Properties

Label 8.4.127027375281.1
Degree $8$
Signature $[4, 2]$
Discriminant $127027375281$
Root discriminant \(24.43\)
Ramified primes $3,199$
Class number $1$
Class group trivial
Galois group $S_4\times C_2$ (as 8T24)

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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 + x^6 - 12*x^5 - 17*x^4 + 29*x^3 + 36*x^2 + 174*x + 61)
 
gp: K = bnfinit(y^8 - 2*y^7 + y^6 - 12*y^5 - 17*y^4 + 29*y^3 + 36*y^2 + 174*y + 61, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^8 - 2*x^7 + x^6 - 12*x^5 - 17*x^4 + 29*x^3 + 36*x^2 + 174*x + 61);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^8 - 2*x^7 + x^6 - 12*x^5 - 17*x^4 + 29*x^3 + 36*x^2 + 174*x + 61)
 

\( x^{8} - 2x^{7} + x^{6} - 12x^{5} - 17x^{4} + 29x^{3} + 36x^{2} + 174x + 61 \) 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:   \(127027375281\) \(\medspace = 3^{4}\cdot 199^{4}\) 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:  \(24.43\)
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:  $3^{1/2}199^{1/2}\approx 24.43358344574123$
Ramified primes:   \(3\), \(199\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7}a^{6}+\frac{2}{7}a^{5}+\frac{3}{7}a^{4}+\frac{2}{7}a^{3}+\frac{1}{7}a^{2}+\frac{2}{7}$, $\frac{1}{72331}a^{7}+\frac{4125}{72331}a^{6}-\frac{35907}{72331}a^{5}-\frac{33647}{72331}a^{4}-\frac{26998}{72331}a^{3}-\frac{1473}{10333}a^{2}-\frac{12500}{72331}a-\frac{2189}{10333}$ 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:   $\frac{44}{10333}a^{7}-\frac{459}{72331}a^{6}-\frac{3046}{72331}a^{5}+\frac{723}{72331}a^{4}-\frac{7652}{72331}a^{3}+\frac{37775}{72331}a^{2}+\frac{7982}{10333}a+\frac{44029}{72331}$, $\frac{33}{72331}a^{7}+\frac{1796}{72331}a^{6}-\frac{6969}{72331}a^{5}+\frac{5613}{72331}a^{4}-\frac{328}{10333}a^{3}-\frac{40606}{72331}a^{2}+\frac{93817}{72331}a+\frac{21324}{72331}$, $\frac{891}{72331}a^{7}-\frac{3173}{72331}a^{6}-\frac{2169}{72331}a^{5}-\frac{492}{10333}a^{4}-\frac{20660}{72331}a^{3}+\frac{81600}{72331}a^{2}+\frac{73805}{72331}a+\frac{38432}{72331}$, $\frac{583}{72331}a^{7}-\frac{2714}{72331}a^{6}+\frac{877}{72331}a^{5}-\frac{4167}{72331}a^{4}-\frac{13008}{72331}a^{3}+\frac{43825}{72331}a^{2}+\frac{17931}{72331}a-\frac{5597}{72331}$, $\frac{907}{10333}a^{7}-\frac{25196}{72331}a^{6}+\frac{24102}{72331}a^{5}-\frac{52026}{72331}a^{4}-\frac{47830}{72331}a^{3}+\frac{470334}{72331}a^{2}-\frac{12532}{10333}a+\frac{226794}{72331}$ 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:  \( 367.02107817 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 367.02107817 \cdot 1}{2\cdot\sqrt{127027375281}}\cr\approx \mathstrut & 0.32523110008 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^7 + x^6 - 12*x^5 - 17*x^4 + 29*x^3 + 36*x^2 + 174*x + 61)
 
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 + x^6 - 12*x^5 - 17*x^4 + 29*x^3 + 36*x^2 + 174*x + 61, 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 + x^6 - 12*x^5 - 17*x^4 + 29*x^3 + 36*x^2 + 174*x + 61);
 
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 + x^6 - 12*x^5 - 17*x^4 + 29*x^3 + 36*x^2 + 174*x + 61);
 
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

$C_2\times S_4$ (as 8T24):

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 48
The 10 conjugacy class representatives for $S_4\times C_2$
Character table for $S_4\times C_2$

Intermediate fields

\(\Q(\sqrt{597}) \), 4.2.1791.1

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 6 siblings: 6.2.23641797.1, 6.0.118803.1
Degree 8 sibling: 8.0.3207681.1
Degree 12 siblings: deg 12, deg 12, deg 12, deg 12, deg 12, deg 12
Degree 16 sibling: deg 16
Degree 24 siblings: deg 24, deg 24, deg 24, deg 24
Minimal sibling: 6.0.118803.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.2.0.1}{2} }$ R ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ ${\href{/padicField/7.2.0.1}{2} }^{4}$ ${\href{/padicField/11.4.0.1}{4} }^{2}$ ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ ${\href{/padicField/17.4.0.1}{4} }^{2}$ ${\href{/padicField/19.4.0.1}{4} }^{2}$ ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ ${\href{/padicField/29.2.0.1}{2} }^{4}$ ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ ${\href{/padicField/59.4.0.1}{4} }^{2}$

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
\(3\) Copy content Toggle raw display 3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
\(199\) Copy content Toggle raw display 199.4.2.1$x^{4} + 386 x^{3} + 37653 x^{2} + 77972 x + 7450967$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
199.4.2.1$x^{4} + 386 x^{3} + 37653 x^{2} + 77972 x + 7450967$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$