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Properties

Label 16.4.135895449600000000.1
Degree $16$
Signature $[4, 6]$
Discriminant $1.359\times 10^{17}$
Root discriminant \(11.77\)
Ramified primes $2,3,5$
Class number $1$
Class group trivial
Galois group $D_4:D_4$ (as 16T115)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^14 - 11*x^12 + 4*x^10 + 6*x^8 - 34*x^6 - 36*x^4 - 8*x^2 + 1)
 
Copy content gp:K = bnfinit(y^16 - 2*y^14 - 11*y^12 + 4*y^10 + 6*y^8 - 34*y^6 - 36*y^4 - 8*y^2 + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^14 - 11*x^12 + 4*x^10 + 6*x^8 - 34*x^6 - 36*x^4 - 8*x^2 + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 2*x^14 - 11*x^12 + 4*x^10 + 6*x^8 - 34*x^6 - 36*x^4 - 8*x^2 + 1)
 

\( x^{16} - 2x^{14} - 11x^{12} + 4x^{10} + 6x^{8} - 34x^{6} - 36x^{4} - 8x^{2} + 1 \) 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:   \(135895449600000000\) \(\medspace = 2^{32}\cdot 3^{4}\cdot 5^{8}\) 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:  \(11.77\)
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:(1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2^{9/4}3^{1/2}5^{1/2}\approx 18.42311740638763$
Ramified primes:   \(2\), \(3\), \(5\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{28}a^{12}-\frac{1}{4}a^{11}-\frac{5}{28}a^{10}-\frac{1}{4}a^{9}-\frac{1}{28}a^{8}+\frac{11}{28}a^{6}-\frac{1}{28}a^{4}-\frac{1}{4}a^{3}+\frac{3}{7}a^{2}-\frac{1}{2}a-\frac{11}{28}$, $\frac{1}{28}a^{13}+\frac{1}{14}a^{11}+\frac{3}{14}a^{9}-\frac{1}{4}a^{8}+\frac{11}{28}a^{7}+\frac{1}{4}a^{6}-\frac{1}{28}a^{5}-\frac{1}{2}a^{4}-\frac{9}{28}a^{3}+\frac{3}{28}a-\frac{1}{4}$, $\frac{1}{1148}a^{14}+\frac{11}{1148}a^{12}-\frac{1}{4}a^{11}+\frac{255}{1148}a^{10}-\frac{83}{574}a^{8}-\frac{1}{4}a^{7}+\frac{11}{41}a^{6}-\frac{1}{2}a^{5}-\frac{65}{574}a^{4}-\frac{1}{4}a^{3}-\frac{127}{1148}a^{2}-\frac{1}{4}a+\frac{391}{1148}$, $\frac{1}{1148}a^{15}+\frac{11}{1148}a^{13}-\frac{8}{287}a^{11}-\frac{1}{4}a^{10}+\frac{121}{1148}a^{9}-\frac{19}{82}a^{7}-\frac{1}{4}a^{6}-\frac{65}{574}a^{5}-\frac{1}{2}a^{4}-\frac{207}{574}a^{3}-\frac{1}{4}a^{2}+\frac{391}{1148}a-\frac{1}{4}$ 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 $1$
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 $1$
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:   $\frac{235}{287}a^{15}-\frac{1185}{574}a^{13}-\frac{4503}{574}a^{11}+\frac{4103}{574}a^{9}+\frac{235}{574}a^{7}-\frac{15713}{574}a^{5}-\frac{4548}{287}a^{3}-\frac{607}{574}a$, $\frac{55}{164}a^{15}-\frac{13}{287}a^{14}-\frac{284}{287}a^{13}-\frac{39}{1148}a^{12}-\frac{3259}{1148}a^{11}+\frac{36}{41}a^{10}+\frac{1222}{287}a^{9}+\frac{33}{41}a^{8}-\frac{1345}{1148}a^{7}-\frac{2117}{1148}a^{6}-\frac{12535}{1148}a^{5}+\frac{2783}{1148}a^{4}-\frac{426}{287}a^{3}+\frac{5825}{1148}a^{2}+\frac{1201}{574}a+\frac{783}{1148}$, $\frac{122}{287}a^{14}-\frac{339}{287}a^{12}-\frac{313}{82}a^{10}+\frac{393}{82}a^{8}-\frac{144}{287}a^{6}-\frac{4134}{287}a^{4}-\frac{2165}{574}a^{2}+\frac{470}{287}$, $\frac{26}{287}a^{15}-\frac{207}{574}a^{13}-\frac{176}{287}a^{11}+\frac{1331}{574}a^{9}-\frac{151}{287}a^{7}-\frac{2537}{574}a^{5}+\frac{2129}{574}a^{3}+\frac{1105}{287}a$, $\frac{1011}{1148}a^{15}-\frac{489}{1148}a^{14}-\frac{2491}{1148}a^{13}+\frac{1345}{1148}a^{12}-\frac{9925}{1148}a^{11}+\frac{2207}{574}a^{10}+\frac{8515}{1148}a^{9}-\frac{1334}{287}a^{8}+\frac{1797}{1148}a^{7}+\frac{555}{1148}a^{6}-\frac{8647}{287}a^{5}+\frac{4023}{287}a^{4}-\frac{20239}{1148}a^{3}+\frac{1473}{287}a^{2}+\frac{5}{287}a-\frac{209}{287}$, $\frac{10}{41}a^{15}-\frac{67}{82}a^{13}-\frac{74}{41}a^{11}+\frac{329}{82}a^{9}-\frac{77}{41}a^{7}-\frac{673}{82}a^{5}+\frac{207}{82}a^{3}+\frac{138}{41}a$, $\frac{523}{574}a^{15}-\frac{1135}{574}a^{13}-\frac{5543}{574}a^{11}+\frac{3013}{574}a^{9}+\frac{339}{82}a^{7}-\frac{9026}{287}a^{5}-\frac{15335}{574}a^{3}-\frac{1504}{287}a$, $\frac{331}{1148}a^{15}+\frac{55}{164}a^{14}-\frac{248}{287}a^{13}-\frac{284}{287}a^{12}-\frac{1319}{574}a^{11}-\frac{3259}{1148}a^{10}+\frac{1965}{574}a^{9}+\frac{1222}{287}a^{8}-\frac{1055}{574}a^{7}-\frac{1345}{1148}a^{6}-\frac{9123}{1148}a^{5}-\frac{12535}{1148}a^{4}-\frac{2349}{1148}a^{3}-\frac{426}{287}a^{2}-\frac{713}{1148}a+\frac{1201}{574}$, $\frac{471}{1148}a^{15}+\frac{219}{574}a^{14}-\frac{197}{164}a^{13}-\frac{543}{574}a^{12}-\frac{1021}{287}a^{11}-\frac{4299}{1148}a^{10}+\frac{6151}{1148}a^{9}+\frac{1899}{574}a^{8}-\frac{569}{574}a^{7}+\frac{793}{1148}a^{6}-\frac{1199}{82}a^{5}-\frac{7437}{574}a^{4}-\frac{117}{82}a^{3}-\frac{7943}{1148}a^{2}+\frac{3761}{1148}a-\frac{573}{1148}$ Copy content Toggle raw display
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:  \( 117.610418303 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 117.610418303 \cdot 1}{2\cdot\sqrt{135895449600000000}}\cr\approx \mathstrut & 0.157040812789 \end{aligned}\]

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 - 2*x^14 - 11*x^12 + 4*x^10 + 6*x^8 - 34*x^6 - 36*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 - 2*x^14 - 11*x^12 + 4*x^10 + 6*x^8 - 34*x^6 - 36*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 - 2*x^14 - 11*x^12 + 4*x^10 + 6*x^8 - 34*x^6 - 36*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 = PolynomialRing(QQ); K, a = NumberField(x^16 - 2*x^14 - 11*x^12 + 4*x^10 + 6*x^8 - 34*x^6 - 36*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

$D_4:D_4$ (as 16T115):

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); G, transitive_group_identification(G)
 
A solvable group of order 64
The 25 conjugacy class representatives for $D_4:D_4$
Character table for $D_4:D_4$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), 4.2.400.1, 4.2.1600.1, \(\Q(\sqrt{2}, \sqrt{5})\), 8.4.368640000.1, 8.4.23040000.1, 8.4.40960000.1

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(b)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 16 siblings: data not computed
Degree 32 siblings: data not computed
Minimal sibling: 16.0.42998169600000000.4

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R R ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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

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 2.4.4.32b25.7$x^{16} + 4 x^{15} + 8 x^{13} + 18 x^{12} + 12 x^{11} + 20 x^{10} + 44 x^{9} + 36 x^{8} + 32 x^{7} + 58 x^{6} + 56 x^{5} + 33 x^{4} + 32 x^{3} + 38 x^{2} + 24 x + 9$$4$$4$$32$$C_2 \times (C_2^2:C_4)$$$[2, 2, 3]^{4}$$
\(3\) Copy content Toggle raw display 3.4.1.0a1.1$x^{4} + 2 x^{3} + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
3.4.1.0a1.1$x^{4} + 2 x^{3} + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
3.4.2.4a1.2$x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{4} + 8 x^{3} + 7$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
\(5\) Copy content Toggle raw display 5.4.2.4a1.2$x^{8} + 8 x^{6} + 8 x^{5} + 20 x^{4} + 32 x^{3} + 32 x^{2} + 16 x + 9$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
5.4.2.4a1.2$x^{8} + 8 x^{6} + 8 x^{5} + 20 x^{4} + 32 x^{3} + 32 x^{2} + 16 x + 9$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$

Spectrum of ring of integers

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