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Properties

Label 11.1.151917743390062651.1
Degree $11$
Signature $[1, 5]$
Discriminant $-1.519\times 10^{17}$
Root discriminant \(36.47\)
Ramified prime $2731$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $D_{11}$ (as 11T2)

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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 - 5*x^10 + 21*x^9 - 95*x^8 + 231*x^7 - 567*x^6 + 1152*x^5 - 1716*x^4 + 2000*x^3 - 1744*x^2 + 960*x - 256)
 
Copy content gp:K = bnfinit(y^11 - 5*y^10 + 21*y^9 - 95*y^8 + 231*y^7 - 567*y^6 + 1152*y^5 - 1716*y^4 + 2000*y^3 - 1744*y^2 + 960*y - 256, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^11 - 5*x^10 + 21*x^9 - 95*x^8 + 231*x^7 - 567*x^6 + 1152*x^5 - 1716*x^4 + 2000*x^3 - 1744*x^2 + 960*x - 256);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^11 - 5*x^10 + 21*x^9 - 95*x^8 + 231*x^7 - 567*x^6 + 1152*x^5 - 1716*x^4 + 2000*x^3 - 1744*x^2 + 960*x - 256)
 

\( x^{11} - 5 x^{10} + 21 x^{9} - 95 x^{8} + 231 x^{7} - 567 x^{6} + 1152 x^{5} - 1716 x^{4} + 2000 x^{3} + \cdots - 256 \) 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:  $[1, 5]$
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:   \(-151917743390062651\) \(\medspace = -\,2731^{5}\) 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:  \(36.47\)
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:  $2731^{1/2}\approx 52.25897052181568$
Ramified primes:   \(2731\) 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:  \(\Q(\sqrt{-2731}) \)
$\Aut(K/\Q)$:   $C_1$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{4}a^{5}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{48}a^{8}-\frac{1}{16}a^{7}+\frac{1}{16}a^{6}+\frac{1}{16}a^{5}+\frac{3}{16}a^{4}-\frac{3}{16}a^{3}-\frac{1}{8}a^{2}+\frac{1}{4}a-\frac{1}{3}$, $\frac{1}{2016}a^{9}+\frac{5}{2016}a^{8}-\frac{5}{224}a^{7}+\frac{73}{672}a^{6}+\frac{9}{224}a^{5}+\frac{13}{672}a^{4}+\frac{5}{48}a^{3}-\frac{17}{56}a^{2}-\frac{8}{63}a+\frac{5}{63}$, $\frac{1}{28224}a^{10}-\frac{1}{4032}a^{9}+\frac{11}{1344}a^{8}+\frac{925}{9408}a^{7}-\frac{3}{3136}a^{6}+\frac{25}{9408}a^{5}-\frac{715}{4704}a^{4}-\frac{381}{784}a^{3}+\frac{1}{882}a^{2}-\frac{277}{882}a-\frac{17}{147}$ 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:  $2$, $3$

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:  $5$
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:   $\frac{17}{3136}a^{10}-\frac{1}{64}a^{9}+\frac{23}{448}a^{8}-\frac{859}{3136}a^{7}+\frac{955}{3136}a^{6}-\frac{3051}{3136}a^{5}+\frac{1217}{784}a^{4}-\frac{669}{784}a^{3}+\frac{125}{196}a^{2}-\frac{101}{196}a-\frac{6}{49}$, $\frac{83}{4032}a^{10}-\frac{109}{1344}a^{9}+\frac{1375}{4032}a^{8}-\frac{2047}{1344}a^{7}+\frac{3767}{1344}a^{6}-\frac{10471}{1344}a^{5}+\frac{157}{12}a^{4}-\frac{5479}{336}a^{3}+\frac{2171}{126}a^{2}-\frac{869}{84}a+\frac{32}{9}$, $\frac{11}{2016}a^{10}-\frac{15}{224}a^{9}+\frac{571}{2016}a^{8}-\frac{715}{672}a^{7}+\frac{341}{96}a^{6}-\frac{4315}{672}a^{5}+\frac{1759}{168}a^{4}-\frac{499}{42}a^{3}+\frac{2465}{252}a^{2}-\frac{13}{2}a+\frac{121}{63}$, $\frac{1269}{1568}a^{10}-\frac{1753}{672}a^{9}+\frac{2661}{224}a^{8}-\frac{85289}{1568}a^{7}+\frac{131273}{1568}a^{6}-\frac{439093}{1568}a^{5}+\frac{309933}{784}a^{4}-\frac{418029}{784}a^{3}+\frac{191361}{392}a^{2}-\frac{170203}{588}a+\frac{2848}{49}$, $\frac{1}{126}a^{10}-\frac{23}{672}a^{9}+\frac{257}{2016}a^{8}-\frac{335}{672}a^{7}+\frac{517}{672}a^{6}-\frac{131}{96}a^{5}-\frac{19}{672}a^{4}+\frac{199}{168}a^{3}-\frac{137}{63}a^{2}+\frac{289}{84}a-\frac{157}{63}$ 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:  \( 165581.985356 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{5}\cdot 165581.985356 \cdot 1}{2\cdot\sqrt{151917743390062651}}\cr\approx \mathstrut & 4.16014221213 \end{aligned}\] (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^11 - 5*x^10 + 21*x^9 - 95*x^8 + 231*x^7 - 567*x^6 + 1152*x^5 - 1716*x^4 + 2000*x^3 - 1744*x^2 + 960*x - 256) 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 - 5*x^10 + 21*x^9 - 95*x^8 + 231*x^7 - 567*x^6 + 1152*x^5 - 1716*x^4 + 2000*x^3 - 1744*x^2 + 960*x - 256, 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 - 5*x^10 + 21*x^9 - 95*x^8 + 231*x^7 - 567*x^6 + 1152*x^5 - 1716*x^4 + 2000*x^3 - 1744*x^2 + 960*x - 256); 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 - 5*x^10 + 21*x^9 - 95*x^8 + 231*x^7 - 567*x^6 + 1152*x^5 - 1716*x^4 + 2000*x^3 - 1744*x^2 + 960*x - 256); 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 11T2):

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
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

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]
 

Sibling fields

Galois closure: deg 22
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.2.0.1}{2} }^{5}{,}\,{\href{/padicField/2.1.0.1}{1} }$ ${\href{/padicField/3.2.0.1}{2} }^{5}{,}\,{\href{/padicField/3.1.0.1}{1} }$ ${\href{/padicField/5.11.0.1}{11} }$ ${\href{/padicField/7.2.0.1}{2} }^{5}{,}\,{\href{/padicField/7.1.0.1}{1} }$ ${\href{/padicField/11.2.0.1}{2} }^{5}{,}\,{\href{/padicField/11.1.0.1}{1} }$ ${\href{/padicField/13.11.0.1}{11} }$ ${\href{/padicField/17.2.0.1}{2} }^{5}{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.11.0.1}{11} }$ ${\href{/padicField/23.11.0.1}{11} }$ ${\href{/padicField/29.11.0.1}{11} }$ ${\href{/padicField/31.11.0.1}{11} }$ ${\href{/padicField/37.11.0.1}{11} }$ ${\href{/padicField/41.11.0.1}{11} }$ ${\href{/padicField/43.11.0.1}{11} }$ ${\href{/padicField/47.11.0.1}{11} }$ ${\href{/padicField/53.11.0.1}{11} }$ ${\href{/padicField/59.2.0.1}{2} }^{5}{,}\,{\href{/padicField/59.1.0.1}{1} }$

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
\(2731\) Copy content Toggle raw display $\Q_{2731}$$x$$1$$1$$0$Trivial$$[\ ]$$
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)