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Properties

Label 16.0.119...000.1
Degree $16$
Signature $[0, 8]$
Discriminant $1.198\times 10^{19}$
Root discriminant \(15.57\)
Ramified primes $2,5,13$
Class number $1$
Class group trivial
Galois group $(C_2^2\times C_4^2):D_8$ (as 16T1276)

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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 - 6*x^15 + 16*x^14 - 20*x^13 + 4*x^12 - 8*x^11 + 104*x^10 - 196*x^9 + 44*x^8 + 272*x^7 - 316*x^6 + 24*x^5 + 168*x^4 - 92*x^3 - 4*x^2 + 8*x + 2)
 
Copy content gp:K = bnfinit(y^16 - 6*y^15 + 16*y^14 - 20*y^13 + 4*y^12 - 8*y^11 + 104*y^10 - 196*y^9 + 44*y^8 + 272*y^7 - 316*y^6 + 24*y^5 + 168*y^4 - 92*y^3 - 4*y^2 + 8*y + 2, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 6*x^15 + 16*x^14 - 20*x^13 + 4*x^12 - 8*x^11 + 104*x^10 - 196*x^9 + 44*x^8 + 272*x^7 - 316*x^6 + 24*x^5 + 168*x^4 - 92*x^3 - 4*x^2 + 8*x + 2);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 6*x^15 + 16*x^14 - 20*x^13 + 4*x^12 - 8*x^11 + 104*x^10 - 196*x^9 + 44*x^8 + 272*x^7 - 316*x^6 + 24*x^5 + 168*x^4 - 92*x^3 - 4*x^2 + 8*x + 2)
 

\( x^{16} - 6 x^{15} + 16 x^{14} - 20 x^{13} + 4 x^{12} - 8 x^{11} + 104 x^{10} - 196 x^{9} + 44 x^{8} + \cdots + 2 \) 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:   \(11979351654400000000\) \(\medspace = 2^{30}\cdot 5^{8}\cdot 13^{4}\) 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:  \(15.57\)
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:  $2^{15/8}5^{3/4}13^{1/2}\approx 44.221189586044886$
Ramified primes:   \(2\), \(5\), \(13\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_2$
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:  \(\Q(\sqrt{-1}) \)

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{13}a^{14}-\frac{4}{13}a^{13}-\frac{1}{13}a^{12}+\frac{1}{13}a^{11}+\frac{2}{13}a^{10}-\frac{5}{13}a^{8}+\frac{2}{13}a^{7}+\frac{2}{13}a^{6}-\frac{2}{13}a^{5}+\frac{3}{13}a^{3}+\frac{5}{13}a^{2}-\frac{5}{13}a+\frac{6}{13}$, $\frac{1}{46066007}a^{15}+\frac{181820}{46066007}a^{14}+\frac{27155}{648817}a^{13}+\frac{20385074}{46066007}a^{12}+\frac{1758206}{46066007}a^{11}+\frac{7193263}{46066007}a^{10}+\frac{5536981}{46066007}a^{9}-\frac{77125}{648817}a^{8}+\frac{17771131}{46066007}a^{7}-\frac{10960147}{46066007}a^{6}+\frac{11209472}{46066007}a^{5}-\frac{6393202}{46066007}a^{4}-\frac{20726046}{46066007}a^{3}-\frac{5118361}{46066007}a^{2}+\frac{16888692}{46066007}a+\frac{762512}{46066007}$ 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:   \( \frac{379421}{3543539} a^{15} - \frac{2834571}{3543539} a^{14} + \frac{113022}{49909} a^{13} - \frac{10566231}{3543539} a^{12} + \frac{713664}{3543539} a^{11} + \frac{2323994}{3543539} a^{10} + \frac{49141234}{3543539} a^{9} - \frac{1497379}{49909} a^{8} + \frac{19871632}{3543539} a^{7} + \frac{169717818}{3543539} a^{6} - \frac{186611832}{3543539} a^{5} + \frac{352252}{3543539} a^{4} + \frac{108913462}{3543539} a^{3} - \frac{51514350}{3543539} a^{2} + \frac{1092072}{3543539} a + \frac{823897}{3543539} \)  (order $4$) 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:   $\frac{3832719}{46066007}a^{15}-\frac{25128855}{46066007}a^{14}+\frac{949111}{648817}a^{13}-\frac{79324619}{46066007}a^{12}-\frac{13769413}{46066007}a^{11}+\frac{26617638}{46066007}a^{10}+\frac{418770642}{46066007}a^{9}-\frac{12100921}{648817}a^{8}+\frac{20510135}{46066007}a^{7}+\frac{1664828718}{46066007}a^{6}-\frac{1619347244}{46066007}a^{5}-\frac{330926861}{46066007}a^{4}+\frac{1266643438}{46066007}a^{3}-\frac{416618360}{46066007}a^{2}-\frac{159502828}{46066007}a-\frac{581193}{46066007}$, $\frac{6915611}{46066007}a^{15}-\frac{28422130}{46066007}a^{14}+\frac{872314}{648817}a^{13}-\frac{45418731}{46066007}a^{12}-\frac{14814855}{46066007}a^{11}-\frac{103282644}{46066007}a^{10}+\frac{482173830}{46066007}a^{9}-\frac{7930886}{648817}a^{8}-\frac{271771261}{46066007}a^{7}+\frac{1110918862}{46066007}a^{6}-\frac{666188227}{46066007}a^{5}-\frac{495066088}{46066007}a^{4}+\frac{581089083}{46066007}a^{3}+\frac{30049769}{46066007}a^{2}-\frac{186673612}{46066007}a-\frac{28034933}{46066007}$, $a-1$, $\frac{12250508}{46066007}a^{15}-\frac{52902528}{46066007}a^{14}+\frac{1691408}{648817}a^{13}-\frac{124677372}{46066007}a^{12}+\frac{53475076}{46066007}a^{11}-\frac{264566189}{46066007}a^{10}+\frac{845897184}{46066007}a^{9}-\frac{14960733}{648817}a^{8}+\frac{222167741}{46066007}a^{7}+\frac{1108704976}{46066007}a^{6}-\frac{1802316109}{46066007}a^{5}+\frac{1176458742}{46066007}a^{4}+\frac{242651115}{46066007}a^{3}-\frac{889620584}{46066007}a^{2}+\frac{464679766}{46066007}a-\frac{31129045}{46066007}$, $\frac{1973466}{3543539}a^{15}-\frac{136898813}{46066007}a^{14}+\frac{4480497}{648817}a^{13}-\frac{300354133}{46066007}a^{12}-\frac{85437858}{46066007}a^{11}-\frac{306605349}{46066007}a^{10}+\frac{193972824}{3543539}a^{9}-\frac{46865172}{648817}a^{8}-\frac{1087294271}{46066007}a^{7}+\frac{5903835732}{46066007}a^{6}-\frac{3602535510}{46066007}a^{5}-\frac{144997270}{3543539}a^{4}+\frac{2506332273}{46066007}a^{3}-\frac{252424063}{46066007}a^{2}-\frac{256085705}{46066007}a-\frac{76142151}{46066007}$, $\frac{1793567}{46066007}a^{15}-\frac{9085769}{46066007}a^{14}+\frac{364690}{648817}a^{13}-\frac{26770702}{46066007}a^{12}-\frac{2422539}{46066007}a^{11}+\frac{2408340}{46066007}a^{10}+\frac{174810188}{46066007}a^{9}-\frac{5262831}{648817}a^{8}-\frac{29401433}{46066007}a^{7}+\frac{651834554}{46066007}a^{6}-\frac{357076186}{46066007}a^{5}-\frac{530593192}{46066007}a^{4}+\frac{429198261}{46066007}a^{3}+\frac{208148549}{46066007}a^{2}-\frac{184179790}{46066007}a-\frac{30234441}{46066007}$, $\frac{24244274}{46066007}a^{15}-\frac{114683327}{46066007}a^{14}+\frac{3342016}{648817}a^{13}-\frac{171725425}{46066007}a^{12}-\frac{120815064}{46066007}a^{11}-\frac{403268717}{46066007}a^{10}+\frac{2081722542}{46066007}a^{9}-\frac{27844561}{648817}a^{8}-\frac{1602512564}{46066007}a^{7}+\frac{4062247867}{46066007}a^{6}-\frac{1606440972}{46066007}a^{5}-\frac{1491054637}{46066007}a^{4}+\frac{1405653138}{46066007}a^{3}-\frac{250369374}{46066007}a^{2}-\frac{54982580}{46066007}a+\frac{38004231}{46066007}$ 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:  \( 2844.26103233 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 2844.26103233 \cdot 1}{4\cdot\sqrt{11979351654400000000}}\cr\approx \mathstrut & 0.499035854716 \end{aligned}\]

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 - 6*x^15 + 16*x^14 - 20*x^13 + 4*x^12 - 8*x^11 + 104*x^10 - 196*x^9 + 44*x^8 + 272*x^7 - 316*x^6 + 24*x^5 + 168*x^4 - 92*x^3 - 4*x^2 + 8*x + 2) 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 - 6*x^15 + 16*x^14 - 20*x^13 + 4*x^12 - 8*x^11 + 104*x^10 - 196*x^9 + 44*x^8 + 272*x^7 - 316*x^6 + 24*x^5 + 168*x^4 - 92*x^3 - 4*x^2 + 8*x + 2, 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 - 6*x^15 + 16*x^14 - 20*x^13 + 4*x^12 - 8*x^11 + 104*x^10 - 196*x^9 + 44*x^8 + 272*x^7 - 316*x^6 + 24*x^5 + 168*x^4 - 92*x^3 - 4*x^2 + 8*x + 2); 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 - 6*x^15 + 16*x^14 - 20*x^13 + 4*x^12 - 8*x^11 + 104*x^10 - 196*x^9 + 44*x^8 + 272*x^7 - 316*x^6 + 24*x^5 + 168*x^4 - 92*x^3 - 4*x^2 + 8*x + 2); 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

$(C_2^2\times C_4^2):D_8$ (as 16T1276):

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 1024
The 43 conjugacy class representatives for $(C_2^2\times C_4^2):D_8$
Character table for $(C_2^2\times C_4^2):D_8$

Intermediate fields

\(\Q(\sqrt{-1}) \), 4.0.320.1, 8.0.26624000.1

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(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: 16.0.194664464384000000000.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.8.0.1}{8} }^{2}$ R ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ R ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ ${\href{/padicField/23.8.0.1}{8} }^{2}$ ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{3}{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$

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 2.1.16.30a1.1$x^{16} + 2 x^{15} + 2$$16$$1$$30$16T166$$[2, 2, 2, 2]^{4}$$
\(5\) Copy content Toggle raw display 5.4.1.0a1.1$x^{4} + 4 x^{2} + 4 x + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
5.2.2.2a1.1$x^{4} + 8 x^{3} + 20 x^{2} + 21 x + 4$$2$$2$$2$$C_4$$$[\ ]_{2}^{2}$$
5.2.4.6a1.3$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 664 x^{4} + 704 x^{3} + 416 x^{2} + 133 x + 31$$4$$2$$6$$C_4\times C_2$$$[\ ]_{4}^{2}$$
\(13\) Copy content Toggle raw display 13.2.1.0a1.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
13.2.1.0a1.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
13.2.1.0a1.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
13.1.2.1a1.1$x^{2} + 13$$2$$1$$1$$C_2$$$[\ ]_{2}$$
13.1.2.1a1.1$x^{2} + 13$$2$$1$$1$$C_2$$$[\ ]_{2}$$
13.2.1.0a1.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
13.2.2.2a1.2$x^{4} + 24 x^{3} + 148 x^{2} + 48 x + 17$$2$$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)