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Properties

Label 8.0.359710104300625.2
Degree $8$
Signature $[0, 4]$
Discriminant $3.597\times 10^{14}$
Root discriminant \(65.99\)
Ramified primes $5,13,67$
Class number $72$
Class group [2, 36]
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 - 3*x^7 + 19*x^6 - 41*x^5 - 32*x^4 + 90*x^3 + 973*x^2 + 3173*x + 5054)
 
gp: K = bnfinit(y^8 - 3*y^7 + 19*y^6 - 41*y^5 - 32*y^4 + 90*y^3 + 973*y^2 + 3173*y + 5054, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^8 - 3*x^7 + 19*x^6 - 41*x^5 - 32*x^4 + 90*x^3 + 973*x^2 + 3173*x + 5054);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^8 - 3*x^7 + 19*x^6 - 41*x^5 - 32*x^4 + 90*x^3 + 973*x^2 + 3173*x + 5054)
 

\( x^{8} - 3x^{7} + 19x^{6} - 41x^{5} - 32x^{4} + 90x^{3} + 973x^{2} + 3173x + 5054 \) 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:  $[0, 4]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(359710104300625\) \(\medspace = 5^{4}\cdot 13^{4}\cdot 67^{4}\) 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:  \(65.99\)
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:  $5^{1/2}13^{1/2}67^{1/2}\approx 65.99242380758567$
Ramified primes:   \(5\), \(13\), \(67\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{6}a^{5}+\frac{1}{6}a^{3}-\frac{1}{6}a^{2}+\frac{1}{6}a-\frac{1}{3}$, $\frac{1}{54}a^{6}-\frac{2}{27}a^{5}-\frac{11}{54}a^{4}+\frac{25}{54}a^{3}-\frac{7}{54}a^{2}-\frac{4}{9}a+\frac{10}{27}$, $\frac{1}{52624566}a^{7}+\frac{188173}{52624566}a^{6}+\frac{54745}{1384857}a^{5}-\frac{3083786}{8770761}a^{4}-\frac{11779595}{52624566}a^{3}-\frac{1320550}{26312283}a^{2}+\frac{12310199}{52624566}a-\frac{607180}{1384857}$ 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

$C_{2}\times C_{36}$, which has order $72$

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:  $3$
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:   $\frac{21365}{26312283}a^{7}-\frac{206489}{52624566}a^{6}+\frac{25213}{1384857}a^{5}-\frac{747991}{17541522}a^{4}-\frac{789437}{52624566}a^{3}+\frac{20618425}{52624566}a^{2}+\frac{10285876}{26312283}a-\frac{1249399}{1384857}$, $\frac{839}{26312283}a^{7}+\frac{3449}{26312283}a^{6}-\frac{71}{1384857}a^{5}+\frac{156082}{8770761}a^{4}+\frac{1567442}{26312283}a^{3}+\frac{3119633}{26312283}a^{2}+\frac{5071264}{26312283}a-\frac{54907}{1384857}$, $\frac{48157424413}{52624566}a^{7}-\frac{306581745970}{26312283}a^{6}+\frac{23248070734}{1384857}a^{5}+\frac{602181429191}{17541522}a^{4}+\frac{1448030440601}{26312283}a^{3}-\frac{13975971502661}{52624566}a^{2}-\frac{72531656444221}{52624566}a-\frac{3295157464972}{1384857}$ 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:  \( 1454.7913466 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 1454.7913466 \cdot 72}{2\cdot\sqrt{359710104300625}}\cr\approx \mathstrut & 4.3037433497 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^8 - 3*x^7 + 19*x^6 - 41*x^5 - 32*x^4 + 90*x^3 + 973*x^2 + 3173*x + 5054)
 
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 - 3*x^7 + 19*x^6 - 41*x^5 - 32*x^4 + 90*x^3 + 973*x^2 + 3173*x + 5054, 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 - 3*x^7 + 19*x^6 - 41*x^5 - 32*x^4 + 90*x^3 + 973*x^2 + 3173*x + 5054);
 
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 - 3*x^7 + 19*x^6 - 41*x^5 - 32*x^4 + 90*x^3 + 973*x^2 + 3173*x + 5054);
 
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

\(\Q(\sqrt{-335}) \), 4.2.56615.1

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: 4.2.56615.1
Degree 6 siblings: 6.2.18966025.1, 6.0.6353618375.1
Degree 12 siblings: deg 12, deg 12
Minimal sibling: 4.2.56615.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.3.0.1}{3} }^{2}{,}\,{\href{/padicField/2.1.0.1}{1} }^{2}$ ${\href{/padicField/3.3.0.1}{3} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ R ${\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ ${\href{/padicField/11.4.0.1}{4} }^{2}$ R ${\href{/padicField/17.4.0.1}{4} }^{2}$ ${\href{/padicField/19.1.0.1}{1} }^{8}$ ${\href{/padicField/23.4.0.1}{4} }^{2}$ ${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{4}$ ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.2.0.1}{2} }^{4}$ ${\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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
\(5\) Copy content Toggle raw display 5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(13\) Copy content Toggle raw display 13.4.2.1$x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(67\) Copy content Toggle raw display 67.2.1.1$x^{2} + 134$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.1$x^{2} + 134$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.1$x^{2} + 134$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.1$x^{2} + 134$$2$$1$$1$$C_2$$[\ ]_{2}$