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Properties

Label 8.6.307235186944.2
Degree $8$
Signature $[6, 1]$
Discriminant $-307235186944$
Root discriminant \(27.29\)
Ramified primes $2,7,101$
Class number $1$
Class group trivial
Galois group $C_2 \wr S_4$ (as 8T44)

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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 - 3*x^6 - 6*x^5 + 8*x^4 + 26*x^3 - 26*x^2 + 2*x + 2)
 
gp: K = bnfinit(y^8 - 2*y^7 - 3*y^6 - 6*y^5 + 8*y^4 + 26*y^3 - 26*y^2 + 2*y + 2, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^8 - 2*x^7 - 3*x^6 - 6*x^5 + 8*x^4 + 26*x^3 - 26*x^2 + 2*x + 2);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^8 - 2*x^7 - 3*x^6 - 6*x^5 + 8*x^4 + 26*x^3 - 26*x^2 + 2*x + 2)
 

\( x^{8} - 2x^{7} - 3x^{6} - 6x^{5} + 8x^{4} + 26x^{3} - 26x^{2} + 2x + 2 \) 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:  $[6, 1]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-307235186944\) \(\medspace = -\,2^{8}\cdot 7^{6}\cdot 101^{2}\) 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:  \(27.29\)
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:  $2^{19/12}7^{3/4}101^{1/2}\approx 129.60299666924007$
Ramified primes:   \(2\), \(7\), \(101\) 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:  \(\Q(\sqrt{-1}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{127}a^{7}-\frac{48}{127}a^{6}+\frac{46}{127}a^{5}+\frac{37}{127}a^{4}-\frac{43}{127}a^{3}-\frac{28}{127}a^{2}-\frac{8}{127}a-\frac{11}{127}$ 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:  $6$
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:   $\frac{46}{127}a^{7}-\frac{49}{127}a^{6}-\frac{170}{127}a^{5}-\frac{457}{127}a^{4}-\frac{73}{127}a^{3}+\frac{998}{127}a^{2}-\frac{241}{127}a-\frac{125}{127}$, $\frac{199}{127}a^{7}-\frac{281}{127}a^{6}-\frac{752}{127}a^{5}-\frac{1654}{127}a^{4}+\frac{587}{127}a^{3}+\frac{5477}{127}a^{2}-\frac{1846}{127}a-\frac{411}{127}$, $\frac{117}{127}a^{7}-\frac{155}{127}a^{6}-\frac{460}{127}a^{5}-\frac{1005}{127}a^{4}+\frac{303}{127}a^{3}+\frac{3201}{127}a^{2}-\frac{1063}{127}a-\frac{271}{127}$, $\frac{82}{127}a^{7}-\frac{126}{127}a^{6}-\frac{292}{127}a^{5}-\frac{649}{127}a^{4}+\frac{284}{127}a^{3}+\frac{2276}{127}a^{2}-\frac{1037}{127}a-\frac{267}{127}$, $\frac{47}{127}a^{7}-\frac{97}{127}a^{6}-\frac{124}{127}a^{5}-\frac{293}{127}a^{4}+\frac{392}{127}a^{3}+\frac{1097}{127}a^{2}-\frac{1138}{127}a+\frac{245}{127}$, $\frac{199}{127}a^{7}-\frac{281}{127}a^{6}-\frac{752}{127}a^{5}-\frac{1654}{127}a^{4}+\frac{587}{127}a^{3}+\frac{5477}{127}a^{2}-\frac{1846}{127}a-\frac{665}{127}$ 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:  \( 2009.20325773 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{1}\cdot 2009.20325773 \cdot 1}{2\cdot\sqrt{307235186944}}\cr\approx \mathstrut & 0.728816579864 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^7 - 3*x^6 - 6*x^5 + 8*x^4 + 26*x^3 - 26*x^2 + 2*x + 2)
 
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 - 3*x^6 - 6*x^5 + 8*x^4 + 26*x^3 - 26*x^2 + 2*x + 2, 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 - 3*x^6 - 6*x^5 + 8*x^4 + 26*x^3 - 26*x^2 + 2*x + 2);
 
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 - 3*x^6 - 6*x^5 + 8*x^4 + 26*x^3 - 26*x^2 + 2*x + 2);
 
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

$C_2\wr S_4$ (as 8T44):

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 384
The 20 conjugacy class representatives for $C_2 \wr S_4$
Character table for $C_2 \wr S_4$

Intermediate fields

4.4.19796.1

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

Degree 8 siblings: data not computed
Degree 16 siblings: data not computed
Degree 24 siblings: data not computed
Degree 32 siblings: data not computed
Minimal sibling: 8.2.307235186944.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.8.0.1}{8} }$ ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ R ${\href{/padicField/11.8.0.1}{8} }$ ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ ${\href{/padicField/29.4.0.1}{4} }^{2}$ ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ ${\href{/padicField/37.2.0.1}{2} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ ${\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ ${\href{/padicField/53.2.0.1}{2} }^{4}$ ${\href{/padicField/59.8.0.1}{8} }$

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 2.2.2.2$x^{2} + 2 x + 6$$2$$1$$2$$C_2$$[2]$
2.6.6.8$x^{6} + 2 x + 2$$6$$1$$6$$S_4$$[4/3, 4/3]_{3}^{2}$
\(7\) Copy content Toggle raw display 7.8.6.3$x^{8} - 154 x^{4} - 1421$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
\(101\) Copy content Toggle raw display 101.2.0.1$x^{2} + 97 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.0.1$x^{2} + 97 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
101.4.2.1$x^{4} + 16556 x^{3} + 69319047 x^{2} + 6570770114 x + 216554003$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$