Properties

Label 6.0.16807.1
Degree 66
Signature [0,3][0, 3]
Discriminant 16807-16807
Root discriminant 5.065.06
Ramified prime 77
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 + x^4 - x^3 + x^2 - x + 1)
 
Copy content gp:K = bnfinit(y^6 - y^5 + y^4 - y^3 + y^2 - y + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^6 - x^5 + x^4 - x^3 + x^2 - x + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^6 - x^5 + x^4 - x^3 + x^2 - x + 1)
 

x6x5+x4x3+x2x+1 x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 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:  [0,3][0, 3]
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:   16807-16807 =75\medspace = -\,7^{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:  5.065.06
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:  75/65.0611401847963867^{5/6}\approx 5.061140184796386
Ramified primes:   77 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(7)\Q(\sqrt{-7})
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:  77
Dirichlet character group:    {\lbraceχ7(1,)\chi_{7}(1,·), χ7(2,)\chi_{7}(2,·), χ7(3,)\chi_{7}(3,·), χ7(4,)\chi_{7}(4,·), χ7(5,)\chi_{7}(5,·), χ7(6,)\chi_{7}(6,·)}\rbrace
This is a CM field.
Reflex fields:  Q(7)\Q(\sqrt{-7}) , Q(ζ7)\Q(\zeta_{7})3^{3}

Integral basis (with respect to field generator aa)

11, aa, a2a^{2}, a3a^{3}, a4a^{4}, a5a^{5} 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:  Yes
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);
 
Relative class number:   11

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:  22
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:   a a  (order 1414) 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:   a1a-1, a4aa^{4}-a 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:  2.10181872849 2.10181872849
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(20(2π)32.1018187284911416807(0.287251117499 \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^{0}\cdot(2\pi)^{3}\cdot 2.10181872849 \cdot 1}{14\cdot\sqrt{16807}}\cr\approx \mathstrut & 0.287251117499 \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 + x^4 - x^3 + x^2 - 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 + x^4 - x^3 + x^2 - 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 + x^4 - x^3 + x^2 - 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 + x^4 - x^3 + x^2 - 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(7)\Q(\sqrt{-7}) , Q(ζ7)+\Q(\zeta_{7})^+

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: Q(ζ7)+\Q(\zeta_{7})^+ ×\times Q(7)\Q(\sqrt{-7}) ×\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 32{\href{/padicField/2.3.0.1}{3} }^{2} 6{\href{/padicField/3.6.0.1}{6} } 6{\href{/padicField/5.6.0.1}{6} } R 32{\href{/padicField/11.3.0.1}{3} }^{2} 23{\href{/padicField/13.2.0.1}{2} }^{3} 6{\href{/padicField/17.6.0.1}{6} } 6{\href{/padicField/19.6.0.1}{6} } 32{\href{/padicField/23.3.0.1}{3} }^{2} 16{\href{/padicField/29.1.0.1}{1} }^{6} 6{\href{/padicField/31.6.0.1}{6} } 32{\href{/padicField/37.3.0.1}{3} }^{2} 23{\href{/padicField/41.2.0.1}{2} }^{3} 16{\href{/padicField/43.1.0.1}{1} }^{6} 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
77 Copy content Toggle raw display 7.1.6.5a1.1x6+7x^{6} + 7661155C6C_6[ ]6[\ ]_{6}

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
* 1.7.2t1.a.a11 7 7 Q(7)\Q(\sqrt{-7}) C2C_2 (as 2T1) 11 1-1
* 1.7.3t1.a.a11 7 7 Q(ζ7)+\Q(\zeta_{7})^+ C3C_3 (as 3T1) 00 11
* 1.7.6t1.a.a11 7 7 Q(ζ7)\Q(\zeta_{7}) C6C_6 (as 6T1) 00 1-1
* 1.7.3t1.a.b11 7 7 Q(ζ7)+\Q(\zeta_{7})^+ C3C_3 (as 3T1) 00 11
* 1.7.6t1.a.b11 7 7 Q(ζ7)\Q(\zeta_{7}) C6C_6 (as 6T1) 00 1-1

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)