}+\frac{443802}{8627}a^{8}-\frac{915587}{8627}a^{7}-\frac{2536762}{8627}a^{6}+\frac{4598070}{8627}a^{5}+\frac{5147105}{8627}a^{4}-\frac{6505843}{8627}a^{3}-\frac{4349192}{8627}a^{2}+\frac{1097163}{8627}a+\frac{283440}{8627}$, $\frac{19465}{8627}a^{10}-\frac{43420}{8627}a^{9}-\frac{353962}{8627}a^{8}+\frac{707489}{8627}a^{7}+\frac{2043477}{8627}a^{6}-\frac{3541663}{8627}a^{5}-\frac{4245420}{8627}a^{4}+\frac{4963473}{8627}a^{3}+\frac{3705663}{8627}a^{2}-\frac{758128}{8627}a-\frac{303587}{8627}$, $\frac{1537}{8627}a^{10}+\frac{2469}{8627}a^{9}+\frac{27182}{8627}a^{8}-\frac{37428}{8627}a^{7}-\frac{142660}{8627}a^{6}+\frac{175370}{8627}a^{5}+\frac{219334}{8627}a^{4}-\frac{211973}{8627}a^{3}-\frac{70057}{8627}a^{2}+\frac{54670}{8627}a+\frac{1239}{8627}$, $\frac{102635}{8627}a^{10}-\frac{230485}{8627}a^{9}-\frac{1867602}{8627}a^{8}+\frac{3762091}{8627}a^{7}+\frac{10799910}{8627}a^{6}-\frac{18874357}{8627}a^{5}-\frac{22503358}{8627}a^{4}+\frac{26618861}{8627}a^{3}+\frac{19673016}{8627}a^{2}-\frac{4212122}{8627}a-\frac{1588846}{8627}$, $\frac{5667}{8627}a^{10}-\frac{11921}{8627}a^{9}-\frac{104201}{8627}a^{8}+\frac{192023}{8627}a^{7}+\frac{612989}{8627}a^{6}-\frac{943301}{8627}a^{5}-\frac{1322213}{8627}a^{4}+\frac{1241217}{8627}a^{3}+\frac{1184682}{8627}a^{2}-\frac{57539}{8627}a-\frac{90900}{8627}$, $\frac{13175}{8627}a^{10}-\frac{31009}{8627}a^{9}-\frac{235701}{8627}a^{8}+\frac{506256}{8627}a^{7}+\frac{1319812}{8627}a^{6}-\frac{2527806}{8627}a^{5}-\frac{2561534}{8627}a^{4}+\frac{3499999}{8627}a^{3}+\frac{2118817}{8627}a^{2}-\frac{534240}{8627}a-\frac{164610}{8627}$ 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:  \( 907140.338432047 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{11}\cdot(2\pi)^{0}\cdot 907140.338432047 \cdot 1}{2\cdot\sqrt{1771197285652216321}}\cr\approx \mathstrut & 0.69797676739439 \end{aligned}\]

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 - x^10 - 21*x^9 + 14*x^8 + 151*x^7 - 53*x^6 - 449*x^5 - 13*x^4 + 516*x^3 + 197*x^2 - 67*x - 19) 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 - x^10 - 21*x^9 + 14*x^8 + 151*x^7 - 53*x^6 - 449*x^5 - 13*x^4 + 516*x^3 + 197*x^2 - 67*x - 19, 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 - x^10 - 21*x^9 + 14*x^8 + 151*x^7 - 53*x^6 - 449*x^5 - 13*x^4 + 516*x^3 + 197*x^2 - 67*x - 19); 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 - x^10 - 21*x^9 + 14*x^8 + 151*x^7 - 53*x^6 - 449*x^5 - 13*x^4 + 516*x^3 + 197*x^2 - 67*x - 19); 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 ${\href{/padicField/2.5.0.1}{5} }^{2}{,}\,{\href{/padicField/2.1.0.1}{1} }$ ${\href{/padicField/3.5.0.1}{5} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }$ ${\href{/padicField/5.11.0.1}{11} }$ ${\href{/padicField/7.5.0.1}{5} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ ${\href{/padicField/11.11.0.1}{11} }$ ${\href{/padicField/13.5.0.1}{5} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.5.0.1}{5} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.5.0.1}{5} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.5.0.1}{5} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.5.0.1}{5} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.11.0.1}{11} }$ ${\href{/padicField/37.11.0.1}{11} }$ ${\href{/padicField/41.11.0.1}{11} }$ ${\href{/padicField/43.5.0.1}{5} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.5.0.1}{5} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.5.0.1}{5} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.5.0.1}{5} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$

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
\(191\) Copy content Toggle raw display $\Q_{191}$$x + 172$$1$$1$$0$Trivial$$[\ ]$$
191.5.4.1$x^{5} + 191$$5$$1$$4$$C_5$$$[\ ]_{5}$$
191.5.4.1$x^{5} + 191$$5$$1$$4$$C_5$$$[\ ]_{5}$$

Spectrum of ring of integers

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