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Properties

Label 6.0.21168.1
Degree $6$
Signature $[0, 3]$
Discriminant $-21168$
Root discriminant \(5.26\)
Ramified primes $2,3,7$
Class number $1$
Class group trivial
Galois group $S_3\times C_3$ (as 6T5)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

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

\( x^{6} - 3x^{5} + 4x^{4} - x^{3} - 2x^{2} + x + 1 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $6$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 3]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-21168\) \(\medspace = -\,2^{4}\cdot 3^{3}\cdot 7^{2}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(5.26\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2^{2/3}3^{1/2}7^{2/3}\approx 10.061112020813587$
Ramified primes:   \(2\), \(3\), \(7\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{-3}) \)
$\card{ \Aut(K/\Q) }$:  $3$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  Yes
Index:  $1$
Inessential primes:  None

Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $2$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -a^{4} + 2 a^{3} - 2 a^{2} + 1 \)  (order $6$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $a^{3}-2a^{2}+2a$, $a^{3}-a^{2}+1$ Copy content Toggle raw display
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 1.11901604193 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{3}\cdot 1.11901604193 \cdot 1}{6\cdot\sqrt{21168}}\cr\approx \mathstrut & 0.317968980042 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^6 - 3*x^5 + 4*x^4 - x^3 - 2*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))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^6 - 3*x^5 + 4*x^4 - x^3 - 2*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)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^6 - 3*x^5 + 4*x^4 - x^3 - 2*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)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^6 - 3*x^5 + 4*x^4 - x^3 - 2*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

$C_3\times S_3$ (as 6T5):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3\times C_3$
Character table for $S_3\times C_3$

Intermediate fields

\(\Q(\sqrt{-3}) \)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Sibling algebras

Galois closure: 18.0.1115906277282951168.1
Twin sextic algebra: \(\Q(\zeta_{7})^+\) $\times$ 3.1.588.1
Degree 9 sibling: 9.3.203297472.1
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 R ${\href{/padicField/5.6.0.1}{6} }$ R ${\href{/padicField/11.6.0.1}{6} }$ ${\href{/padicField/13.3.0.1}{3} }^{2}$ ${\href{/padicField/17.6.0.1}{6} }$ ${\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ ${\href{/padicField/23.6.0.1}{6} }$ ${\href{/padicField/29.2.0.1}{2} }^{3}$ ${\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ ${\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ ${\href{/padicField/41.2.0.1}{2} }^{3}$ ${\href{/padicField/43.3.0.1}{3} }^{2}$ ${\href{/padicField/47.6.0.1}{6} }$ ${\href{/padicField/53.6.0.1}{6} }$ ${\href{/padicField/59.6.0.1}{6} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ 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 2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
\(3\) Copy content Toggle raw display 3.6.3.2$x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
\(7\) Copy content Toggle raw display 7.3.0.1$x^{3} + 6 x^{2} + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.3.2t1.a.a$1$ $ 3 $ \(\Q(\sqrt{-3}) \) $C_2$ (as 2T1) $1$ $-1$
1.7.3t1.a.a$1$ $ 7 $ \(\Q(\zeta_{7})^+\) $C_3$ (as 3T1) $0$ $1$
1.21.6t1.a.a$1$ $ 3 \cdot 7 $ 6.0.64827.1 $C_6$ (as 6T1) $0$ $-1$
1.21.6t1.a.b$1$ $ 3 \cdot 7 $ 6.0.64827.1 $C_6$ (as 6T1) $0$ $-1$
1.7.3t1.a.b$1$ $ 7 $ \(\Q(\zeta_{7})^+\) $C_3$ (as 3T1) $0$ $1$
2.588.3t2.a.a$2$ $ 2^{2} \cdot 3 \cdot 7^{2}$ 3.1.588.1 $S_3$ (as 3T2) $1$ $0$
* 2.84.6t5.b.a$2$ $ 2^{2} \cdot 3 \cdot 7 $ 6.0.21168.1 $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.84.6t5.b.b$2$ $ 2^{2} \cdot 3 \cdot 7 $ 6.0.21168.1 $S_3\times C_3$ (as 6T5) $0$ $0$

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 *.