↑

Properties

Label 16.0.160...000.3
Degree $16$
Signature $[0, 8]$
Discriminant $1.606\times 10^{18}$
Root discriminant \(13.74\)
Ramified primes $2,5,89$
Class number $1$
Class group trivial
Galois group $C_2^6:D_4$ (as 16T969)

Related objects

Downloads

Learn more

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 - 4*x^15 + 10*x^14 - 18*x^13 + 32*x^12 - 48*x^11 + 62*x^10 - 68*x^9 + 73*x^8 - 64*x^7 + 48*x^6 - 26*x^5 + 17*x^4 - 4*x^3 + 5*x^2 + 2*x + 1)
 
Copy content gp:K = bnfinit(y^16 - 4*y^15 + 10*y^14 - 18*y^13 + 32*y^12 - 48*y^11 + 62*y^10 - 68*y^9 + 73*y^8 - 64*y^7 + 48*y^6 - 26*y^5 + 17*y^4 - 4*y^3 + 5*y^2 + 2*y + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^15 + 10*x^14 - 18*x^13 + 32*x^12 - 48*x^11 + 62*x^10 - 68*x^9 + 73*x^8 - 64*x^7 + 48*x^6 - 26*x^5 + 17*x^4 - 4*x^3 + 5*x^2 + 2*x + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 4*x^15 + 10*x^14 - 18*x^13 + 32*x^12 - 48*x^11 + 62*x^10 - 68*x^9 + 73*x^8 - 64*x^7 + 48*x^6 - 26*x^5 + 17*x^4 - 4*x^3 + 5*x^2 + 2*x + 1)
 

\( x^{16} - 4 x^{15} + 10 x^{14} - 18 x^{13} + 32 x^{12} - 48 x^{11} + 62 x^{10} - 68 x^{9} + 73 x^{8} + \cdots + 1 \) 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:  $[0, 8]$
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:   \(1606201369600000000\) \(\medspace = 2^{16}\cdot 5^{8}\cdot 89^{4}\) 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:  \(13.74\)
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:  $2^{3/2}5^{1/2}89^{1/2}\approx 59.665735560705194$
Ramified primes:   \(2\), \(5\), \(89\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{1623523}a^{15}-\frac{391239}{1623523}a^{14}+\frac{641735}{1623523}a^{13}+\frac{521592}{1623523}a^{12}+\frac{430351}{1623523}a^{11}+\frac{702705}{1623523}a^{10}-\frac{276362}{1623523}a^{9}+\frac{725771}{1623523}a^{8}+\frac{60136}{147593}a^{7}+\frac{543237}{1623523}a^{6}+\frac{444760}{1623523}a^{5}+\frac{269468}{1623523}a^{4}-\frac{223435}{1623523}a^{3}+\frac{243332}{1623523}a^{2}+\frac{146659}{1623523}a+\frac{416003}{1623523}$ 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:  $7$
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:   $\frac{21137}{147593}a^{15}-\frac{130546}{147593}a^{14}+\frac{408402}{147593}a^{13}-\frac{897368}{147593}a^{12}+\frac{1648427}{147593}a^{11}-\frac{2750237}{147593}a^{10}+\frac{3957171}{147593}a^{9}-\frac{4917769}{147593}a^{8}+\frac{5426631}{147593}a^{7}-\frac{5500686}{147593}a^{6}+\frac{4678961}{147593}a^{5}-\frac{3263393}{147593}a^{4}+\frac{1853928}{147593}a^{3}-\frac{897938}{147593}a^{2}+\frac{183097}{147593}a-\frac{92750}{147593}$, $\frac{56072}{147593}a^{15}-\frac{215246}{147593}a^{14}+\frac{486706}{147593}a^{13}-\frac{765035}{147593}a^{12}+\frac{1252074}{147593}a^{11}-\frac{1714008}{147593}a^{10}+\frac{1832901}{147593}a^{9}-\frac{1419529}{147593}a^{8}+\frac{1135219}{147593}a^{7}-\frac{301055}{147593}a^{6}-\frac{649269}{147593}a^{5}+\frac{1399844}{147593}a^{4}-\frac{1048666}{147593}a^{3}+\frac{910170}{147593}a^{2}-\frac{270919}{147593}a+\frac{227310}{147593}$, $\frac{757233}{1623523}a^{15}-\frac{3475216}{1623523}a^{14}+\frac{9497171}{1623523}a^{13}-\frac{18612211}{1623523}a^{12}+\frac{33289160}{1623523}a^{11}-\frac{51824244}{1623523}a^{10}+\frac{69876320}{1623523}a^{9}-\frac{80571714}{1623523}a^{8}+\frac{7770234}{147593}a^{7}-\frac{77838962}{1623523}a^{6}+\frac{60161265}{1623523}a^{5}-\frac{34896671}{1623523}a^{4}+\frac{16082274}{1623523}a^{3}-\frac{2722529}{1623523}a^{2}+\frac{1190778}{1623523}a+\frac{655532}{1623523}$, $\frac{545746}{1623523}a^{15}-\frac{2738995}{1623523}a^{14}+\frac{7668888}{1623523}a^{13}-\frac{15022234}{1623523}a^{12}+\frac{26228988}{1623523}a^{11}-\frac{41007067}{1623523}a^{10}+\frac{55406907}{1623523}a^{9}-\frac{62732972}{1623523}a^{8}+\frac{5958103}{147593}a^{7}-\frac{60562056}{1623523}a^{6}+\frac{46643489}{1623523}a^{5}-\frac{25230103}{1623523}a^{4}+\frac{12172635}{1623523}a^{3}-\frac{3468682}{1623523}a^{2}+\frac{2125760}{1623523}a+\frac{140441}{1623523}$, $\frac{263753}{1623523}a^{15}-\frac{961610}{1623523}a^{14}+\frac{2388136}{1623523}a^{13}-\frac{4260721}{1623523}a^{12}+\frac{7497896}{1623523}a^{11}-\frac{10574953}{1623523}a^{10}+\frac{12993729}{1623523}a^{9}-\frac{13435982}{1623523}a^{8}+\frac{1149407}{147593}a^{7}-\frac{8504473}{1623523}a^{6}+\frac{4000484}{1623523}a^{5}+\frac{1650556}{1623523}a^{4}-\frac{2637224}{1623523}a^{3}+\frac{4927852}{1623523}a^{2}-\frac{1931294}{1623523}a+\frac{1107873}{1623523}$, $\frac{1544}{147593}a^{15}+\frac{25133}{147593}a^{14}-\frac{100562}{147593}a^{13}+\frac{218233}{147593}a^{12}-\frac{296928}{147593}a^{11}+\frac{463156}{147593}a^{10}-\frac{601937}{147593}a^{9}+\frac{507147}{147593}a^{8}-\frac{141329}{147593}a^{7}-\frac{13091}{147593}a^{6}+\frac{254397}{147593}a^{5}-\frac{744040}{147593}a^{4}+\frac{826759}{147593}a^{3}-\frac{362356}{147593}a^{2}+\frac{33834}{147593}a-\frac{16104}{147593}$, $\frac{4484}{147593}a^{15}-\frac{25278}{147593}a^{14}+\frac{66612}{147593}a^{13}-\frac{87743}{147593}a^{12}+\frac{63002}{147593}a^{11}-\frac{33737}{147593}a^{10}-\frac{16380}{147593}a^{9}+\frac{226700}{147593}a^{8}-\frac{471236}{147593}a^{7}+\frac{442615}{147593}a^{6}-\frac{415555}{147593}a^{5}+\frac{540993}{147593}a^{4}-\frac{316442}{147593}a^{3}-\frac{54361}{147593}a^{2}+\frac{92141}{147593}a-\frac{70475}{147593}$ 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:  \( 305.381044776 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 305.381044776 \cdot 1}{2\cdot\sqrt{1606201369600000000}}\cr\approx \mathstrut & 0.292651686916 \end{aligned}\]

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 - 4*x^15 + 10*x^14 - 18*x^13 + 32*x^12 - 48*x^11 + 62*x^10 - 68*x^9 + 73*x^8 - 64*x^7 + 48*x^6 - 26*x^5 + 17*x^4 - 4*x^3 + 5*x^2 + 2*x + 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 - 4*x^15 + 10*x^14 - 18*x^13 + 32*x^12 - 48*x^11 + 62*x^10 - 68*x^9 + 73*x^8 - 64*x^7 + 48*x^6 - 26*x^5 + 17*x^4 - 4*x^3 + 5*x^2 + 2*x + 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 - 4*x^15 + 10*x^14 - 18*x^13 + 32*x^12 - 48*x^11 + 62*x^10 - 68*x^9 + 73*x^8 - 64*x^7 + 48*x^6 - 26*x^5 + 17*x^4 - 4*x^3 + 5*x^2 + 2*x + 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 - 4*x^15 + 10*x^14 - 18*x^13 + 32*x^12 - 48*x^11 + 62*x^10 - 68*x^9 + 73*x^8 - 64*x^7 + 48*x^6 - 26*x^5 + 17*x^4 - 4*x^3 + 5*x^2 + 2*x + 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

$C_2^6:D_4$ (as 16T969):

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 512
The 44 conjugacy class representatives for $C_2^6:D_4$
Character table for $C_2^6:D_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.400.1, 8.0.14240000.2, 8.0.14240000.1, 8.4.1267360000.4

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: 16.0.100387585600000000.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.8.0.1}{8} }^{2}$ R ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ ${\href{/padicField/23.8.0.1}{8} }^{2}$ ${\href{/padicField/29.2.0.1}{2} }^{6}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$

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 2.4.2.8a3.1$x^{8} + 2 x^{6} + 4 x^{5} + 2 x^{4} + 2 x^{3} + 5 x^{2} + 4 x + 3$$2$$4$$8$$C_2^2:C_4$$$[2, 2]^{4}$$
2.4.2.8a3.1$x^{8} + 2 x^{6} + 4 x^{5} + 2 x^{4} + 2 x^{3} + 5 x^{2} + 4 x + 3$$2$$4$$8$$C_2^2:C_4$$$[2, 2]^{4}$$
\(5\) 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$$$[\ ]_{2}^{4}$$
\(89\) Copy content Toggle raw display 89.2.1.0a1.1$x^{2} + 82 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
89.2.1.0a1.1$x^{2} + 82 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
89.2.1.0a1.1$x^{2} + 82 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
89.2.1.0a1.1$x^{2} + 82 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
89.2.2.2a1.2$x^{4} + 164 x^{3} + 6730 x^{2} + 492 x + 98$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
89.2.2.2a1.2$x^{4} + 164 x^{3} + 6730 x^{2} + 492 x + 98$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$

Spectrum of ring of integers

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