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Properties

Label 16.0.8084777718513664.1
Degree $16$
Signature $(0, 8)$
Discriminant $8.085\times 10^{15}$
Root discriminant \(9.87\)
Ramified primes $2,7$
Class number $1$
Class group trivial
Galois group $D_4:D_4$ (as 16T141)

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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 - 4*x^15 + 6*x^14 - 14*x^12 + 24*x^11 - 18*x^10 + 15*x^8 - 20*x^7 + 22*x^6 - 24*x^5 + 24*x^4 - 20*x^3 + 12*x^2 - 4*x + 1)
 
Copy content gp:K = bnfinit(y^16 - 4*y^15 + 6*y^14 - 14*y^12 + 24*y^11 - 18*y^10 + 15*y^8 - 20*y^7 + 22*y^6 - 24*y^5 + 24*y^4 - 20*y^3 + 12*y^2 - 4*y + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^15 + 6*x^14 - 14*x^12 + 24*x^11 - 18*x^10 + 15*x^8 - 20*x^7 + 22*x^6 - 24*x^5 + 24*x^4 - 20*x^3 + 12*x^2 - 4*x + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 4*x^15 + 6*x^14 - 14*x^12 + 24*x^11 - 18*x^10 + 15*x^8 - 20*x^7 + 22*x^6 - 24*x^5 + 24*x^4 - 20*x^3 + 12*x^2 - 4*x + 1)
 

\( x^{16} - 4 x^{15} + 6 x^{14} - 14 x^{12} + 24 x^{11} - 18 x^{10} + 15 x^{8} - 20 x^{7} + 22 x^{6} + \cdots + 1 \) 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:   \(8084777718513664\) \(\medspace = 2^{36}\cdot 7^{6}\) 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:  \(9.87\)
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:  $2^{9/4}7^{3/4}\approx 20.471092479852693$
Ramified primes:   \(2\), \(7\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $D_4$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
Maximal CM subfield:  \(\Q(\zeta_{8})\)

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{5549}a^{15}+\frac{2615}{5549}a^{14}+\frac{1225}{5549}a^{13}+\frac{953}{5549}a^{12}-\frac{1157}{5549}a^{11}-\frac{405}{5549}a^{10}-\frac{854}{5549}a^{9}-\frac{379}{5549}a^{8}+\frac{685}{5549}a^{7}+\frac{1668}{5549}a^{6}+\frac{1451}{5549}a^{5}-\frac{920}{5549}a^{4}-\frac{1190}{5549}a^{3}+\frac{1908}{5549}a^{2}-\frac{2585}{5549}a-\frac{339}{5549}$ 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:   \( -\frac{6779}{5549} a^{15} + \frac{24166}{5549} a^{14} - \frac{30716}{5549} a^{13} - \frac{12449}{5549} a^{12} + \frac{91350}{5549} a^{11} - \frac{128887}{5549} a^{10} + \frac{68247}{5549} a^{9} + \frac{38897}{5549} a^{8} - \frac{98984}{5549} a^{7} + \frac{95823}{5549} a^{6} - \frac{97834}{5549} a^{5} + \frac{110584}{5549} a^{4} - \frac{112216}{5549} a^{3} + \frac{83622}{5549} a^{2} - \frac{38870}{5549} a + \frac{6344}{5549} \)  (order $8$) 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:   $\frac{2266}{5549}a^{15}-\frac{6291}{5549}a^{14}+\frac{6899}{5549}a^{13}+\frac{6486}{5549}a^{12}-\frac{24830}{5549}a^{11}+\frac{31149}{5549}a^{10}-\frac{9661}{5549}a^{9}-\frac{15366}{5549}a^{8}+\frac{26235}{5549}a^{7}-\frac{21377}{5549}a^{6}+\frac{25154}{5549}a^{5}-\frac{26041}{5549}a^{4}+\frac{22470}{5549}a^{3}-\frac{10241}{5549}a^{2}+\frac{7683}{5549}a+\frac{3137}{5549}$, $\frac{2568}{5549}a^{15}-\frac{10068}{5549}a^{14}+\frac{10615}{5549}a^{13}+\frac{11293}{5549}a^{12}-\frac{41304}{5549}a^{11}+\frac{42015}{5549}a^{10}-\frac{6766}{5549}a^{9}-\frac{29942}{5549}a^{8}+\frac{38890}{5549}a^{7}-\frac{28149}{5549}a^{6}+\frac{30534}{5549}a^{5}-\frac{31980}{5549}a^{4}+\frac{23775}{5549}a^{3}-\frac{16670}{5549}a^{2}+\frac{3873}{5549}a+\frac{6190}{5549}$, $\frac{5171}{5549}a^{15}-\frac{17395}{5549}a^{14}+\frac{19713}{5549}a^{13}+\frac{11549}{5549}a^{12}-\frac{62064}{5549}a^{11}+\frac{80953}{5549}a^{10}-\frac{43422}{5549}a^{9}-\frac{17659}{5549}a^{8}+\frac{51814}{5549}a^{7}-\frac{64506}{5549}a^{6}+\frac{78559}{5549}a^{5}-\frac{85062}{5549}a^{4}+\frac{78037}{5549}a^{3}-\frac{60893}{5549}a^{2}+\frac{28251}{5549}a-\frac{10583}{5549}$, $\frac{1528}{5549}a^{15}-\frac{5109}{5549}a^{14}+\frac{1787}{5549}a^{13}+\frac{13444}{5549}a^{12}-\frac{25510}{5549}a^{11}+\frac{8197}{5549}a^{10}+\frac{26848}{5549}a^{9}-\frac{40859}{5549}a^{8}+\frac{14566}{5549}a^{7}+\frac{12811}{5549}a^{6}-\frac{8021}{5549}a^{5}-\frac{1863}{5549}a^{4}-\frac{3797}{5549}a^{3}+\frac{13297}{5549}a^{2}-\frac{21188}{5549}a+\frac{9163}{5549}$, $\frac{2721}{5549}a^{15}-\frac{3952}{5549}a^{14}-\frac{1724}{5549}a^{13}+\frac{12828}{5549}a^{12}-\frac{13012}{5549}a^{11}+\frac{2246}{5549}a^{10}+\frac{12395}{5549}a^{9}-\frac{15792}{5549}a^{8}+\frac{10519}{5549}a^{7}-\frac{6003}{5549}a^{6}+\frac{2832}{5549}a^{5}+\frac{4828}{5549}a^{4}+\frac{2626}{5549}a^{3}+\frac{3353}{5549}a^{2}-\frac{8751}{5549}a-\frac{1285}{5549}$, $\frac{754}{5549}a^{15}-\frac{3734}{5549}a^{14}+\frac{2516}{5549}a^{13}+\frac{8290}{5549}a^{12}-\frac{17832}{5549}a^{11}+\frac{5374}{5549}a^{10}+\frac{16415}{5549}a^{9}-\frac{19414}{5549}a^{8}+\frac{433}{5549}a^{7}+\frac{9147}{5549}a^{6}+\frac{6450}{5549}a^{5}-\frac{16702}{5549}a^{4}+\frac{1678}{5549}a^{3}+\frac{6990}{5549}a^{2}-\frac{6940}{5549}a+\frac{5197}{5549}$, $\frac{2868}{5549}a^{15}-\frac{7977}{5549}a^{14}+\frac{6332}{5549}a^{13}+\frac{14194}{5549}a^{12}-\frac{33268}{5549}a^{11}+\frac{25946}{5549}a^{10}+\frac{8935}{5549}a^{9}-\frac{27113}{5549}a^{8}+\frac{22430}{5549}a^{7}-\frac{10512}{5549}a^{6}+\frac{21914}{5549}a^{5}-\frac{24981}{5549}a^{4}+\frac{16362}{5549}a^{3}-\frac{4719}{5549}a^{2}+\frac{5233}{5549}a+\frac{4372}{5549}$ 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:  \( 60.0447869363 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 60.0447869363 \cdot 1}{8\cdot\sqrt{8084777718513664}}\cr\approx \mathstrut & 0.202763708602 \end{aligned}\]

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 + 6*x^14 - 14*x^12 + 24*x^11 - 18*x^10 + 15*x^8 - 20*x^7 + 22*x^6 - 24*x^5 + 24*x^4 - 20*x^3 + 12*x^2 - 4*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 + 6*x^14 - 14*x^12 + 24*x^11 - 18*x^10 + 15*x^8 - 20*x^7 + 22*x^6 - 24*x^5 + 24*x^4 - 20*x^3 + 12*x^2 - 4*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 + 6*x^14 - 14*x^12 + 24*x^11 - 18*x^10 + 15*x^8 - 20*x^7 + 22*x^6 - 24*x^5 + 24*x^4 - 20*x^3 + 12*x^2 - 4*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 = polynomial_ring(QQ); K, a = number_field(x^16 - 4*x^15 + 6*x^14 - 14*x^12 + 24*x^11 - 18*x^10 + 15*x^8 - 20*x^7 + 22*x^6 - 24*x^5 + 24*x^4 - 20*x^3 + 12*x^2 - 4*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

$D_4:D_4$ (as 16T141):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
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 64
The 16 conjugacy class representatives for $D_4:D_4$
Character table for $D_4:D_4$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), 4.2.1792.1, 4.2.448.1, \(\Q(\zeta_{8})\), 8.2.89915392.1 x2, 8.2.22478848.1 x2, 8.0.3211264.1

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

Degree 8 siblings: data not computed
Degree 16 siblings: data not computed
Degree 32 siblings: data not computed
Minimal sibling: 8.2.22478848.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.8.0.1}{8} }^{2}$ ${\href{/padicField/5.8.0.1}{8} }^{2}$ R ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.8.0.1}{8} }^{2}$ ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}$ ${\href{/padicField/19.8.0.1}{8} }^{2}$ ${\href{/padicField/23.2.0.1}{2} }^{8}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.2.0.1}{2} }^{8}$ ${\href{/padicField/59.8.0.1}{8} }^{2}$

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 2.2.8.36b1.27$x^{16} + 8 x^{15} + 40 x^{14} + 140 x^{13} + 380 x^{12} + 824 x^{11} + 1470 x^{10} + 2180 x^{9} + 2715 x^{8} + 2840 x^{7} + 2498 x^{6} + 1832 x^{5} + 1114 x^{4} + 548 x^{3} + 214 x^{2} + 60 x + 17$$8$$2$$36$$D_4\times C_2$$$[2, 2, 3]^{2}$$
\(7\) Copy content Toggle raw display 7.2.1.0a1.1$x^{2} + 6 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
7.2.1.0a1.1$x^{2} + 6 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
7.2.1.0a1.1$x^{2} + 6 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
7.2.1.0a1.1$x^{2} + 6 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
7.2.4.6a1.2$x^{8} + 24 x^{7} + 228 x^{6} + 1080 x^{5} + 2646 x^{4} + 3240 x^{3} + 2052 x^{2} + 648 x + 88$$4$$2$$6$$D_4$$$[\ ]_{4}^{2}$$

Spectrum of ring of integers

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