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Properties

Label 16.0.746...256.3
Degree $16$
Signature $[0, 8]$
Discriminant $7.466\times 10^{18}$
Root discriminant \(15.12\)
Ramified primes $2,3,11$
Class number $1$
Class group trivial
Galois group $D_4\times C_2$ (as 16T9)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^14 + x^12 + 8*x^10 - 20*x^8 + 32*x^6 + 16*x^4 - 64*x^2 + 256)
 
gp: K = bnfinit(y^16 - y^14 + y^12 + 8*y^10 - 20*y^8 + 32*y^6 + 16*y^4 - 64*y^2 + 256, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - x^14 + x^12 + 8*x^10 - 20*x^8 + 32*x^6 + 16*x^4 - 64*x^2 + 256);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - x^14 + x^12 + 8*x^10 - 20*x^8 + 32*x^6 + 16*x^4 - 64*x^2 + 256)
 

\( x^{16} - x^{14} + x^{12} + 8x^{10} - 20x^{8} + 32x^{6} + 16x^{4} - 64x^{2} + 256 \) 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:   \(7465802011608416256\) \(\medspace = 2^{16}\cdot 3^{12}\cdot 11^{8}\) 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:  \(15.12\)
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:  $2\cdot 3^{3/4}11^{1/2}\approx 15.120539229772588$
Ramified primes:   \(2\), \(3\), \(11\) 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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $16$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois over $\Q$.
This is a CM field.
Reflex fields:  unavailable$^{128}$

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{12}a^{8}+\frac{1}{12}a^{6}+\frac{1}{4}a^{4}-\frac{1}{6}a^{2}+\frac{1}{3}$, $\frac{1}{24}a^{9}+\frac{1}{24}a^{7}-\frac{3}{8}a^{5}+\frac{5}{12}a^{3}+\frac{1}{6}a$, $\frac{1}{48}a^{10}+\frac{1}{48}a^{8}-\frac{3}{16}a^{6}-\frac{1}{2}a^{5}+\frac{5}{24}a^{4}-\frac{1}{2}a^{3}-\frac{5}{12}a^{2}-\frac{1}{2}a$, $\frac{1}{96}a^{11}+\frac{1}{96}a^{9}+\frac{5}{32}a^{7}-\frac{1}{4}a^{6}+\frac{17}{48}a^{5}+\frac{1}{4}a^{4}-\frac{11}{24}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{96}a^{12}-\frac{1}{96}a^{10}-\frac{1}{32}a^{8}-\frac{1}{4}a^{7}-\frac{1}{8}a^{6}-\frac{1}{4}a^{5}+\frac{1}{3}a^{4}+\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a+\frac{1}{3}$, $\frac{1}{192}a^{13}-\frac{1}{192}a^{11}-\frac{1}{64}a^{9}-\frac{1}{24}a^{8}-\frac{1}{16}a^{7}-\frac{1}{24}a^{6}+\frac{1}{6}a^{5}-\frac{1}{8}a^{4}-\frac{1}{8}a^{3}+\frac{1}{12}a^{2}+\frac{1}{6}a+\frac{1}{3}$, $\frac{1}{576}a^{14}+\frac{1}{576}a^{12}+\frac{1}{192}a^{10}-\frac{5}{288}a^{8}-\frac{1}{4}a^{7}-\frac{1}{9}a^{6}+\frac{1}{4}a^{5}+\frac{5}{24}a^{4}-\frac{1}{4}a^{3}+\frac{1}{36}a^{2}-\frac{2}{9}$, $\frac{1}{1152}a^{15}+\frac{1}{1152}a^{13}+\frac{1}{384}a^{11}-\frac{5}{576}a^{9}-\frac{1}{24}a^{8}-\frac{1}{18}a^{7}+\frac{5}{24}a^{6}-\frac{19}{48}a^{5}-\frac{3}{8}a^{4}-\frac{35}{72}a^{3}+\frac{1}{3}a^{2}-\frac{1}{9}a+\frac{1}{3}$ 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

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:  $7$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( \frac{1}{576} a^{15} - \frac{5}{576} a^{13} + \frac{1}{64} a^{11} - \frac{1}{36} a^{9} - \frac{1}{36} a^{7} + \frac{1}{4} a^{5} - \frac{5}{36} a^{3} + \frac{5}{18} a \)  (order $12$) 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:   $\frac{1}{1152}a^{15}-\frac{1}{192}a^{14}-\frac{5}{1152}a^{13}-\frac{1}{192}a^{12}+\frac{11}{384}a^{11}+\frac{5}{192}a^{10}-\frac{1}{18}a^{9}-\frac{7}{96}a^{8}-\frac{1}{72}a^{7}+\frac{1}{24}a^{6}+\frac{7}{48}a^{5}+\frac{1}{24}a^{4}-\frac{13}{36}a^{3}-\frac{1}{3}a^{2}+\frac{7}{18}a$, $\frac{1}{576}a^{15}+\frac{1}{576}a^{13}+\frac{1}{64}a^{11}+\frac{1}{48}a^{10}-\frac{1}{144}a^{9}+\frac{1}{48}a^{8}+\frac{13}{288}a^{7}+\frac{1}{16}a^{6}+\frac{1}{16}a^{5}-\frac{1}{24}a^{4}+\frac{5}{72}a^{3}-\frac{1}{6}a^{2}+\frac{7}{9}a$, $\frac{1}{1152}a^{15}-\frac{17}{1152}a^{13}+\frac{1}{128}a^{11}-\frac{1}{72}a^{9}-\frac{19}{288}a^{7}+\frac{1}{8}a^{5}-\frac{5}{72}a^{3}-\frac{5}{18}a-1$, $\frac{1}{1152}a^{15}+\frac{1}{288}a^{14}-\frac{11}{1152}a^{13}+\frac{1}{288}a^{12}+\frac{5}{384}a^{11}-\frac{1}{96}a^{10}-\frac{11}{576}a^{9}+\frac{5}{72}a^{8}+\frac{1}{36}a^{7}-\frac{23}{144}a^{6}+\frac{7}{48}a^{5}-\frac{1}{6}a^{4}-\frac{11}{72}a^{3}-\frac{1}{36}a^{2}+\frac{8}{9}a-\frac{4}{9}$, $\frac{5}{576}a^{14}-\frac{1}{576}a^{12}-\frac{5}{192}a^{10}+\frac{7}{144}a^{8}-\frac{29}{144}a^{6}+\frac{1}{12}a^{4}+\frac{29}{36}a^{2}-\frac{7}{9}$, $\frac{1}{384}a^{15}+\frac{1}{288}a^{14}+\frac{1}{384}a^{13}-\frac{1}{144}a^{12}-\frac{1}{384}a^{11}+\frac{1}{48}a^{10}+\frac{3}{64}a^{9}+\frac{11}{288}a^{8}+\frac{1}{96}a^{7}-\frac{1}{18}a^{6}-\frac{1}{24}a^{5}+\frac{5}{24}a^{4}+\frac{7}{12}a^{3}+\frac{2}{9}a^{2}-\frac{1}{9}$, $\frac{1}{576}a^{15}-\frac{5}{576}a^{14}+\frac{1}{144}a^{13}-\frac{5}{576}a^{12}+\frac{1}{32}a^{11}-\frac{1}{192}a^{10}-\frac{1}{576}a^{9}-\frac{5}{288}a^{8}+\frac{13}{288}a^{7}-\frac{1}{144}a^{6}+\frac{3}{16}a^{5}+\frac{1}{24}a^{4}-\frac{2}{9}a^{3}+\frac{4}{9}a^{2}-\frac{1}{18}a+\frac{1}{9}$ 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:  \( 7645.28430819 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 7645.28430819 \cdot 1}{12\cdot\sqrt{7465802011608416256}}\cr\approx \mathstrut & 0.566386778597 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - x^14 + x^12 + 8*x^10 - 20*x^8 + 32*x^6 + 16*x^4 - 64*x^2 + 256)
 
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 - x^14 + x^12 + 8*x^10 - 20*x^8 + 32*x^6 + 16*x^4 - 64*x^2 + 256, 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 - x^14 + x^12 + 8*x^10 - 20*x^8 + 32*x^6 + 16*x^4 - 64*x^2 + 256);
 
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 - x^14 + x^12 + 8*x^10 - 20*x^8 + 32*x^6 + 16*x^4 - 64*x^2 + 256);
 
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

$C_2\times D_4$ (as 16T9):

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 16
The 10 conjugacy class representatives for $D_4\times C_2$
Character table for $D_4\times C_2$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{11}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{-33}) \), \(\Q(i, \sqrt{11})\), \(\Q(\zeta_{12})\), \(\Q(i, \sqrt{33})\), \(\Q(\sqrt{-3}, \sqrt{-11})\), \(\Q(\sqrt{3}, \sqrt{-11})\), \(\Q(\sqrt{-3}, \sqrt{11})\), \(\Q(\sqrt{3}, \sqrt{11})\), 4.4.4752.1 x2, 4.0.4752.1 x2, 4.0.13068.1 x2, 4.4.13068.1 x2, 8.0.303595776.1, 8.0.22581504.2 x2, 8.0.2732361984.1 x2, 8.0.170772624.1 x2, 8.0.2732361984.2 x2, 8.0.2732361984.4, 8.8.2732361984.1

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: 8.0.2732361984.1, 8.0.170772624.1, 8.0.22581504.2, 8.0.2732361984.2
Minimal sibling: 8.0.22581504.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R ${\href{/padicField/5.4.0.1}{4} }^{4}$ ${\href{/padicField/7.4.0.1}{4} }^{4}$ R ${\href{/padicField/13.2.0.1}{2} }^{8}$ ${\href{/padicField/17.2.0.1}{2} }^{8}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.2.0.1}{2} }^{8}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.2.0.1}{2} }^{8}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

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 2.4.4.1$x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$$2$$2$$4$$C_2^2$$[2]^{2}$
\(3\) Copy content Toggle raw display 3.8.6.2$x^{8} + 6 x^{5} + 6 x^{4} + 18 x^{2} + 18 x + 9$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
3.8.6.2$x^{8} + 6 x^{5} + 6 x^{4} + 18 x^{2} + 18 x + 9$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
\(11\) Copy content Toggle raw display 11.4.2.1$x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$