Properties

Label 8.4.567647723776.3
Degree $8$
Signature $[4, 2]$
Discriminant $567647723776$
Root discriminant \(29.46\)
Ramified primes $2,7,31$
Class number $1$
Class group trivial
Galois group $C_2^3:\GL(3,2)$ (as 8T48)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^7 - 8*x^6 + 4*x^5 + 12*x^4 + 28*x^3 - 56*x + 56)
 
gp: K = bnfinit(y^8 - 2*y^7 - 8*y^6 + 4*y^5 + 12*y^4 + 28*y^3 - 56*y + 56, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^8 - 2*x^7 - 8*x^6 + 4*x^5 + 12*x^4 + 28*x^3 - 56*x + 56);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^8 - 2*x^7 - 8*x^6 + 4*x^5 + 12*x^4 + 28*x^3 - 56*x + 56)
 

\( x^{8} - 2x^{7} - 8x^{6} + 4x^{5} + 12x^{4} + 28x^{3} - 56x + 56 \) 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:  $[4, 2]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(567647723776\) \(\medspace = 2^{8}\cdot 7^{4}\cdot 31^{4}\) 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:  \(29.46\)
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:  $2^{31/28}7^{3/4}31^{1/2}\approx 51.616385159869715$
Ramified primes:   \(2\), \(7\), \(31\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{4}a^{6}$, $\frac{1}{68}a^{7}+\frac{3}{68}a^{6}-\frac{5}{34}a^{5}-\frac{3}{17}a^{4}-\frac{7}{34}a^{3}-\frac{2}{17}a^{2}+\frac{7}{17}a+\frac{4}{17}$ 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

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:  $5$
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:   $\frac{6}{17}a^{7}-\frac{16}{17}a^{6}-\frac{26}{17}a^{5}+\frac{9}{34}a^{4}+\frac{35}{17}a^{3}+\frac{190}{17}a^{2}-\frac{53}{17}a+\frac{11}{17}$, $\frac{63}{68}a^{7}-\frac{253}{68}a^{6}-\frac{9}{34}a^{5}+\frac{183}{34}a^{4}+\frac{26}{17}a^{3}+\frac{367}{17}a^{2}-\frac{783}{17}a+\frac{507}{17}$, $\frac{6}{17}a^{7}-\frac{115}{68}a^{6}+\frac{8}{17}a^{5}+\frac{111}{34}a^{4}+\frac{53}{34}a^{3}+\frac{139}{17}a^{2}-\frac{393}{17}a+\frac{215}{17}$, $\frac{21}{34}a^{7}-\frac{73}{34}a^{6}-\frac{20}{17}a^{5}+\frac{44}{17}a^{4}+\frac{63}{34}a^{3}+\frac{256}{17}a^{2}-\frac{403}{17}a+\frac{151}{17}$, $\frac{1}{68}a^{7}+\frac{3}{68}a^{6}-\frac{5}{34}a^{5}-\frac{23}{34}a^{4}-\frac{12}{17}a^{3}+\frac{32}{17}a^{2}+\frac{58}{17}a-\frac{13}{17}$ 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:  \( 3004.84581333 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 3004.84581333 \cdot 1}{2\cdot\sqrt{567647723776}}\cr\approx \mathstrut & 1.25959945906 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^7 - 8*x^6 + 4*x^5 + 12*x^4 + 28*x^3 - 56*x + 56)
 
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 - 2*x^7 - 8*x^6 + 4*x^5 + 12*x^4 + 28*x^3 - 56*x + 56, 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 - 2*x^7 - 8*x^6 + 4*x^5 + 12*x^4 + 28*x^3 - 56*x + 56);
 
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 - 2*x^7 - 8*x^6 + 4*x^5 + 12*x^4 + 28*x^3 - 56*x + 56);
 
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

$C_2^3:\GL(3,2)$ (as 8T48):

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 1344
The 11 conjugacy class representatives for $C_2^3:\GL(3,2)$
Character table for $C_2^3:\GL(3,2)$

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 fields

Degree 8 sibling: 8.0.28943536384.2
Degree 14 siblings: deg 14, deg 14
Degree 28 siblings: deg 28, deg 28, deg 28
Degree 42 siblings: deg 42, some data not computed
Minimal sibling: 8.0.28943536384.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.7.0.1}{7} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ ${\href{/padicField/5.7.0.1}{7} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ R ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ ${\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ ${\href{/padicField/19.4.0.1}{4} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{2}$ ${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ R ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }$

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 2.8.8.13$x^{8} + 2 x + 2$$8$$1$$8$$C_2^3:(C_7: C_3)$$[8/7, 8/7, 8/7]_{7}^{3}$
\(7\) Copy content Toggle raw display $\Q_{7}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 4$$1$$1$$0$Trivial$[\ ]$
7.2.1.2$x^{2} + 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.4.3.2$x^{4} + 21$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
\(31\) Copy content Toggle raw display 31.8.4.1$x^{8} + 130 x^{6} + 32 x^{5} + 5967 x^{4} - 1888 x^{3} + 113114 x^{2} - 89184 x + 766577$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$