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Properties

Label 8.8.321368580683536.1
Degree $8$
Signature $[8, 0]$
Discriminant $3.214\times 10^{14}$
Root discriminant \(65.07\)
Ramified primes $2,29,73$
Class number $2$
Class group [2]
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 - 4*x^7 - 88*x^6 + 230*x^5 + 1937*x^4 - 1942*x^3 - 14522*x^2 - 8352*x + 1363)
 
gp: K = bnfinit(y^8 - 4*y^7 - 88*y^6 + 230*y^5 + 1937*y^4 - 1942*y^3 - 14522*y^2 - 8352*y + 1363, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^8 - 4*x^7 - 88*x^6 + 230*x^5 + 1937*x^4 - 1942*x^3 - 14522*x^2 - 8352*x + 1363);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^8 - 4*x^7 - 88*x^6 + 230*x^5 + 1937*x^4 - 1942*x^3 - 14522*x^2 - 8352*x + 1363)
 

\( x^{8} - 4x^{7} - 88x^{6} + 230x^{5} + 1937x^{4} - 1942x^{3} - 14522x^{2} - 8352x + 1363 \) 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:   \(321368580683536\) \(\medspace = 2^{4}\cdot 29^{4}\cdot 73^{4}\) 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:  \(65.07\)
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:  $2^{2/3}29^{1/2}73^{1/2}\approx 73.03770071172075$
Ramified primes:   \(2\), \(29\), \(73\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{12518918198}a^{7}+\frac{1506517815}{6259459099}a^{6}+\frac{6157568791}{12518918198}a^{5}+\frac{4984829625}{12518918198}a^{4}-\frac{631480639}{12518918198}a^{3}-\frac{245330313}{12518918198}a^{2}+\frac{807380763}{6259459099}a-\frac{3745891699}{12518918198}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$

Class group and class number

$C_{2}$, which has order $2$

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:   $\frac{2240783}{6259459099}a^{7}+\frac{23112207}{6259459099}a^{6}-\frac{299395139}{6259459099}a^{5}-\frac{1907600838}{6259459099}a^{4}+\frac{8502678702}{6259459099}a^{3}+\frac{23778991893}{6259459099}a^{2}-\frac{31528630379}{6259459099}a-\frac{27387187980}{6259459099}$, $\frac{15859759}{6259459099}a^{7}-\frac{38504521}{6259459099}a^{6}-\frac{1249399734}{6259459099}a^{5}+\frac{1234206971}{6259459099}a^{4}+\frac{16329612197}{6259459099}a^{3}+\frac{6395822932}{6259459099}a^{2}-\frac{9491207000}{6259459099}a+\frac{12365587191}{6259459099}$, $\frac{13618976}{6259459099}a^{7}-\frac{61616728}{6259459099}a^{6}-\frac{950004595}{6259459099}a^{5}+\frac{3141807809}{6259459099}a^{4}+\frac{7826933495}{6259459099}a^{3}-\frac{17383168961}{6259459099}a^{2}+\frac{22037423379}{6259459099}a+\frac{33493316072}{6259459099}$, $\frac{157454329}{12518918198}a^{7}+\frac{240549743}{12518918198}a^{6}-\frac{6906207671}{6259459099}a^{5}-\frac{36942071063}{12518918198}a^{4}+\frac{100402799490}{6259459099}a^{3}+\frac{705158720119}{12518918198}a^{2}+\frac{370423536011}{12518918198}a-\frac{30055257176}{6259459099}$, $\frac{56160599}{12518918198}a^{7}+\frac{26236439}{6259459099}a^{6}-\frac{4745582563}{12518918198}a^{5}-\frac{10593261749}{12518918198}a^{4}+\frac{61738916311}{12518918198}a^{3}+\frac{211216171947}{12518918198}a^{2}+\frac{78984992739}{6259459099}a+\frac{22799419917}{12518918198}$, $\frac{8077939}{12518918198}a^{7}+\frac{4675772}{6259459099}a^{6}-\frac{1156328603}{12518918198}a^{5}-\frac{823396917}{12518918198}a^{4}+\frac{46554452435}{12518918198}a^{3}+\frac{19004233087}{12518918198}a^{2}-\frac{233600852167}{6259459099}a-\frac{483793016421}{12518918198}$, $\frac{17121479}{12518918198}a^{7}-\frac{93689383}{12518918198}a^{6}-\frac{758917116}{6259459099}a^{5}+\frac{6041659563}{12518918198}a^{4}+\frac{18692971963}{6259459099}a^{3}-\frac{57849381769}{12518918198}a^{2}-\frac{343153169839}{12518918198}a-\frac{128039162703}{6259459099}$ 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:  \( 30662.021125 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{0}\cdot 30662.021125 \cdot 2}{2\cdot\sqrt{321368580683536}}\cr\approx \mathstrut & 0.43786379465 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^8 - 4*x^7 - 88*x^6 + 230*x^5 + 1937*x^4 - 1942*x^3 - 14522*x^2 - 8352*x + 1363)
 
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 - 4*x^7 - 88*x^6 + 230*x^5 + 1937*x^4 - 1942*x^3 - 14522*x^2 - 8352*x + 1363, 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 - 4*x^7 - 88*x^6 + 230*x^5 + 1937*x^4 - 1942*x^3 - 14522*x^2 - 8352*x + 1363);
 
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 - 4*x^7 - 88*x^6 + 230*x^5 + 1937*x^4 - 1942*x^3 - 14522*x^2 - 8352*x + 1363);
 
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

\(\Q(\sqrt{2117}) \), 4.4.8468.1

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: 4.4.8468.1
Degree 6 siblings: 6.6.71707024.1, 6.6.151803769808.2
Degree 12 siblings: deg 12, deg 12
Minimal sibling: 4.4.8468.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.4.0.1}{4} }^{2}$ ${\href{/padicField/5.4.0.1}{4} }^{2}$ ${\href{/padicField/7.4.0.1}{4} }^{2}$ ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }^{2}$ ${\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ ${\href{/padicField/19.4.0.1}{4} }^{2}$ ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ R ${\href{/padicField/31.2.0.1}{2} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }^{2}$ ${\href{/padicField/43.2.0.1}{2} }^{4}$ ${\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.2.0.1}{2} }^{4}$ ${\href{/padicField/59.2.0.1}{2} }^{4}$

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
\(2\) Copy content Toggle raw display 2.2.0.1$x^{2} + x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
\(29\) Copy content Toggle raw display 29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
\(73\) Copy content Toggle raw display 73.2.1.2$x^{2} + 365$$2$$1$$1$$C_2$$[\ ]_{2}$
73.2.1.2$x^{2} + 365$$2$$1$$1$$C_2$$[\ ]_{2}$
73.2.1.2$x^{2} + 365$$2$$1$$1$$C_2$$[\ ]_{2}$
73.2.1.2$x^{2} + 365$$2$$1$$1$$C_2$$[\ ]_{2}$