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Properties

Label 8.0.1945064112500161.2
Degree $8$
Signature $[0, 4]$
Discriminant $1.945\times 10^{15}$
Root discriminant \(81.49\)
Ramified primes $29,229$
Class number $4$ (GRH)
Class group [2, 2] (GRH)
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 + 88*x^6 - 399*x^5 + 26075*x^4 - 17556*x^3 + 1020564*x^2 - 5489792*x + 144276560)
 
gp: K = bnfinit(y^8 + 88*y^6 - 399*y^5 + 26075*y^4 - 17556*y^3 + 1020564*y^2 - 5489792*y + 144276560, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^8 + 88*x^6 - 399*x^5 + 26075*x^4 - 17556*x^3 + 1020564*x^2 - 5489792*x + 144276560);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^8 + 88*x^6 - 399*x^5 + 26075*x^4 - 17556*x^3 + 1020564*x^2 - 5489792*x + 144276560)
 

\( x^{8} + 88x^{6} - 399x^{5} + 26075x^{4} - 17556x^{3} + 1020564x^{2} - 5489792x + 144276560 \) 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:   \(1945064112500161\) \(\medspace = 29^{4}\cdot 229^{4}\) 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:  \(81.49\)
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:  $29^{1/2}229^{1/2}\approx 81.49233092751734$
Ramified primes:   \(29\), \(229\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{6}-\frac{1}{4}a^{5}+\frac{1}{8}a^{3}+\frac{1}{8}a^{2}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{87\!\cdots\!96}a^{7}+\frac{10\!\cdots\!45}{43\!\cdots\!48}a^{6}-\frac{40\!\cdots\!21}{10\!\cdots\!62}a^{5}-\frac{22\!\cdots\!15}{87\!\cdots\!96}a^{4}-\frac{87\!\cdots\!99}{87\!\cdots\!96}a^{3}-\frac{39\!\cdots\!59}{43\!\cdots\!48}a^{2}-\frac{89\!\cdots\!39}{21\!\cdots\!24}a-\frac{10\!\cdots\!66}{54\!\cdots\!31}$ 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

$C_{2}\times C_{2}$, which has order $4$ (assuming GRH)

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:   $\frac{73967535921143}{87\!\cdots\!96}a^{7}+\frac{12\!\cdots\!61}{43\!\cdots\!48}a^{6}-\frac{494798522384759}{10\!\cdots\!62}a^{5}+\frac{19\!\cdots\!87}{87\!\cdots\!96}a^{4}+\frac{15\!\cdots\!51}{87\!\cdots\!96}a^{3}+\frac{18\!\cdots\!17}{43\!\cdots\!48}a^{2}-\frac{27\!\cdots\!01}{21\!\cdots\!24}a+\frac{65\!\cdots\!67}{54\!\cdots\!31}$, $\frac{2392157652766}{33483773}a^{7}-\frac{9910367418602}{33483773}a^{6}+\frac{251567109891902}{33483773}a^{5}-\frac{19\!\cdots\!28}{33483773}a^{4}+\frac{37\!\cdots\!48}{33483773}a^{3}-\frac{19\!\cdots\!52}{33483773}a^{2}+\frac{17\!\cdots\!76}{33483773}a-\frac{23\!\cdots\!03}{33483773}$, $\frac{57\!\cdots\!55}{54\!\cdots\!31}a^{7}+\frac{13\!\cdots\!55}{21\!\cdots\!24}a^{6}+\frac{37\!\cdots\!21}{54\!\cdots\!31}a^{5}-\frac{44\!\cdots\!93}{54\!\cdots\!31}a^{4}+\frac{27\!\cdots\!55}{21\!\cdots\!24}a^{3}+\frac{64\!\cdots\!73}{21\!\cdots\!24}a^{2}+\frac{20\!\cdots\!73}{54\!\cdots\!31}a-\frac{63\!\cdots\!31}{54\!\cdots\!31}$ Copy content Toggle raw display (assuming GRH)
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:  \( 5019.55943964 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 5019.55943964 \cdot 4}{2\cdot\sqrt{1945064112500161}}\cr\approx \mathstrut & 0.354770998226 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^8 + 88*x^6 - 399*x^5 + 26075*x^4 - 17556*x^3 + 1020564*x^2 - 5489792*x + 144276560)
 
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 + 88*x^6 - 399*x^5 + 26075*x^4 - 17556*x^3 + 1020564*x^2 - 5489792*x + 144276560, 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 + 88*x^6 - 399*x^5 + 26075*x^4 - 17556*x^3 + 1020564*x^2 - 5489792*x + 144276560);
 
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 + 88*x^6 - 399*x^5 + 26075*x^4 - 17556*x^3 + 1020564*x^2 - 5489792*x + 144276560);
 
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

\(\Q(\sqrt{6641}) \), 4.0.229.1

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: 6.2.1278983549.1, 6.2.292887232721.1
Degree 8 sibling: 8.0.37090522921.1
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: 6.2.1278983549.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.4.0.1}{4} }^{2}$ ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ ${\href{/padicField/5.3.0.1}{3} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ ${\href{/padicField/7.4.0.1}{4} }^{2}$ ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ ${\href{/padicField/13.4.0.1}{4} }^{2}$ ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ ${\href{/padicField/23.2.0.1}{2} }^{4}$ R ${\href{/padicField/31.4.0.1}{4} }^{2}$ ${\href{/padicField/37.2.0.1}{2} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{2}$ ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ ${\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ ${\href{/padicField/53.2.0.1}{2} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{2}$

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
\(29\) 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}$
29.4.2.1$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(229\) Copy content Toggle raw display Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $4$$2$$2$$2$