Properties

Label 6.0.273791.1
Degree 66
Signature [0,3][0, 3]
Discriminant 273791-273791
Root discriminant 8.068.06
Ramified primes 7,391137,39113
Class number 11
Class group trivial
Galois group S6S_6 (as 6T16)

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

x63x5+5x43x32x2+2x+1 x^{6} - 3x^{5} + 5x^{4} - 3x^{3} - 2x^{2} + 2x + 1 Copy content Toggle raw display

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

Invariants

Degree:  66
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  [0,3][0, 3]
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   273791-273791 =739113\medspace = -\,7\cdot 39113 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:  8.068.06
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:  71/2391131/2523.25041806003377^{1/2}39113^{1/2}\approx 523.2504180600337
Ramified primes:   77, 3911339113 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(273791)\Q(\sqrt{-273791})
Aut(K/Q)\Aut(K/\Q):   C1C_1
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over Q\Q.
This is not a CM field.

Integral basis (with respect to field generator aa)

11, aa, a2a^{2}, a3a^{3}, a4a^{4}, a5a^{5} Copy content Toggle raw display

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

Monogenic:  Yes
Index:  11
Inessential primes:  None

Class group and class number

Trivial group, which has order 11

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:  22
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   1 -1  (order 22) 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:   aa, a1a-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:  2.23197126062 2.23197126062
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

lims1(s1)ζK(s)=(2r1(2π)r2RhwD(20(2π)32.2319712606212273791(0.529040138799 \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 2.23197126062 \cdot 1}{2\cdot\sqrt{273791}}\cr\approx \mathstrut & 0.529040138799 \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 + 5*x^4 - 3*x^3 - 2*x^2 + 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 + 5*x^4 - 3*x^3 - 2*x^2 + 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(Rationals()); K<a> := NumberField(x^6 - 3*x^5 + 5*x^4 - 3*x^3 - 2*x^2 + 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 + 5*x^4 - 3*x^3 - 2*x^2 + 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

S6S_6 (as 6T16):

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 non-solvable group of order 720
The 11 conjugacy class representatives for S6S_6
Character table for S6S_6

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and Q\Q.
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

Twin sextic algebra: 6.4.20523787244652671.1
Degree 6 sibling: 6.4.20523787244652671.1
Degree 10 sibling: deg 10
Degree 12 siblings: deg 12, deg 12
Degree 15 siblings: deg 15, deg 15
Degree 20 siblings: deg 20, deg 20, deg 20
Degree 30 siblings: deg 30, deg 30, deg 30, deg 30, deg 30, deg 30
Degree 36 sibling: deg 36
Degree 40 siblings: deg 40, deg 40, deg 40
Degree 45 sibling: deg 45
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 4,2{\href{/padicField/2.4.0.1}{4} }{,}\,{\href{/padicField/2.2.0.1}{2} } 5,1{\href{/padicField/3.5.0.1}{5} }{,}\,{\href{/padicField/3.1.0.1}{1} } 32{\href{/padicField/5.3.0.1}{3} }^{2} R 3,2,1{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} } 6{\href{/padicField/13.6.0.1}{6} } 4,12{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2} 3,2,1{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} } 5,1{\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.1.0.1}{1} } 4,12{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2} 5,1{\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.1.0.1}{1} } 5,1{\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.1.0.1}{1} } 6{\href{/padicField/41.6.0.1}{6} } 6{\href{/padicField/43.6.0.1}{6} } 3,2,1{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} } 22,12{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2} 5,1{\href{/padicField/59.5.0.1}{5} }{,}\,{\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.

# to obtain a list of [ei,fi][e_i,f_i] for the factorization of the ideal pOKp\mathcal{O}_K for p=7p=7 in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of [ei,fi][e_i,f_i] for the factorization of the ideal pOKp\mathcal{O}_K for p=7p=7 in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of [ei,fi][e_i,f_i] for the factorization of the ideal pOKp\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 [ei,fi][e_i,f_i] for the factorization of the ideal pOKp\mathcal{O}_K for p=7p=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 7.1.2.1a1.2x2+21x^{2} + 21221111C2C_2[ ]2[\ ]_{2}
7.4.1.0a1.1x4+5x2+4x+3x^{4} + 5 x^{2} + 4 x + 3114400C4C_4[ ]4[\ ]^{4}
3911339113 Copy content Toggle raw display Deg 22112200C2C_2[ ]2[\ ]^{2}
Deg 22112200C2C_2[ ]2[\ ]^{2}
Deg 22221111C2C_2[ ]2[\ ]_{2}

Spectrum of ring of integers

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