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Properties

Label 16.0.441...625.1
Degree $16$
Signature $[0, 8]$
Discriminant $4.413\times 10^{22}$
Root discriminant \(26.02\)
Ramified primes $5,29,109$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^6:D_4$ (as 16T969)

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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 + 11*x^14 + 78*x^12 + 304*x^10 + 313*x^8 - 412*x^6 - 78*x^4 + 303*x^2 + 1)
 
Copy content gp:K = bnfinit(y^16 + 11*y^14 + 78*y^12 + 304*y^10 + 313*y^8 - 412*y^6 - 78*y^4 + 303*y^2 + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 11*x^14 + 78*x^12 + 304*x^10 + 313*x^8 - 412*x^6 - 78*x^4 + 303*x^2 + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 + 11*x^14 + 78*x^12 + 304*x^10 + 313*x^8 - 412*x^6 - 78*x^4 + 303*x^2 + 1)
 

\( x^{16} + 11x^{14} + 78x^{12} + 304x^{10} + 313x^{8} - 412x^{6} - 78x^{4} + 303x^{2} + 1 \) 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:  $[0, 8]$
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:   \(44133664812763327950625\) \(\medspace = 5^{4}\cdot 29^{8}\cdot 109^{4}\) 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:  \(26.02\)
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:  $5^{1/2}29^{1/2}109^{1/2}\approx 125.7179382586272$
Ramified primes:   \(5\), \(29\), \(109\) 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.
Maximal CM subfield:  8.0.1927340725.1

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{60}a^{12}+\frac{7}{30}a^{10}+\frac{11}{60}a^{8}+\frac{11}{60}a^{6}-\frac{13}{60}a^{4}-\frac{1}{2}a^{2}-\frac{11}{60}$, $\frac{1}{60}a^{13}+\frac{7}{30}a^{11}+\frac{11}{60}a^{9}+\frac{11}{60}a^{7}-\frac{13}{60}a^{5}-\frac{1}{2}a^{3}-\frac{11}{60}a$, $\frac{1}{204650520}a^{14}-\frac{1}{120}a^{13}-\frac{271597}{204650520}a^{12}-\frac{7}{60}a^{11}-\frac{37586563}{204650520}a^{10}-\frac{11}{120}a^{9}+\frac{2807477}{20465052}a^{8}-\frac{11}{120}a^{7}+\frac{18316333}{102325260}a^{6}+\frac{13}{120}a^{5}-\frac{12701309}{68216840}a^{4}+\frac{1}{4}a^{3}+\frac{19879879}{204650520}a^{2}-\frac{49}{120}a-\frac{28672693}{68216840}$, $\frac{1}{204650520}a^{15}+\frac{179228}{25581315}a^{13}-\frac{1}{120}a^{12}-\frac{4570223}{68216840}a^{11}-\frac{7}{60}a^{10}+\frac{15611467}{68216840}a^{9}-\frac{11}{120}a^{8}-\frac{15644321}{68216840}a^{7}-\frac{11}{120}a^{6}+\frac{2102543}{10232526}a^{5}+\frac{13}{120}a^{4}-\frac{31282751}{204650520}a^{3}+\frac{1}{4}a^{2}-\frac{245245}{20465052}a+\frac{11}{120}$ 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:  $7$
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:   $\frac{697691}{13643368}a^{15}-\frac{1090703}{204650520}a^{14}+\frac{22921085}{40930104}a^{13}-\frac{991913}{17054210}a^{12}+\frac{162178201}{40930104}a^{11}-\frac{84599129}{204650520}a^{10}+\frac{157179973}{10232526}a^{9}-\frac{328475207}{204650520}a^{8}+\frac{78370703}{5116263}a^{7}-\frac{68219779}{40930104}a^{6}-\frac{884630021}{40930104}a^{5}+\frac{51129244}{25581315}a^{4}-\frac{39899931}{13643368}a^{3}+\frac{139392733}{204650520}a^{2}+\frac{646376135}{40930104}a-\frac{145101703}{102325260}$, $\frac{697691}{13643368}a^{15}+\frac{1090703}{204650520}a^{14}+\frac{22921085}{40930104}a^{13}+\frac{991913}{17054210}a^{12}+\frac{162178201}{40930104}a^{11}+\frac{84599129}{204650520}a^{10}+\frac{157179973}{10232526}a^{9}+\frac{328475207}{204650520}a^{8}+\frac{78370703}{5116263}a^{7}+\frac{68219779}{40930104}a^{6}-\frac{884630021}{40930104}a^{5}-\frac{51129244}{25581315}a^{4}-\frac{39899931}{13643368}a^{3}-\frac{139392733}{204650520}a^{2}+\frac{646376135}{40930104}a+\frac{145101703}{102325260}$, $\frac{3082487}{40930104}a^{15}+\frac{1090703}{204650520}a^{14}+\frac{56307091}{68216840}a^{13}+\frac{991913}{17054210}a^{12}+\frac{1196078227}{204650520}a^{11}+\frac{84599129}{204650520}a^{10}+\frac{2322173579}{102325260}a^{9}+\frac{328475207}{204650520}a^{8}+\frac{2341883549}{102325260}a^{7}+\frac{68219779}{40930104}a^{6}-\frac{6375871489}{204650520}a^{5}-\frac{51129244}{25581315}a^{4}-\frac{182667439}{40930104}a^{3}-\frac{139392733}{204650520}a^{2}+\frac{4369312927}{204650520}a+\frac{196264333}{102325260}$, $\frac{27237739}{102325260}a^{15}+\frac{150096121}{51162630}a^{13}+\frac{2131039133}{102325260}a^{11}+\frac{1665400057}{20465052}a^{9}+\frac{8713056389}{102325260}a^{7}-\frac{1832044883}{17054210}a^{5}-\frac{2315778089}{102325260}a^{3}+\frac{672682667}{8527105}a$, $\frac{697691}{6821684}a^{15}+\frac{22921085}{20465052}a^{13}+\frac{162178201}{20465052}a^{11}+\frac{157179973}{5116263}a^{9}+\frac{156741406}{5116263}a^{7}-\frac{884630021}{20465052}a^{5}-\frac{39899931}{6821684}a^{3}+\frac{625911083}{20465052}a$, $\frac{815281}{13643368}a^{15}-\frac{208477}{25581315}a^{14}+\frac{33291587}{51162630}a^{13}-\frac{18398447}{204650520}a^{12}+\frac{939168277}{204650520}a^{11}-\frac{21887793}{34108420}a^{10}+\frac{3613672663}{204650520}a^{9}-\frac{172039763}{68216840}a^{8}+\frac{3427405723}{204650520}a^{7}-\frac{38051871}{13643368}a^{6}-\frac{677615578}{25581315}a^{5}+\frac{600268399}{204650520}a^{4}-\frac{23012283}{13643368}a^{3}+\frac{160117823}{102325260}a^{2}+\frac{2037373141}{102325260}a+\frac{44010113}{204650520}$, $\frac{2256503}{51162630}a^{15}-\frac{30617}{40930104}a^{14}+\frac{6880061}{13643368}a^{13}-\frac{4148077}{204650520}a^{12}+\frac{73101425}{20465052}a^{11}-\frac{9555611}{68216840}a^{10}+\frac{2940005249}{204650520}a^{9}-\frac{14054691}{17054210}a^{8}+\frac{3217401941}{204650520}a^{7}-\frac{23848911}{17054210}a^{6}-\frac{4050333611}{204650520}a^{5}+\frac{268993121}{204650520}a^{4}-\frac{493367261}{102325260}a^{3}+\frac{9321241}{40930104}a^{2}+\frac{2880047843}{204650520}a-\frac{259000103}{204650520}$ 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:  \( 47218.1659606 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 47218.1659606 \cdot 1}{2\cdot\sqrt{44133664812763327950625}}\cr\approx \mathstrut & 0.272981407918 \end{aligned}\] (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 + 11*x^14 + 78*x^12 + 304*x^10 + 313*x^8 - 412*x^6 - 78*x^4 + 303*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 + 11*x^14 + 78*x^12 + 304*x^10 + 313*x^8 - 412*x^6 - 78*x^4 + 303*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 + 11*x^14 + 78*x^12 + 304*x^10 + 313*x^8 - 412*x^6 - 78*x^4 + 303*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 + 11*x^14 + 78*x^12 + 304*x^10 + 313*x^8 - 412*x^6 - 78*x^4 + 303*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

$C_2^6:D_4$ (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

\(\Q(\sqrt{29}) \), 4.4.4205.1, 8.4.1927340725.1, 8.0.1927340725.1, 8.4.210080139025.3

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 ${\href{/padicField/2.8.0.1}{8} }^{2}$ ${\href{/padicField/3.4.0.1}{4} }^{4}$ R ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{4}$ ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ ${\href{/padicField/17.8.0.1}{8} }^{2}$ ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ R ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.8.0.1}{8} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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
\(5\) Copy content Toggle raw display 5.2.1.0a1.1$x^{2} + 4 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
5.2.1.0a1.1$x^{2} + 4 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
5.2.1.0a1.1$x^{2} + 4 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
5.2.1.0a1.1$x^{2} + 4 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
5.2.2.2a1.2$x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
5.2.2.2a1.2$x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
\(29\) Copy content Toggle raw display 29.4.2.4a1.2$x^{8} + 4 x^{6} + 30 x^{5} + 8 x^{4} + 60 x^{3} + 233 x^{2} + 60 x + 33$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
29.4.2.4a1.2$x^{8} + 4 x^{6} + 30 x^{5} + 8 x^{4} + 60 x^{3} + 233 x^{2} + 60 x + 33$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
\(109\) Copy content Toggle raw display $\Q_{109}$$x + 103$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{109}$$x + 103$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{109}$$x + 103$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{109}$$x + 103$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{109}$$x + 103$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{109}$$x + 103$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{109}$$x + 103$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{109}$$x + 103$$1$$1$$0$Trivial$$[\ ]$$
109.2.2.2a1.2$x^{4} + 216 x^{3} + 11676 x^{2} + 1296 x + 145$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
109.2.2.2a1.2$x^{4} + 216 x^{3} + 11676 x^{2} + 1296 x + 145$$2$$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)