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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 + 57*x^4 - 57*x^3 + 953*x^2 - 953*x + 4537)
 
Copy content gp:K = bnfinit(y^6 - y^5 + 57*y^4 - 57*y^3 + 953*y^2 - 953*y + 4537, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^6 - x^5 + 57*x^4 - 57*x^3 + 953*x^2 - 953*x + 4537);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^6 - x^5 + 57*x^4 - 57*x^3 + 953*x^2 - 953*x + 4537)
 

\( x^{6} - x^{5} + 57x^{4} - 57x^{3} + 953x^{2} - 953x + 4537 \) 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:  $6$
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:  $(0, 3)$
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:   \(-603993159\) \(\medspace = -\,3^{3}\cdot 7^{5}\cdot 11^{3}\) 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:  \(29.07\)
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:  $3^{1/2}7^{5/6}11^{1/2}\approx 29.074036854473896$
Ramified primes:   \(3\), \(7\), \(11\) 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:  \(\Q(\sqrt{-231}) \)
$\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$:   $C_6$
Copy content comment:Automorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(231=3\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{231}(1,·)$, $\chi_{231}(67,·)$, $\chi_{231}(164,·)$, $\chi_{231}(131,·)$, $\chi_{231}(230,·)$, $\chi_{231}(100,·)$$\rbrace$
This is a CM field.
Reflex fields:  \(\Q(\sqrt{-231}) \), 6.0.603993159.1$^{3}$

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{937}a^{4}-\frac{106}{937}a^{3}+\frac{32}{937}a^{2}+\frac{267}{937}a+\frac{128}{937}$, $\frac{1}{937}a^{5}+\frac{40}{937}a^{3}-\frac{89}{937}a^{2}+\frac{320}{937}a+\frac{450}{937}$ 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:  $C_{2}\times C_{18}$, which has order $36$
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:  $C_{2}\times C_{18}$, which has order $36$
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:   $36$

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:  $2$
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:   $\frac{9}{937}a^{4}-\frac{17}{937}a^{3}+\frac{288}{937}a^{2}-\frac{408}{937}a+\frac{1152}{937}$, $\frac{1}{937}a^{5}+\frac{40}{937}a^{3}-\frac{89}{937}a^{2}+\frac{320}{937}a-\frac{1424}{937}$ 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:  \( 2.10181872849 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{3}\cdot 2.10181872849 \cdot 36}{2\cdot\sqrt{603993159}}\cr\approx \mathstrut & 0.38184873647 \end{aligned}\]

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 + 57*x^4 - 57*x^3 + 953*x^2 - 953*x + 4537) 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 + 57*x^4 - 57*x^3 + 953*x^2 - 953*x + 4537, 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 + 57*x^4 - 57*x^3 + 953*x^2 - 953*x + 4537); 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^6 - x^5 + 57*x^4 - 57*x^3 + 953*x^2 - 953*x + 4537); 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

$C_6$ (as 6T1):

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 cyclic group of order 6
The 6 conjugacy class representatives for $C_6$
Character table for $C_6$

Intermediate fields

\(\Q(\sqrt{-231}) \), \(\Q(\zeta_{7})^+\)

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

Sibling algebras

Twin sextic algebra: \(\Q(\zeta_{7})^+\) $\times$ \(\Q(\sqrt{-231}) \) $\times$ \(\Q\)
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 ${\href{/padicField/2.3.0.1}{3} }^{2}$ R ${\href{/padicField/5.3.0.1}{3} }^{2}$ R R ${\href{/padicField/13.1.0.1}{1} }^{6}$ ${\href{/padicField/17.6.0.1}{6} }$ ${\href{/padicField/19.3.0.1}{3} }^{2}$ ${\href{/padicField/23.6.0.1}{6} }$ ${\href{/padicField/29.1.0.1}{1} }^{6}$ ${\href{/padicField/31.6.0.1}{6} }$ ${\href{/padicField/37.3.0.1}{3} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{3}$ ${\href{/padicField/43.2.0.1}{2} }^{3}$ ${\href{/padicField/47.3.0.1}{3} }^{2}$ ${\href{/padicField/53.6.0.1}{6} }$ ${\href{/padicField/59.3.0.1}{3} }^{2}$

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
\(3\) Copy content Toggle raw display 3.3.2.3a1.1$x^{6} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 7 x + 1$$2$$3$$3$$C_6$$$[\ ]_{2}^{3}$$
\(7\) Copy content Toggle raw display 7.1.6.5a1.6$x^{6} + 42$$6$$1$$5$$C_6$$$[\ ]_{6}$$
\(11\) Copy content Toggle raw display 11.3.2.3a1.1$x^{6} + 4 x^{4} + 18 x^{3} + 4 x^{2} + 47 x + 81$$2$$3$$3$$C_6$$$[\ ]_{2}^{3}$$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
*6 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
*6 1.231.2t1.a.a$1$ $ 3 \cdot 7 \cdot 11 $ \(\Q(\sqrt{-231}) \) $C_2$ (as 2T1) $1$ $-1$
*6 1.7.3t1.a.a$1$ $ 7 $ \(\Q(\zeta_{7})^+\) $C_3$ (as 3T1) $0$ $1$
*6 1.231.6t1.a.a$1$ $ 3 \cdot 7 \cdot 11 $ 6.0.603993159.1 $C_6$ (as 6T1) $0$ $-1$
*6 1.7.3t1.a.b$1$ $ 7 $ \(\Q(\zeta_{7})^+\) $C_3$ (as 3T1) $0$ $1$
*6 1.231.6t1.a.b$1$ $ 3 \cdot 7 \cdot 11 $ 6.0.603993159.1 $C_6$ (as 6T1) $0$ $-1$

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)