Properties

Label 6.6.286315237.1
Degree 66
Signature [6,0][6, 0]
Discriminant 286315237286315237
Root discriminant 25.6725.67
Ramified primes 13,1913,19
Class number 11
Class group trivial
Galois group C6C_6 (as 6T1)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^6 - x^5 - 22*x^4 + 5*x^3 + 73*x^2 - 58*x + 1)
 
Copy content gp:K = bnfinit(y^6 - y^5 - 22*y^4 + 5*y^3 + 73*y^2 - 58*y + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^6 - x^5 - 22*x^4 + 5*x^3 + 73*x^2 - 58*x + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^6 - x^5 - 22*x^4 + 5*x^3 + 73*x^2 - 58*x + 1)
 

x6x522x4+5x3+73x258x+1 x^{6} - x^{5} - 22x^{4} + 5x^{3} + 73x^{2} - 58x + 1 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:  66
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:  [6,0][6, 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:   286315237286315237 =133194\medspace = 13^{3}\cdot 19^{4} 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:  25.6725.67
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:(1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  131/2192/325.6728496126612413^{1/2}19^{2/3}\approx 25.67284961266124
Ramified primes:   1313, 1919 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(13)\Q(\sqrt{13})
Aut(K/Q)\Aut(K/\Q) == Gal(K/Q)\Gal(K/\Q):   C6C_6
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is Galois and abelian over Q\Q.
Conductor:  247=1319247=13\cdot 19
Dirichlet character group:    {\lbraceχ247(64,)\chi_{247}(64,·), χ247(1,)\chi_{247}(1,·), χ247(144,)\chi_{247}(144,·), χ247(235,)\chi_{247}(235,·), χ247(220,)\chi_{247}(220,·), χ247(77,)\chi_{247}(77,·)}\rbrace
This is not a CM field.
This field has no CM subfields.

Integral basis (with respect to field generator aa)

11, aa, a2a^{2}, a3a^{3}, a4a^{4}, 1107a5+19107a4+37107a34107a27107a+16107\frac{1}{107}a^{5}+\frac{19}{107}a^{4}+\frac{37}{107}a^{3}-\frac{4}{107}a^{2}-\frac{7}{107}a+\frac{16}{107} 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:  11
Inessential primes:  None

Class group and class number

Ideal class group:  Trivial group, which has order 11
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 11
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:  55
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 -1  (order 22) 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:   22107a510107a4470107a3195107a2+1237107a397107\frac{22}{107}a^{5}-\frac{10}{107}a^{4}-\frac{470}{107}a^{3}-\frac{195}{107}a^{2}+\frac{1237}{107}a-\frac{397}{107}, 62107a5+1107a41344107a31104107a2+3097107a78107\frac{62}{107}a^{5}+\frac{1}{107}a^{4}-\frac{1344}{107}a^{3}-\frac{1104}{107}a^{2}+\frac{3097}{107}a-\frac{78}{107}, 40107a5+11107a4874107a3909107a2+1967107a+319107\frac{40}{107}a^{5}+\frac{11}{107}a^{4}-\frac{874}{107}a^{3}-\frac{909}{107}a^{2}+\frac{1967}{107}a+\frac{319}{107}, 22107a510107a4470107a3195107a2+1130107a397107\frac{22}{107}a^{5}-\frac{10}{107}a^{4}-\frac{470}{107}a^{3}-\frac{195}{107}a^{2}+\frac{1130}{107}a-\frac{397}{107}, 62107a5+1107a41344107a31104107a2+3204107a78107\frac{62}{107}a^{5}+\frac{1}{107}a^{4}-\frac{1344}{107}a^{3}-\frac{1104}{107}a^{2}+\frac{3204}{107}a-\frac{78}{107} 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:  81.4434827116 81.4434827116
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

lims1(s1)ζK(s)=(2r1(2π)r2RhwD(26(2π)081.443482711612286315237(0.154022470373 \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^{6}\cdot(2\pi)^{0}\cdot 81.4434827116 \cdot 1}{2\cdot\sqrt{286315237}}\cr\approx \mathstrut & 0.154022470373 \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^6 - x^5 - 22*x^4 + 5*x^3 + 73*x^2 - 58*x + 1) 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^6 - x^5 - 22*x^4 + 5*x^3 + 73*x^2 - 58*x + 1, 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^6 - x^5 - 22*x^4 + 5*x^3 + 73*x^2 - 58*x + 1); 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 = PolynomialRing(QQ); K, a = NumberField(x^6 - x^5 - 22*x^4 + 5*x^3 + 73*x^2 - 58*x + 1); 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

C6C_6 (as 6T1):

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); G, transitive_group_identification(G)
 
A cyclic group of order 6
The 6 conjugacy class representatives for C6C_6
Character table for C6C_6

Intermediate fields

Q(13)\Q(\sqrt{13}) , 3.3.361.1

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(b)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling algebras

Twin sextic algebra: 3.3.361.1 ×\times Q(13)\Q(\sqrt{13}) ×\times Q\Q
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

pp 22 33 55 77 1111 1313 1717 1919 2323 2929 3131 3737 4141 4343 4747 5353 5959
Cycle type 6{\href{/padicField/2.6.0.1}{6} } 32{\href{/padicField/3.3.0.1}{3} }^{2} 6{\href{/padicField/5.6.0.1}{6} } 23{\href{/padicField/7.2.0.1}{2} }^{3} 23{\href{/padicField/11.2.0.1}{2} }^{3} R 32{\href{/padicField/17.3.0.1}{3} }^{2} R 32{\href{/padicField/23.3.0.1}{3} }^{2} 32{\href{/padicField/29.3.0.1}{3} }^{2} 23{\href{/padicField/31.2.0.1}{2} }^{3} 23{\href{/padicField/37.2.0.1}{2} }^{3} 6{\href{/padicField/41.6.0.1}{6} } 32{\href{/padicField/43.3.0.1}{3} }^{2} 6{\href{/padicField/47.6.0.1}{6} } 32{\href{/padicField/53.3.0.1}{3} }^{2} 6{\href{/padicField/59.6.0.1}{6} }

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

ppLabelPolynomial ee ff cc Galois group Slope content
1313 Copy content Toggle raw display 13.3.2.3a1.2x6+4x4+22x3+4x2+44x+134x^{6} + 4 x^{4} + 22 x^{3} + 4 x^{2} + 44 x + 134223333C6C_6[ ]23[\ ]_{2}^{3}
1919 Copy content Toggle raw display 19.2.3.4a1.2x6+54x5+978x4+6048x3+1956x2+216x+27x^{6} + 54 x^{5} + 978 x^{4} + 6048 x^{3} + 1956 x^{2} + 216 x + 27332244C6C_6[ ]32[\ ]_{3}^{2}

Artin representations

Label Dimension Conductor Artin stem field GG Ind χ(c)\chi(c)
* 1.1.1t1.a.a11 11 Q\Q C1C_1 11 11
* 1.13.2t1.a.a11 13 13 Q(13)\Q(\sqrt{13}) C2C_2 (as 2T1) 11 11
* 1.19.3t1.a.a11 19 19 3.3.361.1 C3C_3 (as 3T1) 00 11
* 1.247.6t1.b.a11 1319 13 \cdot 19 6.6.286315237.1 C6C_6 (as 6T1) 00 11
* 1.19.3t1.a.b11 19 19 3.3.361.1 C3C_3 (as 3T1) 00 11
* 1.247.6t1.b.b11 1319 13 \cdot 19 6.6.286315237.1 C6C_6 (as 6T1) 00 11

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.

Spectrum of ring of integers

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