Properties

Label 16.6.188...784.2
Degree $16$
Signature $(6, 5)$
Discriminant $-1.889\times 10^{22}$
Root discriminant \(24.68\)
Ramified prime $2$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^7.D_8$ (as 16T1476)

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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 + 8*x^10 - 74*x^8 + 96*x^6 - 40*x^4 + 8*x^2 - 1)
 
Copy content gp:K = bnfinit(y^16 + 8*y^10 - 74*y^8 + 96*y^6 - 40*y^4 + 8*y^2 - 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 8*x^10 - 74*x^8 + 96*x^6 - 40*x^4 + 8*x^2 - 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 + 8*x^10 - 74*x^8 + 96*x^6 - 40*x^4 + 8*x^2 - 1)
 

\( x^{16} + 8x^{10} - 74x^{8} + 96x^{6} - 40x^{4} + 8x^{2} - 1 \) 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:  $(6, 5)$
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:   \(-18889465931478580854784\) \(\medspace = -\,2^{74}\) 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:  \(24.68\)
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:  $2^{5525/1024}\approx 42.09298197761502$
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:  \(\Q(\sqrt{-1}) \)
$\Aut(K/\Q)$:   $C_2$
Copy content comment:Automorphisms
 
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.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{62087}a^{14}+\frac{19476}{62087}a^{12}+\frac{25093}{62087}a^{10}+\frac{24499}{62087}a^{8}+\frac{3855}{62087}a^{6}+\frac{16893}{62087}a^{4}+\frac{9015}{62087}a^{2}-\frac{5888}{62087}$, $\frac{1}{62087}a^{15}+\frac{19476}{62087}a^{13}+\frac{25093}{62087}a^{11}+\frac{24499}{62087}a^{9}+\frac{3855}{62087}a^{7}+\frac{16893}{62087}a^{5}+\frac{9015}{62087}a^{3}-\frac{5888}{62087}a$ 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:  $10$
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:   $\frac{8556}{62087}a^{14}-\frac{4852}{62087}a^{12}-\frac{1138}{62087}a^{10}+\frac{69819}{62087}a^{8}-\frac{667774}{62087}a^{6}+\frac{1177625}{62087}a^{4}-\frac{724758}{62087}a^{2}+\frac{99003}{62087}$, $a$, $\frac{51673}{62087}a^{14}+\frac{15165}{62087}a^{12}+\frac{5681}{62087}a^{10}+\frac{417506}{62087}a^{8}-\frac{3700901}{62087}a^{6}+\frac{3880250}{62087}a^{4}-\frac{1000058}{62087}a^{2}+\frac{37763}{62087}$, $\frac{39094}{62087}a^{15}+\frac{21863}{62087}a^{13}+\frac{11142}{62087}a^{11}+\frac{320279}{62087}a^{9}-\frac{2709607}{62087}a^{7}+\frac{2230655}{62087}a^{5}-\frac{221750}{62087}a^{3}+\frac{95211}{62087}a$, $\frac{62614}{62087}a^{15}+\frac{19497}{62087}a^{13}-\frac{520}{62087}a^{11}+\frac{493573}{62087}a^{9}-\frac{4487550}{62087}a^{7}+\frac{4556521}{62087}a^{5}-\frac{650664}{62087}a^{3}+\frac{125548}{62087}a$, $\frac{56117}{62087}a^{15}+\frac{32396}{62087}a^{14}+\frac{17231}{62087}a^{13}+\frac{16402}{62087}a^{12}+\frac{10721}{62087}a^{11}+\frac{7737}{62087}a^{10}+\frac{452551}{62087}a^{9}+\frac{259831}{62087}a^{8}-\frac{4015728}{62087}a^{7}-\frac{2267596}{62087}a^{6}+\frac{4199994}{62087}a^{5}+\frac{1955507}{62087}a^{4}-\frac{1356035}{62087}a^{3}-\frac{317743}{62087}a^{2}+\frac{444727}{62087}a+\frac{169877}{62087}$, $\frac{29810}{62087}a^{15}-\frac{15165}{62087}a^{14}+\frac{4023}{62087}a^{13}-\frac{5681}{62087}a^{12}-\frac{1846}{62087}a^{11}-\frac{4122}{62087}a^{10}+\frac{234157}{62087}a^{9}-\frac{122901}{62087}a^{8}-\frac{2178532}{62087}a^{7}+\frac{1080358}{62087}a^{6}+\frac{2538240}{62087}a^{5}-\frac{1066862}{62087}a^{4}-\frac{720430}{62087}a^{3}+\frac{375621}{62087}a^{2}+\frac{122843}{62087}a-\frac{113760}{62087}$, $\frac{89920}{62087}a^{15}-\frac{728}{62087}a^{14}-\frac{6089}{62087}a^{13}-\frac{22692}{62087}a^{12}-\frac{3194}{62087}a^{11}-\frac{14126}{62087}a^{10}+\frac{724190}{62087}a^{9}-\frac{16303}{62087}a^{8}-\frac{6695517}{62087}a^{7}-\frac{136699}{62087}a^{6}+\frac{9062720}{62087}a^{5}+\frac{1485210}{62087}a^{4}-\frac{3890553}{62087}a^{3}-\frac{1223438}{62087}a^{2}+\frac{401498}{62087}a+\frac{188722}{62087}$, $\frac{132127}{62087}a^{15}+\frac{2586}{62087}a^{14}+\frac{47650}{62087}a^{13}+\frac{12379}{62087}a^{12}+\frac{17011}{62087}a^{11}+\frac{9583}{62087}a^{10}+\frac{1067020}{62087}a^{9}+\frac{25674}{62087}a^{8}-\frac{9387300}{62087}a^{7}-\frac{89064}{62087}a^{6}+\frac{9306811}{62087}a^{5}-\frac{582733}{62087}a^{4}-\frac{1876800}{62087}a^{3}+\frac{402687}{62087}a^{2}+\frac{110508}{62087}a+\frac{47034}{62087}$, $\frac{45904}{62087}a^{15}+\frac{30739}{62087}a^{14}+\frac{35591}{62087}a^{13}+\frac{29910}{62087}a^{12}+\frac{31048}{62087}a^{11}+\frac{26926}{62087}a^{10}+\frac{392787}{62087}a^{9}+\frac{269886}{62087}a^{8}-\frac{3092380}{62087}a^{7}-\frac{2012022}{62087}a^{6}+\frac{2038513}{62087}a^{5}+\frac{971651}{62087}a^{4}-\frac{481991}{62087}a^{3}-\frac{168457}{62087}a^{2}+\frac{230307}{62087}a+\frac{116547}{62087}$ 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:  \( 283316.581727 \) (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)
 
Unit signature rank:  \( 6 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{5}\cdot 283316.581727 \cdot 1}{2\cdot\sqrt{18889465931478580854784}}\cr\approx \mathstrut & 0.64596869822 \end{aligned}\] (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^10 - 74*x^8 + 96*x^6 - 40*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^10 - 74*x^8 + 96*x^6 - 40*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^10 - 74*x^8 + 96*x^6 - 40*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^10 - 74*x^8 + 96*x^6 - 40*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

$C_2^7.D_8$ (as 16T1476):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
Copy content gp:polgalois(K.pol)
 
Copy content magma: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 2048
The 47 conjugacy class representatives for $C_2^7.D_8$
Character table for $C_2^7.D_8$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{1 + \sqrt{2}})\), 8.4.536870912.1

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 16 siblings: 16.2.18889465931478580854784.16, 16.2.18889465931478580854784.17, 16.6.18889465931478580854784.6, 16.6.18889465931478580854784.8, 16.4.9444732965739290427392.12, 16.0.9444732965739290427392.9, 16.4.9444732965739290427392.14, 16.0.9444732965739290427392.15, some data not computed
Degree 32 siblings: data not computed
Minimal sibling: 16.4.9444732965739290427392.5

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R $16$ ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{4}$ ${\href{/padicField/7.4.0.1}{4} }^{3}{,}\,{\href{/padicField/7.1.0.1}{1} }^{4}$ $16$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}$ $16$ ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{6}$ ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ $16$ ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{6}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ $16$

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 2.1.16.74a1.12$x^{16} + 16 x^{13} + 20 x^{12} + 16 x^{11} + 48 x^{10} + 32 x^{9} + 8 x^{8} + 32 x^{7} + 16 x^{6} + 32 x^{5} + 8 x^{4} + 32 x^{3} + 16 x^{2} + 32 x + 2$$16$$1$$74$16T1476$$[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{9}{2}, \frac{19}{4}, \frac{39}{8}, \frac{21}{4}, \frac{43}{8}, \frac{45}{8}]$$

Spectrum of ring of integers

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