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Properties

Label 16.4.461...904.1
Degree $16$
Signature $[4, 6]$
Discriminant $4.612\times 10^{18}$
Root discriminant \(14.67\)
Ramified prime $2$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $D_4:C_4$ (as 16T26)

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Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^14 + 12*x^12 + 8*x^10 - 90*x^8 + 8*x^6 + 12*x^4 - 8*x^2 + 1)
 
Copy content gp:K = bnfinit(y^16 - 8*y^14 + 12*y^12 + 8*y^10 - 90*y^8 + 8*y^6 + 12*y^4 - 8*y^2 + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^14 + 12*x^12 + 8*x^10 - 90*x^8 + 8*x^6 + 12*x^4 - 8*x^2 + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 8*x^14 + 12*x^12 + 8*x^10 - 90*x^8 + 8*x^6 + 12*x^4 - 8*x^2 + 1)
 

\( x^{16} - 8x^{14} + 12x^{12} + 8x^{10} - 90x^{8} + 8x^{6} + 12x^{4} - 8x^{2} + 1 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $16$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $[4, 6]$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(4611686018427387904\) \(\medspace = 2^{62}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(14.67\)
Copy content comment:Root discriminant
 
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  $2^{63/16}\approx 15.32165249117718$
Ramified primes:   \(2\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant(OK))
 
Discriminant root field:  \(\Q\)
$\Aut(K/\Q)$:   $C_2^2$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{4}-\frac{1}{4}a^{2}+\frac{1}{4}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{5}-\frac{1}{4}a^{3}+\frac{1}{4}a$, $\frac{1}{8}a^{8}-\frac{1}{4}a^{4}+\frac{1}{8}$, $\frac{1}{16}a^{9}-\frac{1}{16}a^{8}-\frac{1}{8}a^{5}+\frac{1}{8}a^{4}-\frac{7}{16}a+\frac{7}{16}$, $\frac{1}{16}a^{10}-\frac{1}{16}a^{8}-\frac{1}{8}a^{6}+\frac{1}{8}a^{4}-\frac{7}{16}a^{2}+\frac{7}{16}$, $\frac{1}{32}a^{11}-\frac{1}{32}a^{10}-\frac{1}{32}a^{9}+\frac{1}{32}a^{8}-\frac{1}{16}a^{7}+\frac{1}{16}a^{6}-\frac{3}{16}a^{5}+\frac{3}{16}a^{4}-\frac{7}{32}a^{3}+\frac{7}{32}a^{2}+\frac{15}{32}a-\frac{15}{32}$, $\frac{1}{32}a^{12}+\frac{1}{32}a^{8}+\frac{3}{32}a^{4}-\frac{5}{32}$, $\frac{1}{32}a^{13}-\frac{1}{32}a^{9}-\frac{1}{16}a^{8}+\frac{7}{32}a^{5}+\frac{1}{8}a^{4}+\frac{9}{32}a+\frac{7}{16}$, $\frac{1}{64}a^{14}-\frac{1}{64}a^{12}+\frac{1}{64}a^{10}-\frac{1}{64}a^{8}+\frac{3}{64}a^{6}-\frac{3}{64}a^{4}+\frac{27}{64}a^{2}-\frac{27}{64}$, $\frac{1}{64}a^{15}-\frac{1}{64}a^{13}-\frac{1}{64}a^{11}-\frac{1}{32}a^{10}+\frac{1}{64}a^{9}+\frac{1}{32}a^{8}+\frac{7}{64}a^{7}+\frac{1}{16}a^{6}+\frac{9}{64}a^{5}+\frac{3}{16}a^{4}+\frac{9}{64}a^{3}-\frac{9}{32}a^{2}-\frac{25}{64}a+\frac{1}{32}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Ideal class group:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $9$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $\frac{31}{64}a^{15}-\frac{235}{64}a^{13}+\frac{271}{64}a^{11}+\frac{381}{64}a^{9}-\frac{2659}{64}a^{7}-\frac{881}{64}a^{5}+\frac{213}{64}a^{3}-\frac{129}{64}a$, $\frac{3}{8}a^{15}-\frac{49}{16}a^{13}+\frac{79}{16}a^{11}+\frac{11}{4}a^{9}-35a^{7}+\frac{131}{16}a^{5}+\frac{147}{16}a^{3}-\frac{27}{8}a$, $\frac{9}{64}a^{15}-\frac{63}{64}a^{13}+\frac{37}{64}a^{11}+\frac{173}{64}a^{9}-\frac{733}{64}a^{7}-\frac{725}{64}a^{5}+\frac{111}{64}a^{3}-\frac{89}{64}a$, $\frac{45}{64}a^{15}-\frac{7}{32}a^{14}-\frac{349}{64}a^{13}+\frac{57}{32}a^{12}+\frac{455}{64}a^{11}-\frac{23}{8}a^{10}+\frac{469}{64}a^{9}-\frac{11}{8}a^{8}-\frac{3933}{64}a^{7}+\frac{637}{32}a^{6}-\frac{603}{64}a^{5}-\frac{139}{32}a^{4}+\frac{377}{64}a^{3}-\frac{41}{16}a^{2}-\frac{269}{64}a+\frac{19}{16}$, $\frac{45}{64}a^{15}+\frac{11}{64}a^{14}-\frac{349}{64}a^{13}-\frac{85}{64}a^{12}+\frac{455}{64}a^{11}+\frac{109}{64}a^{10}+\frac{469}{64}a^{9}+\frac{117}{64}a^{8}-\frac{3933}{64}a^{7}-\frac{963}{64}a^{6}-\frac{603}{64}a^{5}-\frac{163}{64}a^{4}+\frac{377}{64}a^{3}+\frac{91}{64}a^{2}-\frac{269}{64}a-\frac{45}{64}$, $a$, $\frac{45}{64}a^{15}-\frac{11}{64}a^{14}-\frac{349}{64}a^{13}+\frac{85}{64}a^{12}+\frac{455}{64}a^{11}-\frac{109}{64}a^{10}+\frac{469}{64}a^{9}-\frac{117}{64}a^{8}-\frac{3933}{64}a^{7}+\frac{963}{64}a^{6}-\frac{603}{64}a^{5}+\frac{163}{64}a^{4}+\frac{377}{64}a^{3}-\frac{91}{64}a^{2}-\frac{269}{64}a+\frac{45}{64}$, $\frac{27}{64}a^{15}+\frac{1}{16}a^{14}-\frac{205}{64}a^{13}-\frac{15}{32}a^{12}+\frac{237}{64}a^{11}+\frac{17}{32}a^{10}+\frac{341}{64}a^{9}+\frac{5}{8}a^{8}-\frac{2339}{64}a^{7}-5a^{6}-\frac{747}{64}a^{5}-\frac{67}{32}a^{4}+\frac{331}{64}a^{3}-\frac{59}{32}a^{2}-\frac{173}{64}a-\frac{5}{16}$, $\frac{13}{64}a^{15}-\frac{9}{64}a^{14}-\frac{97}{64}a^{13}+\frac{75}{64}a^{12}+\frac{103}{64}a^{11}-\frac{131}{64}a^{10}+\frac{165}{64}a^{9}-\frac{43}{64}a^{8}-\frac{1085}{64}a^{7}+\frac{841}{64}a^{6}-\frac{495}{64}a^{5}-\frac{339}{64}a^{4}-\frac{71}{64}a^{3}-\frac{173}{64}a^{2}-\frac{69}{64}a+\frac{99}{64}$ Copy content Toggle raw display (assuming GRH)
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 1482.43828312 \) (assuming GRH)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 1482.43828312 \cdot 1}{2\cdot\sqrt{4611686018427387904}}\cr\approx \mathstrut & 0.339794193634 \end{aligned}\] (assuming GRH)

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^14 + 12*x^12 + 8*x^10 - 90*x^8 + 8*x^6 + 12*x^4 - 8*x^2 + 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))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^16 - 8*x^14 + 12*x^12 + 8*x^10 - 90*x^8 + 8*x^6 + 12*x^4 - 8*x^2 + 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)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^14 + 12*x^12 + 8*x^10 - 90*x^8 + 8*x^6 + 12*x^4 - 8*x^2 + 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)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 8*x^14 + 12*x^12 + 8*x^10 - 90*x^8 + 8*x^6 + 12*x^4 - 8*x^2 + 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

$D_4:C_4$ (as 16T26):

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A solvable group of order 32
The 14 conjugacy class representatives for $D_4:C_4$
Character table for $D_4:C_4$

Intermediate fields

\(\Q(\sqrt{2}) \), 4.2.1024.1, \(\Q(\zeta_{16})^+\), 4.2.2048.1, 8.2.268435456.2, 8.2.268435456.1, 8.4.67108864.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Galois closure: deg 32
Degree 16 sibling: 16.0.4611686018427387904.4
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.8.0.1}{8} }^{2}$ ${\href{/padicField/5.4.0.1}{4} }^{4}$ ${\href{/padicField/7.2.0.1}{2} }^{8}$ ${\href{/padicField/11.8.0.1}{8} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.8.0.1}{8} }^{2}$ ${\href{/padicField/23.2.0.1}{2} }^{8}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.2.0.1}{2} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.2.0.1}{2} }^{6}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.8.0.1}{8} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_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 2.1.16.62l1.21$x^{16} + 8 x^{15} + 4 x^{12} + 8 x^{10} + 8 x^{6} + 16 x^{3} + 2$$16$$1$$62$16T26$$[2, 3, \frac{7}{2}, 4, \frac{9}{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)