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Properties

Label 6.6.616133159929744.1
Degree $6$
Signature $[6, 0]$
Discriminant $6.161\times 10^{14}$
Root discriminant \(291.71\)
Ramified primes $2,33769$
Class number $4$ (GRH)
Class group [4] (GRH)
Galois group $\PGL(2,5)$ (as 6T14)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^6 - x^5 - 89*x^4 + 495*x^3 - 872*x^2 + 480*x - 76)
 
gp: K = bnfinit(y^6 - y^5 - 89*y^4 + 495*y^3 - 872*y^2 + 480*y - 76, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^6 - x^5 - 89*x^4 + 495*x^3 - 872*x^2 + 480*x - 76);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^6 - x^5 - 89*x^4 + 495*x^3 - 872*x^2 + 480*x - 76)
 

\( x^{6} - x^{5} - 89x^{4} + 495x^{3} - 872x^{2} + 480x - 76 \) 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:  $[6, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(616133159929744\) \(\medspace = 2^{4}\cdot 33769^{3}\) 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:  \(291.71\)
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:  $2^{2/3}33769^{1/2}\approx 291.70626984657$
Ramified primes:   \(2\), \(33769\) 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:  \(\Q(\sqrt{33769}) \)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{46}a^{5}+\frac{15}{46}a^{4}+\frac{13}{46}a^{3}+\frac{13}{46}a^{2}-\frac{10}{23}a+\frac{11}{23}$ Copy content Toggle raw display

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

$C_{4}$, which has order $4$ (assuming GRH)

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:  $5$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) 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:   $\frac{7}{46}a^{5}-\frac{33}{46}a^{4}-\frac{783}{46}a^{3}+\frac{4875}{46}a^{2}-\frac{3750}{23}a+\frac{859}{23}$, $\frac{113}{46}a^{5}-\frac{191}{46}a^{4}-\frac{10491}{46}a^{3}+\frac{61453}{46}a^{2}-\frac{48349}{23}a+\frac{11133}{23}$, $\frac{2303}{46}a^{5}-\frac{1105}{46}a^{4}-\frac{205535}{46}a^{3}+\frac{1033061}{46}a^{2}-\frac{735685}{23}a+\frac{171337}{23}$, $\frac{44343}{46}a^{5}-\frac{32123}{46}a^{4}-\frac{3955553}{46}a^{3}+\frac{20859055}{46}a^{2}-\frac{16453224}{23}a+\frac{6094713}{23}$, $\frac{3079}{46}a^{5}+\frac{8465}{46}a^{4}-\frac{241861}{46}a^{3}+\frac{613601}{46}a^{2}-\frac{188179}{23}a+\frac{31109}{23}$ Copy content Toggle raw display (assuming GRH)
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:  \( 384040.90998 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{0}\cdot 384040.90998 \cdot 4}{2\cdot\sqrt{616133159929744}}\cr\approx \mathstrut & 1.9803874627 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^6 - x^5 - 89*x^4 + 495*x^3 - 872*x^2 + 480*x - 76)
 
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 - x^5 - 89*x^4 + 495*x^3 - 872*x^2 + 480*x - 76, 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 - x^5 - 89*x^4 + 495*x^3 - 872*x^2 + 480*x - 76);
 
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 - x^5 - 89*x^4 + 495*x^3 - 872*x^2 + 480*x - 76);
 
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

$S_5$ (as 6T14):

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 120
The 7 conjugacy class representatives for $\PGL(2,5)$
Character table for $\PGL(2,5)$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.
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: \(\Q\) $\times$ 5.5.135076.1
Degree 5 sibling: 5.5.135076.1
Degree 10 siblings: deg 10, 10.10.2464532639718976.1
Degree 12 sibling: deg 12
Degree 15 sibling: deg 15
Degree 20 siblings: deg 20, deg 20, deg 20
Degree 24 sibling: deg 24
Degree 30 siblings: deg 30, deg 30, deg 30
Degree 40 sibling: deg 40
Minimal sibling: 5.5.135076.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.5.0.1}{5} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ ${\href{/padicField/5.5.0.1}{5} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ ${\href{/padicField/7.5.0.1}{5} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ ${\href{/padicField/13.6.0.1}{6} }$ ${\href{/padicField/17.6.0.1}{6} }$ ${\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.3.0.1}{3} }^{2}$ ${\href{/padicField/41.6.0.1}{6} }$ ${\href{/padicField/43.3.0.1}{3} }^{2}$ ${\href{/padicField/47.2.0.1}{2} }^{3}$ ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.1.0.1}{1} }$

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 $[e_i,f_i]$ 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 2.3.2.1$x^{3} + 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} + 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
\(33769\) Copy content Toggle raw display Deg $6$$2$$3$$3$