Properties

Label 16.0.687...000.6
Degree $16$
Signature $[0, 8]$
Discriminant $6.872\times 10^{18}$
Root discriminant \(15.04\)
Ramified primes $2,5$
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 + 42*x^12 + 83*x^8 + 42*x^4 + 1)
 
gp: K = bnfinit(y^16 + 42*y^12 + 83*y^8 + 42*y^4 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 42*x^12 + 83*x^8 + 42*x^4 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 + 42*x^12 + 83*x^8 + 42*x^4 + 1)
 

\( x^{16} + 42x^{12} + 83x^{8} + 42x^{4} + 1 \) 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:   \(6871947673600000000\) \(\medspace = 2^{44}\cdot 5^{8}\) 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:  \(15.04\)
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:  $2^{11/4}5^{1/2}\approx 15.042412372345574$
Ramified primes:   \(2\), \(5\) 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 not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{6}a^{8}-\frac{1}{2}a^{4}+\frac{1}{6}$, $\frac{1}{6}a^{9}-\frac{1}{2}a^{5}+\frac{1}{6}a$, $\frac{1}{6}a^{10}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}+\frac{1}{6}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{6}a^{11}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{3}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{156}a^{12}-\frac{1}{12}a^{10}+\frac{1}{39}a^{8}-\frac{1}{4}a^{6}-\frac{1}{2}a^{5}-\frac{14}{39}a^{4}-\frac{1}{2}a^{3}-\frac{1}{12}a^{2}-\frac{1}{2}a+\frac{25}{156}$, $\frac{1}{156}a^{13}-\frac{1}{12}a^{11}+\frac{1}{39}a^{9}-\frac{1}{4}a^{7}+\frac{11}{78}a^{5}-\frac{1}{2}a^{4}+\frac{5}{12}a^{3}-\frac{1}{2}a^{2}-\frac{53}{156}a-\frac{1}{2}$, $\frac{1}{156}a^{14}-\frac{3}{52}a^{10}-\frac{1}{12}a^{8}-\frac{17}{156}a^{6}+\frac{1}{4}a^{4}-\frac{11}{26}a^{2}-\frac{1}{12}$, $\frac{1}{156}a^{15}-\frac{3}{52}a^{11}-\frac{1}{12}a^{9}-\frac{17}{156}a^{7}+\frac{1}{4}a^{5}-\frac{11}{26}a^{3}-\frac{1}{12}a$ 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 $1$

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:   \( \frac{7}{78} a^{14} + \frac{7}{52} a^{12} + \frac{589}{156} a^{10} + \frac{72}{13} a^{8} + \frac{1205}{156} a^{6} + \frac{84}{13} a^{4} + \frac{727}{156} a^{2} + \frac{45}{52} \)  (order $8$) 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:   $\frac{5}{78}a^{13}+\frac{215}{78}a^{9}+\frac{617}{78}a^{5}+\frac{160}{39}a$, $\frac{7}{156}a^{14}-\frac{1}{39}a^{12}+\frac{275}{156}a^{10}-\frac{53}{52}a^{8}-\frac{197}{156}a^{6}+\frac{29}{156}a^{4}-\frac{109}{39}a^{2}+\frac{23}{52}$, $\frac{31}{156}a^{15}-\frac{1}{6}a^{14}-\frac{5}{156}a^{13}-\frac{7}{52}a^{12}+\frac{647}{78}a^{11}-\frac{83}{12}a^{10}-\frac{215}{156}a^{9}-\frac{72}{13}a^{8}+\frac{1121}{78}a^{7}-\frac{125}{12}a^{6}-\frac{617}{156}a^{5}-\frac{84}{13}a^{4}+\frac{853}{156}a^{3}-\frac{41}{12}a^{2}-\frac{80}{39}a-\frac{71}{52}$, $\frac{31}{156}a^{15}-\frac{1}{6}a^{14}-\frac{5}{156}a^{13}+\frac{7}{52}a^{12}+\frac{647}{78}a^{11}-\frac{83}{12}a^{10}-\frac{215}{156}a^{9}+\frac{72}{13}a^{8}+\frac{1121}{78}a^{7}-\frac{125}{12}a^{6}-\frac{617}{156}a^{5}+\frac{84}{13}a^{4}+\frac{853}{156}a^{3}-\frac{41}{12}a^{2}-\frac{80}{39}a+\frac{19}{52}$, $\frac{4}{13}a^{15}+\frac{15}{52}a^{13}+\frac{1999}{156}a^{11}+\frac{461}{39}a^{9}+\frac{1093}{52}a^{7}+\frac{154}{13}a^{5}+\frac{979}{156}a^{3}-\frac{97}{156}a$, $\frac{10}{39}a^{15}-\frac{17}{39}a^{14}+\frac{7}{26}a^{13}-\frac{11}{52}a^{12}+\frac{139}{13}a^{11}-\frac{2807}{156}a^{10}+\frac{144}{13}a^{9}-\frac{677}{78}a^{8}+\frac{1415}{78}a^{7}-\frac{3641}{156}a^{6}+\frac{168}{13}a^{5}-\frac{119}{13}a^{4}+\frac{105}{13}a^{3}-\frac{1037}{156}a^{2}+\frac{42}{13}a-\frac{97}{156}$, $\frac{1}{13}a^{15}-\frac{1}{6}a^{14}+\frac{3}{52}a^{12}+\frac{245}{78}a^{11}-\frac{83}{12}a^{10}+\frac{187}{78}a^{8}+\frac{35}{13}a^{7}-\frac{125}{12}a^{6}+\frac{49}{13}a^{4}-\frac{97}{78}a^{3}-\frac{35}{12}a^{2}-\frac{1}{2}a+\frac{173}{156}$ 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:  \( 2592.42198789 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 2592.42198789 \cdot 1}{8\cdot\sqrt{6871947673600000000}}\cr\approx \mathstrut & 0.300271929185 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 + 42*x^12 + 83*x^8 + 42*x^4 + 1)
 
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 + 42*x^12 + 83*x^8 + 42*x^4 + 1, 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 + 42*x^12 + 83*x^8 + 42*x^4 + 1);
 
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 + 42*x^12 + 83*x^8 + 42*x^4 + 1);
 
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

\(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{-2}, \sqrt{-5})\), \(\Q(\zeta_{8})\), \(\Q(\sqrt{-2}, \sqrt{5})\), \(\Q(i, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-5})\), \(\Q(i, \sqrt{10})\), \(\Q(\sqrt{2}, \sqrt{5})\), 4.0.12800.1 x2, 4.0.512.1 x2, 4.2.25600.3 x2, 4.2.1024.1 x2, 8.0.40960000.1, 8.0.2621440000.11, 8.0.4194304.1, 8.0.163840000.3 x2, 8.0.2621440000.12 x2, 8.0.2621440000.8 x2, 8.4.655360000.2 x2

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: 8.4.655360000.2, 8.0.2621440000.12, 8.0.163840000.3, 8.0.2621440000.8
Minimal sibling: 8.0.163840000.3

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.4.0.1}{4} }^{4}$ R ${\href{/padicField/7.2.0.1}{2} }^{8}$ ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.2.0.1}{2} }^{8}$ ${\href{/padicField/17.2.0.1}{2} }^{8}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.2.0.1}{2} }^{8}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.2.0.1}{2} }^{8}$ ${\href{/padicField/41.1.0.1}{1} }^{16}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.2.0.1}{2} }^{8}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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 2.16.44.2$x^{16} + 8 x^{15} + 20 x^{14} + 16 x^{13} + 16 x^{12} + 8 x^{11} + 72 x^{10} + 136 x^{9} + 136 x^{8} + 160 x^{7} + 136 x^{6} + 208 x^{5} + 240 x^{4} + 208 x^{3} + 160 x^{2} + 48 x + 36$$8$$2$$44$$D_4\times C_2$$[2, 3, 7/2]^{2}$
\(5\) Copy content Toggle raw display 5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$