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Properties

Label 11.3.117...216.2
Degree $11$
Signature $[3, 4]$
Discriminant $1.176\times 10^{19}$
Root discriminant \(54.16\)
Ramified primes $2,11$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $\PSL(2,11)$ (as 11T5)

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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 - 22*x^9 - 11*x^8 + 143*x^7 + 143*x^6 - 209*x^5 - 1188*x^4 - 3619*x^3 - 4059*x^2 - 2948*x - 3486)
 
Copy content gp:K = bnfinit(y^11 - 22*y^9 - 11*y^8 + 143*y^7 + 143*y^6 - 209*y^5 - 1188*y^4 - 3619*y^3 - 4059*y^2 - 2948*y - 3486, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^11 - 22*x^9 - 11*x^8 + 143*x^7 + 143*x^6 - 209*x^5 - 1188*x^4 - 3619*x^3 - 4059*x^2 - 2948*x - 3486);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^11 - 22*x^9 - 11*x^8 + 143*x^7 + 143*x^6 - 209*x^5 - 1188*x^4 - 3619*x^3 - 4059*x^2 - 2948*x - 3486)
 

\( x^{11} - 22 x^{9} - 11 x^{8} + 143 x^{7} + 143 x^{6} - 209 x^{5} - 1188 x^{4} - 3619 x^{3} - 4059 x^{2} + \cdots - 3486 \) 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:  $[3, 4]$
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:   \(11763130845074473216\) \(\medspace = 2^{8}\cdot 11^{16}\) 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:  \(54.16\)
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:  $2\cdot 11^{84/55}\approx 77.89698382370653$
Ramified primes:   \(2\), \(11\) 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: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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}-\frac{1}{3}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{892268465877}a^{10}+\frac{105634823782}{892268465877}a^{9}+\frac{115467063307}{297422821959}a^{8}-\frac{27718931528}{892268465877}a^{7}+\frac{76553251057}{297422821959}a^{6}-\frac{31390975258}{892268465877}a^{5}-\frac{130557733147}{297422821959}a^{4}-\frac{135320001709}{297422821959}a^{3}-\frac{389363731681}{892268465877}a^{2}-\frac{225594330856}{892268465877}a+\frac{36296589622}{297422821959}$ 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 $1$ (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:  $6$
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:   $\frac{10753587575}{892268465877}a^{10}-\frac{22217020738}{892268465877}a^{9}-\frac{19859840107}{99140940653}a^{8}+\frac{213957922331}{892268465877}a^{7}+\frac{304169004449}{297422821959}a^{6}+\frac{117992656843}{892268465877}a^{5}-\frac{539079911020}{297422821959}a^{4}-\frac{3554658454630}{297422821959}a^{3}-\frac{18855550285094}{892268465877}a^{2}-\frac{12169954947812}{892268465877}a-\frac{3822358403545}{297422821959}$, $\frac{29779661713}{892268465877}a^{10}-\frac{79626280838}{892268465877}a^{9}-\frac{153426047557}{297422821959}a^{8}+\frac{962076937039}{892268465877}a^{7}+\frac{652282021447}{297422821959}a^{6}-\frac{1790905974670}{892268465877}a^{5}-\frac{292235847032}{99140940653}a^{4}-\frac{2847631466955}{99140940653}a^{3}-\frac{36125925463750}{892268465877}a^{2}-\frac{11829683135587}{892268465877}a-\frac{16829334274691}{297422821959}$, $\frac{130972212011}{892268465877}a^{10}-\frac{257634432190}{892268465877}a^{9}-\frac{791395064830}{297422821959}a^{8}+\frac{3172630440665}{892268465877}a^{7}+\frac{4219995300758}{297422821959}a^{6}-\frac{5246750702939}{892268465877}a^{5}-\frac{6563366414477}{297422821959}a^{4}-\frac{40609113693848}{297422821959}a^{3}-\frac{222276855973130}{892268465877}a^{2}-\frac{80422554510698}{892268465877}a-\frac{73220987340127}{297422821959}$, $\frac{14426359328}{297422821959}a^{10}-\frac{33489408040}{297422821959}a^{9}-\frac{78960921433}{99140940653}a^{8}+\frac{393345415550}{297422821959}a^{7}+\frac{352806714493}{99140940653}a^{6}-\frac{407623789346}{297422821959}a^{5}-\frac{410808002474}{99140940653}a^{4}-\frac{4751927055116}{99140940653}a^{3}-\frac{22307179830023}{297422821959}a^{2}-\frac{9426220221782}{297422821959}a-\frac{7873008856697}{99140940653}$, $\frac{18691816939}{297422821959}a^{10}+\frac{4371380003}{297422821959}a^{9}-\frac{417351885634}{297422821959}a^{8}-\frac{109738317712}{99140940653}a^{7}+\frac{933004538879}{99140940653}a^{6}+\frac{3926103700445}{297422821959}a^{5}-\frac{4606896066094}{297422821959}a^{4}-\frac{27652325613349}{297422821959}a^{3}-\frac{71475277058636}{297422821959}a^{2}-\frac{25308883531537}{99140940653}a-\frac{6933444048371}{99140940653}$, $\frac{6595542890}{892268465877}a^{10}+\frac{20348398043}{892268465877}a^{9}-\frac{45246413681}{297422821959}a^{8}-\frac{608498546341}{892268465877}a^{7}-\frac{92662425037}{297422821959}a^{6}+\frac{2105042680132}{892268465877}a^{5}+\frac{781825448108}{99140940653}a^{4}+\frac{1710706050106}{99140940653}a^{3}+\frac{17146786763395}{892268465877}a^{2}+\frac{12084314467663}{892268465877}a+\frac{4211091680609}{297422821959}$ 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:  \( 1258534.69335 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{4}\cdot 1258534.69335 \cdot 1}{2\cdot\sqrt{11763130845074473216}}\cr\approx \mathstrut & 2.28761635519 \end{aligned}\] (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 - 22*x^9 - 11*x^8 + 143*x^7 + 143*x^6 - 209*x^5 - 1188*x^4 - 3619*x^3 - 4059*x^2 - 2948*x - 3486) 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 - 22*x^9 - 11*x^8 + 143*x^7 + 143*x^6 - 209*x^5 - 1188*x^4 - 3619*x^3 - 4059*x^2 - 2948*x - 3486, 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 - 22*x^9 - 11*x^8 + 143*x^7 + 143*x^6 - 209*x^5 - 1188*x^4 - 3619*x^3 - 4059*x^2 - 2948*x - 3486); 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 - 22*x^9 - 11*x^8 + 143*x^7 + 143*x^6 - 209*x^5 - 1188*x^4 - 3619*x^3 - 4059*x^2 - 2948*x - 3486); 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

$\PSL(2,11)$ (as 11T5):

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 non-solvable group of order 660
The 8 conjugacy class representatives for $\PSL(2,11)$
Character table for $\PSL(2,11)$

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

Degree 12 sibling: 12.0.188210093521191571456.1
Arithmetically equivalent sibling: 11.3.11763130845074473216.1
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 R ${\href{/padicField/3.3.0.1}{3} }^{3}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ ${\href{/padicField/5.5.0.1}{5} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }$ ${\href{/padicField/7.5.0.1}{5} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ R ${\href{/padicField/13.3.0.1}{3} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ ${\href{/padicField/17.11.0.1}{11} }$ ${\href{/padicField/19.3.0.1}{3} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.5.0.1}{5} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ ${\href{/padicField/31.11.0.1}{11} }$ ${\href{/padicField/37.5.0.1}{5} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ ${\href{/padicField/43.3.0.1}{3} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.11.0.1}{11} }$ ${\href{/padicField/53.5.0.1}{5} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.5.0.1}{5} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$

In the table, R denotes a ramified prime. 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
\(2\) Copy content Toggle raw display 2.1.2.2a1.2$x^{2} + 2 x + 6$$2$$1$$2$$C_2$$$[2]$$
2.3.1.0a1.1$x^{3} + x + 1$$1$$3$$0$$C_3$$$[\ ]^{3}$$
2.3.2.6a1.2$x^{6} + 2 x^{4} + 4 x^{3} + x^{2} + 4 x + 9$$2$$3$$6$$C_6$$$[2]^{3}$$
\(11\) Copy content Toggle raw display 11.1.11.16a2.1$x^{11} + 22 x^{6} + 11$$11$$1$$16$$C_{11}:C_5$$$[\frac{8}{5}]_{5}$$

Spectrum of ring of integers

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