Properties

Label 8.8.14667743362801.2
Degree $8$
Signature $[8, 0]$
Discriminant $1.467\times 10^{13}$
Root discriminant \(44.24\)
Ramified primes $19,103$
Class number $3$
Class group [3]
Galois group $S_4$ (as 8T14)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 56*x^6 - 34*x^5 + 1007*x^4 + 952*x^3 - 5955*x^2 - 3791*x + 11943)
 
gp: K = bnfinit(y^8 - 56*y^6 - 34*y^5 + 1007*y^4 + 952*y^3 - 5955*y^2 - 3791*y + 11943, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^8 - 56*x^6 - 34*x^5 + 1007*x^4 + 952*x^3 - 5955*x^2 - 3791*x + 11943);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^8 - 56*x^6 - 34*x^5 + 1007*x^4 + 952*x^3 - 5955*x^2 - 3791*x + 11943)
 

\( x^{8} - 56x^{6} - 34x^{5} + 1007x^{4} + 952x^{3} - 5955x^{2} - 3791x + 11943 \) Copy content Toggle raw display

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

Invariants

Degree:  $8$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[8, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(14667743362801\) \(\medspace = 19^{4}\cdot 103^{4}\) 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:  \(44.24\)
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:  $19^{1/2}103^{1/2}\approx 44.23799272118933$
Ramified primes:   \(19\), \(103\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{9768443}a^{7}+\frac{3601766}{9768443}a^{6}+\frac{1340511}{9768443}a^{5}-\frac{2305446}{9768443}a^{4}-\frac{2276036}{9768443}a^{3}-\frac{3333923}{9768443}a^{2}+\frac{4338865}{9768443}a+\frac{1783513}{9768443}$ 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

$C_{3}$, which has order $3$

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:  $7$
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:   $\frac{65232}{9768443}a^{7}-\frac{191324}{9768443}a^{6}-\frac{2888184}{9768443}a^{5}+\frac{6094956}{9768443}a^{4}+\frac{39258577}{9768443}a^{3}-\frac{52460842}{9768443}a^{2}-\frac{134784004}{9768443}a+\frac{185564303}{9768443}$, $\frac{171992}{9768443}a^{7}-\frac{643416}{9768443}a^{6}-\frac{7392217}{9768443}a^{5}+\frac{21906710}{9768443}a^{4}+\frac{98285500}{9768443}a^{3}-\frac{205617819}{9768443}a^{2}-\frac{342260967}{9768443}a+\frac{636269605}{9768443}$, $\frac{106760}{9768443}a^{7}-\frac{452092}{9768443}a^{6}-\frac{4504033}{9768443}a^{5}+\frac{15811754}{9768443}a^{4}+\frac{59026923}{9768443}a^{3}-\frac{153156977}{9768443}a^{2}-\frac{207476963}{9768443}a+\frac{450705302}{9768443}$, $\frac{747152}{9768443}a^{7}-\frac{2617866}{9768443}a^{6}-\frac{32059890}{9768443}a^{5}+\frac{87722600}{9768443}a^{4}+\frac{424361675}{9768443}a^{3}-\frac{805053065}{9768443}a^{2}-\frac{1477940165}{9768443}a+\frac{2454800767}{9768443}$, $\frac{8139}{9768443}a^{7}-\frac{323969}{9768443}a^{6}-\frac{931802}{9768443}a^{5}+\frac{10922452}{9768443}a^{4}+\frac{25616253}{9768443}a^{3}-\frac{76212187}{9768443}a^{2}-\frac{106352083}{9768443}a+\frac{175937983}{9768443}$, $\frac{867356}{9768443}a^{7}-\frac{3099805}{9768443}a^{6}-\frac{37511374}{9768443}a^{5}+\frac{104387969}{9768443}a^{4}+\frac{498660976}{9768443}a^{3}-\frac{952085927}{9768443}a^{2}-\frac{1690857664}{9768443}a+\frac{2794074403}{9768443}$, $\frac{142880}{9768443}a^{7}-\frac{788046}{9768443}a^{6}-\frac{7418664}{9768443}a^{5}+\frac{19078809}{9768443}a^{4}+\frac{128201992}{9768443}a^{3}-\frac{31869117}{9768443}a^{2}-\frac{447015287}{9768443}a-\frac{69414202}{9768443}$ 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:  \( 6402.557564 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{0}\cdot 6402.557564 \cdot 3}{2\cdot\sqrt{14667743362801}}\cr\approx \mathstrut & 0.6419527518 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^8 - 56*x^6 - 34*x^5 + 1007*x^4 + 952*x^3 - 5955*x^2 - 3791*x + 11943)
 
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^8 - 56*x^6 - 34*x^5 + 1007*x^4 + 952*x^3 - 5955*x^2 - 3791*x + 11943, 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^8 - 56*x^6 - 34*x^5 + 1007*x^4 + 952*x^3 - 5955*x^2 - 3791*x + 11943);
 
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^8 - 56*x^6 - 34*x^5 + 1007*x^4 + 952*x^3 - 5955*x^2 - 3791*x + 11943);
 
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

$S_4$ (as 8T14):

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 24
The 5 conjugacy class representatives for $S_4$
Character table for $S_4$

Intermediate fields

\(\Q(\sqrt{1957}) \), 4.4.1957.1

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 fields

Galois closure: deg 24
Degree 4 sibling: 4.4.1957.1
Degree 6 siblings: 6.6.3829849.1, 6.6.7495014493.2
Degree 12 siblings: 12.12.28704773761001557.1, deg 12
Minimal sibling: 4.4.1957.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.4.0.1}{4} }^{2}$ ${\href{/padicField/3.3.0.1}{3} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ ${\href{/padicField/5.4.0.1}{4} }^{2}$ ${\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ ${\href{/padicField/11.4.0.1}{4} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }^{2}$ ${\href{/padicField/17.2.0.1}{2} }^{4}$ R ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.4.0.1}{4} }^{2}$ ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }^{2}$ ${\href{/padicField/43.2.0.1}{2} }^{4}$ ${\href{/padicField/47.2.0.1}{2} }^{4}$ ${\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{4}$

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
\(19\) Copy content Toggle raw display 19.4.2.1$x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(103\) Copy content Toggle raw display 103.4.2.1$x^{4} + 204 x^{3} + 10620 x^{2} + 22032 x + 1081216$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
103.4.2.1$x^{4} + 204 x^{3} + 10620 x^{2} + 22032 x + 1081216$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$