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Properties

Label 8.2.143864627200.6
Degree $8$
Signature $[2, 3]$
Discriminant $-143864627200$
Root discriminant \(24.82\)
Ramified primes $2,5,7$
Class number $4$
Class group [4]
Galois group $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T30)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^8 + 12*x^6 - 8*x^5 + 48*x^4 + 48*x^3 + 120*x^2 + 208*x + 68)
 
gp: K = bnfinit(y^8 + 12*y^6 - 8*y^5 + 48*y^4 + 48*y^3 + 120*y^2 + 208*y + 68, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^8 + 12*x^6 - 8*x^5 + 48*x^4 + 48*x^3 + 120*x^2 + 208*x + 68);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^8 + 12*x^6 - 8*x^5 + 48*x^4 + 48*x^3 + 120*x^2 + 208*x + 68)
 

\( x^{8} + 12x^{6} - 8x^{5} + 48x^{4} + 48x^{3} + 120x^{2} + 208x + 68 \) 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:  $[2, 3]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-143864627200\) \(\medspace = -\,2^{24}\cdot 5^{2}\cdot 7^{3}\) 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:  \(24.82\)
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:  $2^{3}5^{1/2}7^{1/2}\approx 47.32863826479693$
Ramified primes:   \(2\), \(5\), \(7\) 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:  \(\Q(\sqrt{-7}) \)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{4}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{8}a^{5}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{4}a$, $\frac{1}{8}a^{6}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}$, $\frac{1}{16136}a^{7}+\frac{107}{8068}a^{6}-\frac{583}{16136}a^{5}+\frac{71}{4034}a^{4}+\frac{1087}{4034}a^{3}-\frac{671}{2017}a^{2}+\frac{529}{8068}a-\frac{919}{2017}$ 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

$C_{4}$, which has order $4$

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:  $4$
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:   $\frac{85}{8068}a^{7}+\frac{37}{8068}a^{6}+\frac{435}{4034}a^{5}-\frac{16}{2017}a^{4}+\frac{1243}{4034}a^{3}+\frac{3815}{4034}a^{2}+\frac{1304}{2017}a+\frac{5130}{2017}$, $\frac{647}{16136}a^{7}-\frac{715}{16136}a^{6}+\frac{4023}{8068}a^{5}-\frac{6959}{8068}a^{4}+\frac{20899}{8068}a^{3}-\frac{3945}{8068}a^{2}+\frac{9424}{2017}a+\frac{4456}{2017}$, $\frac{73}{4034}a^{7}-\frac{39}{16136}a^{6}+\frac{1613}{8068}a^{5}-\frac{893}{8068}a^{4}+\frac{1376}{2017}a^{3}+\frac{12987}{8068}a^{2}+\frac{2605}{4034}a+\frac{9998}{2017}$, $\frac{209}{16136}a^{7}+\frac{44}{2017}a^{6}+\frac{3207}{16136}a^{5}+\frac{360}{2017}a^{4}+\frac{1648}{2017}a^{3}+\frac{3919}{4034}a^{2}+\frac{9711}{8068}a+\frac{3578}{2017}$ 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:  \( 174.513256564 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 174.513256564 \cdot 4}{2\cdot\sqrt{143864627200}}\cr\approx \mathstrut & 0.913021496075 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^8 + 12*x^6 - 8*x^5 + 48*x^4 + 48*x^3 + 120*x^2 + 208*x + 68)
 
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 + 12*x^6 - 8*x^5 + 48*x^4 + 48*x^3 + 120*x^2 + 208*x + 68, 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(Rationals()); K<a> := NumberField(x^8 + 12*x^6 - 8*x^5 + 48*x^4 + 48*x^3 + 120*x^2 + 208*x + 68);
 
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 + 12*x^6 - 8*x^5 + 48*x^4 + 48*x^3 + 120*x^2 + 208*x + 68);
 
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

$C_4^2:C_4$ (as 8T30):

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 64
The 13 conjugacy class representatives for $(((C_4 \times C_2): C_2):C_2):C_2$
Character table for $(((C_4 \times C_2): C_2):C_2):C_2$

Intermediate fields

\(\Q(\sqrt{2}) \), 4.2.14336.1

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 32 siblings: data not computed
Minimal sibling: 8.2.112394240000.6

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ R R ${\href{/padicField/11.4.0.1}{4} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{2}$ ${\href{/padicField/29.4.0.1}{4} }^{2}$ ${\href{/padicField/31.2.0.1}{2} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$

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
\(2\) Copy content Toggle raw display 2.1.8.24c1.61$x^{8} + 8 x^{7} + 4 x^{6} + 2 x^{4} + 8 x^{3} + 4 x^{2} + 8 x + 14$$8$$1$$24$$C_4\times C_2$$$[2, 3, 4]$$
\(5\) Copy content Toggle raw display 5.2.1.0a1.1$x^{2} + 4 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
5.2.1.0a1.1$x^{2} + 4 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
5.2.2.2a1.1$x^{4} + 8 x^{3} + 20 x^{2} + 21 x + 4$$2$$2$$2$$C_4$$$[\ ]_{2}^{2}$$
\(7\) Copy content Toggle raw display $\Q_{7}$$x + 4$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{7}$$x + 4$$1$$1$$0$Trivial$$[\ ]$$
7.1.2.1a1.1$x^{2} + 7$$2$$1$$1$$C_2$$$[\ ]_{2}$$
7.2.2.2a1.2$x^{4} + 12 x^{3} + 42 x^{2} + 36 x + 16$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)