Properties

Label 16.0.149...296.34
Degree $16$
Signature $(0, 8)$
Discriminant $1.496\times 10^{20}$
Root discriminant \(18.24\)
Ramified primes $2,3$
Class number $1$
Class group trivial
Galois group $D_8:C_4$ (as 16T114)

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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 + 84*x^12 - 396*x^10 + 1197*x^8 - 2052*x^6 + 1782*x^4 - 648*x^2 + 81)
 
Copy content gp:K = bnfinit(y^16 - 12*y^14 + 84*y^12 - 396*y^10 + 1197*y^8 - 2052*y^6 + 1782*y^4 - 648*y^2 + 81, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 12*x^14 + 84*x^12 - 396*x^10 + 1197*x^8 - 2052*x^6 + 1782*x^4 - 648*x^2 + 81);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 12*x^14 + 84*x^12 - 396*x^10 + 1197*x^8 - 2052*x^6 + 1782*x^4 - 648*x^2 + 81)
 

\( x^{16} - 12x^{14} + 84x^{12} - 396x^{10} + 1197x^{8} - 2052x^{6} + 1782x^{4} - 648x^{2} + 81 \) 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:   \(149587343098087735296\) \(\medspace = 2^{48}\cdot 3^{12}\) 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:  \(18.24\)
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:  $2^{29/8}3^{3/4}\approx 28.12384367863563$
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:  \(\Q\)
$\Aut(K/\Q)$:   $C_8$
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:  \(\Q(\zeta_{12})\)

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3}a^{4}$, $\frac{1}{3}a^{5}$, $\frac{1}{3}a^{6}$, $\frac{1}{3}a^{7}$, $\frac{1}{9}a^{8}$, $\frac{1}{9}a^{9}$, $\frac{1}{9}a^{10}$, $\frac{1}{9}a^{11}$, $\frac{1}{27}a^{12}$, $\frac{1}{27}a^{13}$, $\frac{1}{31239}a^{14}+\frac{16}{2403}a^{12}-\frac{59}{1157}a^{10}-\frac{95}{10413}a^{8}+\frac{38}{267}a^{6}-\frac{102}{1157}a^{4}-\frac{391}{1157}a^{2}-\frac{426}{1157}$, $\frac{1}{31239}a^{15}+\frac{16}{2403}a^{13}-\frac{59}{1157}a^{11}-\frac{95}{10413}a^{9}+\frac{38}{267}a^{7}-\frac{102}{1157}a^{5}-\frac{391}{1157}a^{3}-\frac{426}{1157}a$ 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:   \( \frac{1702}{31239} a^{14} - \frac{505}{801} a^{12} + \frac{44978}{10413} a^{10} - \frac{206813}{10413} a^{8} + \frac{15370}{267} a^{6} - \frac{314866}{3471} a^{4} + \frac{77312}{1157} a^{2} - \frac{16968}{1157} \)  (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:   $\frac{157}{10413}a^{14}+\frac{158}{801}a^{12}-\frac{14852}{10413}a^{10}+\frac{8057}{1157}a^{8}-\frac{1961}{89}a^{6}+\frac{138341}{3471}a^{4}-\frac{37983}{1157}a^{2}+\frac{7427}{1157}$, $\frac{742}{31239}a^{14}+\frac{677}{2403}a^{12}-\frac{20204}{10413}a^{10}+\frac{31210}{3471}a^{8}-\frac{2338}{89}a^{6}+\frac{47916}{1157}a^{4}-\frac{32681}{1157}a^{2}+\frac{3702}{1157}$, $\frac{202}{10413}a^{14}-\frac{539}{2403}a^{12}+\frac{16058}{10413}a^{10}-\frac{73768}{10413}a^{8}+\frac{5495}{267}a^{6}-\frac{37515}{1157}a^{4}+\frac{26850}{1157}a^{2}-\frac{4773}{1157}$, $\frac{3823}{31239}a^{15}-\frac{275}{3471}a^{14}+\frac{373}{267}a^{13}+\frac{2141}{2403}a^{12}-\frac{98867}{10413}a^{11}-\frac{62605}{10413}a^{10}+\frac{449960}{10413}a^{9}+\frac{282562}{10413}a^{8}-\frac{32867}{267}a^{7}-\frac{6786}{89}a^{6}+\frac{648031}{3471}a^{5}+\frac{391738}{3471}a^{4}-\frac{145833}{1157}a^{3}-\frac{86302}{1157}a^{2}+\frac{23839}{1157}a+\frac{14207}{1157}$, $\frac{205}{1157}a^{15}+\frac{98}{2403}a^{14}+\frac{1630}{801}a^{13}-\frac{355}{801}a^{12}-\frac{144320}{10413}a^{11}+\frac{2341}{801}a^{10}+\frac{658880}{10413}a^{9}-\frac{10289}{801}a^{8}-\frac{48350}{267}a^{7}+\frac{9163}{267}a^{6}+\frac{320443}{1157}a^{5}-\frac{4033}{89}a^{4}-\frac{219235}{1157}a^{3}+\frac{2088}{89}a^{2}+\frac{40439}{1157}a-\frac{274}{89}$, $\frac{478}{2403}a^{15}+\frac{1706}{10413}a^{14}+\frac{5596}{2403}a^{13}-\frac{4531}{2403}a^{12}-\frac{12826}{801}a^{11}+\frac{134347}{10413}a^{10}+\frac{19735}{267}a^{9}-\frac{615794}{10413}a^{8}-\frac{57509}{267}a^{7}+\frac{45587}{267}a^{6}+\frac{90910}{267}a^{5}-\frac{926287}{3471}a^{4}-\frac{21629}{89}a^{3}+\frac{222616}{1157}a^{2}+\frac{4179}{89}a-\frac{45603}{1157}$, $\frac{2729}{10413}a^{15}-\frac{4387}{31239}a^{14}-\frac{2438}{801}a^{13}+\frac{1315}{801}a^{12}+\frac{72356}{3471}a^{11}-\frac{13062}{1157}a^{10}-\frac{997595}{10413}a^{9}+\frac{542878}{10413}a^{8}+\frac{24670}{89}a^{7}-\frac{13531}{89}a^{6}-\frac{500701}{1157}a^{5}+\frac{837970}{3471}a^{4}+\frac{353187}{1157}a^{3}-\frac{202989}{1157}a^{2}-\frac{67570}{1157}a+\frac{39645}{1157}$ 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:  \( 25544.612875827956 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 25544.612875827956 \cdot 1}{12\cdot\sqrt{149587343098087735296}}\cr\approx \mathstrut & 0.422775295705510 \end{aligned}\]

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 + 84*x^12 - 396*x^10 + 1197*x^8 - 2052*x^6 + 1782*x^4 - 648*x^2 + 81) 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 + 84*x^12 - 396*x^10 + 1197*x^8 - 2052*x^6 + 1782*x^4 - 648*x^2 + 81, 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 + 84*x^12 - 396*x^10 + 1197*x^8 - 2052*x^6 + 1782*x^4 - 648*x^2 + 81); 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 + 84*x^12 - 396*x^10 + 1197*x^8 - 2052*x^6 + 1782*x^4 - 648*x^2 + 81); 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

$D_8:C_4$ (as 16T114):

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 64
The 28 conjugacy class representatives for $D_8:C_4$
Character table for $D_8:C_4$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{12})\), 8.0.5308416.2

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(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 16 sibling: 16.0.149587343098087735296.38
Degree 32 siblings: deg 32, deg 32, deg 32
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 R ${\href{/padicField/5.8.0.1}{8} }^{2}$ ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.8.0.1}{8} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{8}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.8.0.1}{8} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.8.0.1}{8} }^{2}$ ${\href{/padicField/31.8.0.1}{8} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{8}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.8.0.1}{8} }^{2}$ ${\href{/padicField/59.8.0.1}{8} }^{2}$

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 2.2.8.48c3.32$x^{16} + 16 x^{15} + 100 x^{14} + 392 x^{13} + 1106 x^{12} + 2416 x^{11} + 4248 x^{10} + 6160 x^{9} + 7469 x^{8} + 7624 x^{7} + 6568 x^{6} + 4760 x^{5} + 2888 x^{4} + 1440 x^{3} + 584 x^{2} + 176 x + 57$$8$$2$$48$16T114$$[2, 3, 3, 4, 4]^{2}$$
\(3\) Copy content Toggle raw display 3.4.4.12a1.1$x^{16} + 8 x^{15} + 24 x^{14} + 32 x^{13} + 24 x^{12} + 48 x^{11} + 96 x^{10} + 64 x^{9} + 24 x^{8} + 96 x^{7} + 96 x^{6} + 32 x^{4} + 64 x^{3} + 3 x^{2} + 16$$4$$4$$12$$C_8: C_2$$$[\ ]_{4}^{4}$$

Spectrum of ring of integers

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