Properties

Label 16.0.861...496.23
Degree $16$
Signature $(0, 8)$
Discriminant $8.616\times 10^{22}$
Root discriminant \(27.13\)
Ramified primes $2,3$
Class number $1$
Class group trivial
Galois group $D_8:C_2$ (as 16T45)

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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 - 12*x^14 + 78*x^12 - 336*x^10 + 870*x^8 - 1152*x^6 + 684*x^4 - 144*x^2 + 36)
 
Copy content gp:K = bnfinit(y^16 - 12*y^14 + 78*y^12 - 336*y^10 + 870*y^8 - 1152*y^6 + 684*y^4 - 144*y^2 + 36, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 12*x^14 + 78*x^12 - 336*x^10 + 870*x^8 - 1152*x^6 + 684*x^4 - 144*x^2 + 36);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 12*x^14 + 78*x^12 - 336*x^10 + 870*x^8 - 1152*x^6 + 684*x^4 - 144*x^2 + 36)
 

\( x^{16} - 12x^{14} + 78x^{12} - 336x^{10} + 870x^{8} - 1152x^{6} + 684x^{4} - 144x^{2} + 36 \) 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:   \(86162309624498535530496\) \(\medspace = 2^{54}\cdot 3^{14}\) 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:  \(27.13\)
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:  $2^{29/8}3^{7/8}\approx 32.263749133641326$
Ramified primes:   \(2\), \(3\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_4$
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.
Maximal CM subfield:  8.0.191102976.5

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{6}a^{8}$, $\frac{1}{6}a^{9}$, $\frac{1}{30}a^{10}+\frac{1}{30}a^{8}+\frac{2}{5}a^{4}+\frac{1}{5}$, $\frac{1}{30}a^{11}+\frac{1}{30}a^{9}+\frac{2}{5}a^{5}+\frac{1}{5}a$, $\frac{1}{30}a^{12}-\frac{1}{30}a^{8}+\frac{2}{5}a^{6}-\frac{2}{5}a^{4}+\frac{1}{5}a^{2}-\frac{1}{5}$, $\frac{1}{30}a^{13}-\frac{1}{30}a^{9}+\frac{2}{5}a^{7}-\frac{2}{5}a^{5}+\frac{1}{5}a^{3}-\frac{1}{5}a$, $\frac{1}{112230}a^{14}-\frac{227}{37410}a^{12}-\frac{49}{7482}a^{10}+\frac{561}{12470}a^{8}+\frac{2696}{18705}a^{6}-\frac{293}{1247}a^{4}-\frac{19}{6235}a^{2}-\frac{2261}{6235}$, $\frac{1}{112230}a^{15}-\frac{227}{37410}a^{13}-\frac{49}{7482}a^{11}+\frac{561}{12470}a^{9}+\frac{2696}{18705}a^{7}-\frac{293}{1247}a^{5}-\frac{19}{6235}a^{3}-\frac{2261}{6235}a$ 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 $1$
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:   \( -\frac{1106}{56115} a^{14} + \frac{4156}{18705} a^{12} - \frac{25816}{18705} a^{10} + \frac{211489}{37410} a^{8} - \frac{247304}{18705} a^{6} + \frac{17134}{1247} a^{4} - \frac{27804}{6235} a^{2} + \frac{862}{6235} \)  (order $12$) 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:   $\frac{1663}{56115}a^{14}-\frac{393}{1247}a^{12}+\frac{2350}{1247}a^{10}-\frac{276959}{37410}a^{8}+\frac{295258}{18705}a^{6}-\frac{80369}{6235}a^{4}+\frac{19108}{6235}a^{2}-\frac{8158}{6235}$, $\frac{1661}{112230}a^{14}-\frac{5441}{37410}a^{12}+\frac{1967}{2494}a^{10}-\frac{50659}{18705}a^{8}+\frac{67417}{18705}a^{6}+\frac{28213}{6235}a^{4}-\frac{71463}{6235}a^{2}+\frac{25383}{6235}$, $\frac{1286}{18705}a^{15}-\frac{1589}{112230}a^{14}+\frac{5112}{6235}a^{13}+\frac{437}{2494}a^{12}-\frac{39661}{7482}a^{11}-\frac{7234}{6235}a^{10}+\frac{849437}{37410}a^{9}+\frac{95036}{18705}a^{8}-\frac{362422}{6235}a^{7}-\frac{254887}{18705}a^{6}+\frac{93502}{1247}a^{5}+\frac{119448}{6235}a^{4}-\frac{258671}{6235}a^{3}-\frac{80792}{6235}a^{2}+\frac{37756}{6235}a+\frac{28803}{6235}$, $\frac{91}{12470}a^{15}+\frac{1093}{56115}a^{14}+\frac{3851}{37410}a^{13}-\frac{1233}{6235}a^{12}-\frac{26299}{37410}a^{11}+\frac{7173}{6235}a^{10}+\frac{58388}{18705}a^{9}-\frac{165463}{37410}a^{8}-\frac{52652}{6235}a^{7}+\frac{33197}{3741}a^{6}+\frac{62571}{6235}a^{5}-\frac{42592}{6235}a^{4}+\frac{1844}{6235}a^{3}+\frac{4662}{1247}a^{2}-\frac{13753}{6235}a+\frac{4303}{6235}$, $\frac{1729}{56115}a^{14}-\frac{1432}{3741}a^{12}+\frac{46514}{18705}a^{10}-\frac{133823}{12470}a^{8}+\frac{516454}{18705}a^{6}-\frac{212646}{6235}a^{4}+\frac{93914}{6235}a^{2}+\frac{7634}{6235}$, $\frac{9143}{112230}a^{15}+\frac{1919}{37410}a^{14}-\frac{5687}{6235}a^{13}-\frac{1329}{2494}a^{12}+\frac{211567}{37410}a^{11}+\frac{11711}{3741}a^{10}-\frac{868223}{37410}a^{9}-\frac{451573}{37410}a^{8}+\frac{1021372}{18705}a^{7}+\frac{153202}{6235}a^{6}-\frac{372074}{6235}a^{5}-\frac{109033}{6235}a^{4}+\frac{190407}{6235}a^{3}+\frac{30281}{6235}a^{2}-\frac{13379}{1247}a-\frac{31566}{6235}$, $\frac{19}{774}a^{15}-\frac{2519}{112230}a^{14}+\frac{61}{215}a^{13}+\frac{4708}{18705}a^{12}-\frac{77}{43}a^{11}-\frac{19573}{12470}a^{10}+\frac{4832}{645}a^{9}+\frac{80747}{12470}a^{8}-\frac{11794}{645}a^{7}-\frac{289366}{18705}a^{6}+\frac{4628}{215}a^{5}+\frac{108951}{6235}a^{4}-\frac{2409}{215}a^{3}-\frac{53146}{6235}a^{2}+\frac{569}{215}a+\frac{14127}{6235}$ 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:  \( 695883.8930798479 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 695883.8930798479 \cdot 1}{12\cdot\sqrt{86162309624498535530496}}\cr\approx \mathstrut & 0.479883501764229 \end{aligned}\]

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 - 12*x^14 + 78*x^12 - 336*x^10 + 870*x^8 - 1152*x^6 + 684*x^4 - 144*x^2 + 36) 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 - 12*x^14 + 78*x^12 - 336*x^10 + 870*x^8 - 1152*x^6 + 684*x^4 - 144*x^2 + 36, 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 - 12*x^14 + 78*x^12 - 336*x^10 + 870*x^8 - 1152*x^6 + 684*x^4 - 144*x^2 + 36); 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 - 12*x^14 + 78*x^12 - 336*x^10 + 870*x^8 - 1152*x^6 + 684*x^4 - 144*x^2 + 36); 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

$D_8:C_2$ (as 16T45):

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 32
The 11 conjugacy class representatives for $D_8:C_2$
Character table for $D_8:C_2$

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{3 + \sqrt{3}})\), \(\Q(\sqrt{-3 + \sqrt{3}})\), \(\Q(\zeta_{12})\), 8.0.191102976.5

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 8 siblings: 8.4.587068342272.3, 8.4.587068342272.4
Degree 16 siblings: 16.0.344649238497994142121984.83, 16.8.1378596953991976568487936.31, 16.0.1378596953991976568487936.48
Minimal sibling: 8.4.587068342272.3

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 ${\href{/padicField/5.2.0.1}{2} }^{8}$ ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.2.0.1}{2} }^{6}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ ${\href{/padicField/17.8.0.1}{8} }^{2}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.8.0.1}{8} }^{2}$ ${\href{/padicField/37.2.0.1}{2} }^{6}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ ${\href{/padicField/41.8.0.1}{8} }^{2}$ ${\href{/padicField/43.2.0.1}{2} }^{8}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.2.0.1}{2} }^{8}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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 2.2.8.54a1.1031$x^{16} + 8 x^{15} + 40 x^{14} + 144 x^{13} + 402 x^{12} + 904 x^{11} + 1676 x^{10} + 2608 x^{9} + 3433 x^{8} + 3856 x^{7} + 3680 x^{6} + 2976 x^{5} + 1996 x^{4} + 1104 x^{3} + 472 x^{2} + 152 x + 25$$8$$2$$54$16T45$$[2, 3, \frac{7}{2}, \frac{9}{2}]^{2}$$
\(3\) Copy content Toggle raw display 3.2.8.14a1.2$x^{16} + 16 x^{15} + 128 x^{14} + 672 x^{13} + 2576 x^{12} + 7616 x^{11} + 17920 x^{10} + 34176 x^{9} + 53344 x^{8} + 68352 x^{7} + 71680 x^{6} + 60928 x^{5} + 41216 x^{4} + 21504 x^{3} + 8192 x^{2} + 2048 x + 259$$8$$2$$14$$QD_{16}$$$[\ ]_{8}^{2}$$

Spectrum of ring of integers

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