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Properties

Label 16.8.429...000.1
Degree $16$
Signature $[8, 4]$
Discriminant $4.295\times 10^{21}$
Root discriminant \(22.49\)
Ramified primes $2,5$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^5:C_4$ (as 16T227)

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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 - 12*x^14 + 62*x^12 - 220*x^10 + 551*x^8 - 740*x^6 + 398*x^4 - 96*x^2 + 1)
 
Copy content gp:K = bnfinit(y^16 - 12*y^14 + 62*y^12 - 220*y^10 + 551*y^8 - 740*y^6 + 398*y^4 - 96*y^2 + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 12*x^14 + 62*x^12 - 220*x^10 + 551*x^8 - 740*x^6 + 398*x^4 - 96*x^2 + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 12*x^14 + 62*x^12 - 220*x^10 + 551*x^8 - 740*x^6 + 398*x^4 - 96*x^2 + 1)
 

\( x^{16} - 12x^{14} + 62x^{12} - 220x^{10} + 551x^{8} - 740x^{6} + 398x^{4} - 96x^{2} + 1 \) 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:  $[8, 4]$
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:   \(4294967296000000000000\) \(\medspace = 2^{44}\cdot 5^{12}\) 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:  \(22.49\)
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:  $2^{49/16}5^{3/4}\approx 27.93391870957562$
Ramified primes:   \(2\), \(5\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{10}-\frac{1}{4}a^{6}-\frac{1}{4}a^{2}-\frac{1}{4}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{11}-\frac{1}{4}a^{7}-\frac{1}{4}a^{3}-\frac{1}{4}a$, $\frac{1}{2024804}a^{14}-\frac{197153}{2024804}a^{12}+\frac{160327}{1012402}a^{10}-\frac{175755}{2024804}a^{8}+\frac{35479}{1012402}a^{6}+\frac{132817}{2024804}a^{4}-\frac{935275}{2024804}a^{2}-\frac{67283}{1012402}$, $\frac{1}{2024804}a^{15}-\frac{197153}{2024804}a^{13}+\frac{160327}{1012402}a^{11}-\frac{175755}{2024804}a^{9}+\frac{35479}{1012402}a^{7}+\frac{132817}{2024804}a^{5}-\frac{935275}{2024804}a^{3}-\frac{67283}{1012402}a$ 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:  $C_{2}$, which has order $2$ (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:  $11$
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:   $\frac{10276}{506201}a^{14}-\frac{127826}{506201}a^{12}+\frac{684396}{506201}a^{10}-\frac{2464217}{506201}a^{8}+\frac{6309380}{506201}a^{6}-\frac{9002022}{506201}a^{4}+\frac{4908296}{506201}a^{2}-\frac{365285}{506201}$, $\frac{6681}{2024804}a^{14}-\frac{44191}{2024804}a^{12}+\frac{23371}{1012402}a^{10}+\frac{167165}{2024804}a^{8}-\frac{879271}{1012402}a^{6}+\frac{6560637}{2024804}a^{4}-\frac{8126347}{2024804}a^{2}+\frac{247483}{506201}$, $\frac{81859}{2024804}a^{15}-\frac{1059547}{2024804}a^{13}+\frac{1485886}{506201}a^{11}-\frac{22156567}{2024804}a^{9}+\frac{14779991}{506201}a^{7}-\frac{93059259}{2024804}a^{5}+\frac{67018753}{2024804}a^{3}-\frac{3922616}{506201}a$, $\frac{111233}{2024804}a^{15}-\frac{1292329}{2024804}a^{13}+\frac{3229167}{1012402}a^{11}-\frac{22546139}{2024804}a^{9}+\frac{27427465}{1012402}a^{7}-\frac{68179959}{2024804}a^{5}+\frac{31357505}{2024804}a^{3}-\frac{2439159}{1012402}a$, $\frac{999309}{2024804}a^{15}-\frac{17671}{506201}a^{14}-\frac{5971747}{1012402}a^{13}+\frac{215381}{506201}a^{12}+\frac{61358593}{2024804}a^{11}-\frac{2256685}{1012402}a^{10}-\frac{216683559}{2024804}a^{9}+\frac{8039955}{1012402}a^{8}+\frac{539436007}{2024804}a^{7}-\frac{10162961}{506201}a^{6}-\frac{711384951}{2024804}a^{5}+\frac{13912257}{506201}a^{4}+\frac{90115574}{506201}a^{3}-\frac{7304939}{506201}a^{2}-\frac{81205203}{2024804}a+\frac{3110383}{1012402}$, $\frac{154747}{1012402}a^{15}-\frac{17671}{506201}a^{14}-\frac{3745449}{2024804}a^{13}+\frac{215381}{506201}a^{12}+\frac{19524865}{2024804}a^{11}-\frac{2256685}{1012402}a^{10}-\frac{34813325}{1012402}a^{9}+\frac{8039955}{1012402}a^{8}+\frac{175702815}{2024804}a^{7}-\frac{10162961}{506201}a^{6}-\frac{120216541}{1012402}a^{5}+\frac{13912257}{506201}a^{4}+\frac{134072647}{2024804}a^{3}-\frac{7304939}{506201}a^{2}-\frac{33091595}{2024804}a+\frac{3110383}{1012402}$, $\frac{58201}{1012402}a^{15}+\frac{23615}{1012402}a^{14}-\frac{1393573}{2024804}a^{13}-\frac{118749}{506201}a^{12}+\frac{7122987}{2024804}a^{11}+\frac{995853}{1012402}a^{10}-\frac{6230986}{506201}a^{9}-\frac{1574766}{506201}a^{8}+\frac{61727921}{2024804}a^{7}+\frac{3111136}{506201}a^{6}-\frac{19798466}{506201}a^{5}-\frac{2985147}{1012402}a^{4}+\frac{35696389}{2024804}a^{3}-\frac{2994299}{1012402}a^{2}-\frac{12522813}{2024804}a+\frac{583095}{506201}$, $\frac{156095}{1012402}a^{15}+\frac{47785}{2024804}a^{14}-\frac{3758887}{2024804}a^{13}-\frac{555495}{2024804}a^{12}+\frac{19416741}{2024804}a^{11}+\frac{1392163}{1012402}a^{10}-\frac{17161398}{506201}a^{9}-\frac{9689703}{2024804}a^{8}+\frac{171611997}{2024804}a^{7}+\frac{11739489}{1012402}a^{6}-\frac{56896883}{506201}a^{5}-\frac{29447451}{2024804}a^{4}+\frac{115228991}{2024804}a^{3}+\frac{11506837}{2024804}a^{2}-\frac{26354801}{2024804}a-\frac{624003}{506201}$, $\frac{5138}{506201}a^{15}-\frac{2257}{506201}a^{14}-\frac{63913}{506201}a^{13}+\frac{23642}{506201}a^{12}+\frac{342198}{506201}a^{11}-\frac{203497}{1012402}a^{10}-\frac{2464217}{1012402}a^{9}+\frac{323652}{506201}a^{8}+\frac{3154690}{506201}a^{7}-\frac{698891}{506201}a^{6}-\frac{4501011}{506201}a^{5}+\frac{409224}{506201}a^{4}+\frac{5414497}{1012402}a^{3}+\frac{621211}{1012402}a^{2}-\frac{941944}{506201}a-\frac{516477}{1012402}$, $\frac{227635}{2024804}a^{15}-\frac{23615}{1012402}a^{14}-\frac{1342951}{1012402}a^{13}+\frac{118749}{506201}a^{12}+\frac{13581321}{2024804}a^{11}-\frac{995853}{1012402}a^{10}-\frac{47470083}{2024804}a^{9}+\frac{1574766}{506201}a^{8}+\frac{116582851}{2024804}a^{7}-\frac{3111136}{506201}a^{6}-\frac{147373823}{2024804}a^{5}+\frac{2985147}{1012402}a^{4}+\frac{33526947}{1012402}a^{3}+\frac{2994299}{1012402}a^{2}-\frac{15376327}{2024804}a-\frac{583095}{506201}$, $\frac{107025}{2024804}a^{15}+\frac{13831}{1012402}a^{14}-\frac{333986}{506201}a^{13}-\frac{342913}{2024804}a^{12}+\frac{7184369}{2024804}a^{11}+\frac{1795629}{2024804}a^{10}-\frac{26072167}{2024804}a^{9}-\frac{3127409}{1012402}a^{8}+\frac{67577281}{2024804}a^{7}+\frac{15484949}{2024804}a^{6}-\frac{98600051}{2024804}a^{5}-\frac{9629321}{1012402}a^{4}+\frac{15666598}{506201}a^{3}+\frac{3899467}{2024804}a^{2}-\frac{16199931}{2024804}a+\frac{743663}{2024804}$ 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:  \( 131049.544859 \) (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)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 131049.544859 \cdot 1}{2\cdot\sqrt{4294967296000000000000}}\cr\approx \mathstrut & 0.398919282660 \end{aligned}\] (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 - 12*x^14 + 62*x^12 - 220*x^10 + 551*x^8 - 740*x^6 + 398*x^4 - 96*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 - 12*x^14 + 62*x^12 - 220*x^10 + 551*x^8 - 740*x^6 + 398*x^4 - 96*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 - 12*x^14 + 62*x^12 - 220*x^10 + 551*x^8 - 740*x^6 + 398*x^4 - 96*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 - 12*x^14 + 62*x^12 - 220*x^10 + 551*x^8 - 740*x^6 + 398*x^4 - 96*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

$C_2^5:C_4$ (as 16T227):

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 128
The 26 conjugacy class representatives for $C_2^5:C_4$
Character table for $C_2^5:C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.1600.1, 4.4.8000.1, 4.2.2000.1, 8.4.3276800000.2, 8.4.3276800000.1, 8.4.1024000000.2

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: This field is its own minimal sibling

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 ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.8.0.1}{8} }^{2}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

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 2.2.8.44d13.77$x^{16} + 12 x^{15} + 68 x^{14} + 256 x^{13} + 716 x^{12} + 1580 x^{11} + 2844 x^{10} + 4268 x^{9} + 5401 x^{8} + 5796 x^{7} + 5272 x^{6} + 4044 x^{5} + 2592 x^{4} + 1360 x^{3} + 576 x^{2} + 188 x + 43$$8$$2$$44$16T227$$[2, 2, 3, 3, \frac{7}{2}]^{4}$$
\(5\) Copy content Toggle raw display 5.1.4.3a1.4$x^{4} + 20$$4$$1$$3$$C_4$$$[\ ]_{4}$$
5.1.4.3a1.4$x^{4} + 20$$4$$1$$3$$C_4$$$[\ ]_{4}$$
5.2.4.6a1.2$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 664 x^{4} + 704 x^{3} + 416 x^{2} + 128 x + 21$$4$$2$$6$$C_4\times C_2$$$[\ ]_{4}^{2}$$

Spectrum of ring of integers

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