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Properties

Label 16.0.405...864.3
Degree $16$
Signature $[0, 8]$
Discriminant $4.057\times 10^{20}$
Root discriminant \(19.41\)
Ramified primes $2,7$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $D_8:C_2$ (as 16T45)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^12 + 13*x^8 + 12*x^4 + 4)
 
gp: K = bnfinit(y^16 - 8*y^12 + 13*y^8 + 12*y^4 + 4, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^12 + 13*x^8 + 12*x^4 + 4);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^12 + 13*x^8 + 12*x^4 + 4)
 

\( x^{16} - 8x^{12} + 13x^{8} + 12x^{4} + 4 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(405661806804141604864\) \(\medspace = 2^{46}\cdot 7^{8}\) 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:  \(19.41\)
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:  $2^{3}7^{1/2}\approx 21.166010488516726$
Ramified primes:   \(2\), \(7\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $4$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{8}a^{12}-\frac{1}{2}a^{10}+\frac{1}{4}a^{8}+\frac{1}{8}a^{4}-\frac{1}{2}a^{2}-\frac{1}{4}$, $\frac{1}{8}a^{13}-\frac{1}{2}a^{11}+\frac{1}{4}a^{9}+\frac{1}{8}a^{5}-\frac{1}{2}a^{3}-\frac{1}{4}a$, $\frac{1}{16}a^{14}+\frac{1}{8}a^{10}-\frac{1}{2}a^{8}+\frac{1}{16}a^{6}+\frac{3}{8}a^{2}-\frac{1}{2}$, $\frac{1}{16}a^{15}+\frac{1}{8}a^{11}-\frac{1}{2}a^{9}+\frac{1}{16}a^{7}+\frac{3}{8}a^{3}-\frac{1}{2}a$ 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 $1$ (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:  $7$
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:   $\frac{3}{16}a^{14}-\frac{13}{8}a^{10}+\frac{1}{2}a^{8}+\frac{51}{16}a^{6}-2a^{4}+\frac{9}{8}a^{2}-\frac{1}{2}$, $\frac{1}{8}a^{14}-\frac{3}{4}a^{10}+\frac{9}{8}a^{6}-\frac{1}{4}a^{2}$, $\frac{1}{8}a^{14}-\frac{1}{4}a^{12}-\frac{3}{4}a^{10}+\frac{3}{2}a^{8}+\frac{9}{8}a^{6}-\frac{5}{4}a^{4}-\frac{5}{4}a^{2}-\frac{3}{2}$, $\frac{3}{16}a^{15}-\frac{1}{16}a^{14}-\frac{1}{8}a^{13}-\frac{9}{8}a^{11}-\frac{1}{8}a^{10}+\frac{5}{4}a^{9}-\frac{1}{2}a^{8}+\frac{19}{16}a^{7}+\frac{31}{16}a^{6}-\frac{25}{8}a^{5}+2a^{4}+\frac{5}{8}a^{3}-\frac{3}{8}a^{2}-\frac{1}{4}a+\frac{1}{2}$, $\frac{3}{16}a^{15}+\frac{3}{16}a^{14}-\frac{3}{8}a^{12}-\frac{13}{8}a^{11}-\frac{9}{8}a^{10}+\frac{1}{2}a^{9}+\frac{7}{4}a^{8}+\frac{51}{16}a^{7}+\frac{19}{16}a^{6}-2a^{5}-\frac{3}{8}a^{4}+\frac{9}{8}a^{3}+\frac{5}{8}a^{2}+\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{4}a^{15}-\frac{1}{8}a^{14}-\frac{3}{8}a^{13}-2a^{11}+\frac{3}{4}a^{10}+\frac{9}{4}a^{9}+a^{8}+\frac{17}{4}a^{7}-\frac{1}{8}a^{6}-\frac{19}{8}a^{5}-4a^{4}-a^{3}-\frac{11}{4}a^{2}-\frac{9}{4}a-2$, $\frac{5}{16}a^{15}-\frac{1}{16}a^{14}-\frac{1}{8}a^{13}+\frac{1}{8}a^{12}-\frac{23}{8}a^{11}+\frac{3}{8}a^{10}+\frac{5}{4}a^{9}-\frac{5}{4}a^{8}+\frac{101}{16}a^{7}-\frac{1}{16}a^{6}-\frac{25}{8}a^{5}+\frac{25}{8}a^{4}+\frac{11}{8}a^{3}-\frac{15}{8}a^{2}-\frac{1}{4}a+\frac{1}{4}$ 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:  \( 10856.0619541 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 10856.0619541 \cdot 1}{2\cdot\sqrt{405661806804141604864}}\cr\approx \mathstrut & 0.654634974201 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^12 + 13*x^8 + 12*x^4 + 4)
 
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^16 - 8*x^12 + 13*x^8 + 12*x^4 + 4, 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^16 - 8*x^12 + 13*x^8 + 12*x^4 + 4);
 
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^16 - 8*x^12 + 13*x^8 + 12*x^4 + 4);
 
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

$D_8:C_2$ (as 16T45):

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 32
The 11 conjugacy class representatives for $D_8:C_2$
Character table for $D_8:C_2$

Intermediate fields

\(\Q(\sqrt{-7}) \), \(\Q(\sqrt{14}) \), \(\Q(\sqrt{-2}) \), 4.0.1568.1, 4.0.392.1, \(\Q(\sqrt{-2}, \sqrt{-7})\), 8.0.39337984.3

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 32
Degree 8 siblings: 8.2.5754585088.4, 8.2.5754585088.5
Degree 16 siblings: 16.4.1622647227216566419456.1, 16.0.1622647227216566419456.6, 16.0.33115249535031967744.3
Minimal sibling: 8.2.5754585088.4

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.8.0.1}{8} }^{2}$ ${\href{/padicField/5.8.0.1}{8} }^{2}$ R ${\href{/padicField/11.2.0.1}{2} }^{6}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ ${\href{/padicField/13.8.0.1}{8} }^{2}$ ${\href{/padicField/17.2.0.1}{2} }^{8}$ ${\href{/padicField/19.8.0.1}{8} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.2.0.1}{2} }^{6}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.8.0.1}{8} }^{2}$

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 2.8.22.10$x^{8} - 8 x^{7} + 88 x^{6} - 96 x^{5} + 204 x^{4} - 240 x^{3} + 720 x^{2} + 900$$4$$2$$22$$Q_8:C_2$$[2, 3, 4]^{2}$
2.8.24.29$x^{8} + 8 x^{7} + 14 x^{4} + 12 x^{2} + 8 x + 6$$8$$1$$24$$Q_8:C_2$$[2, 3, 4]^{2}$
\(7\) Copy content Toggle raw display 7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$