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Properties

Label 16.8.980...656.110
Degree $16$
Signature $[8, 4]$
Discriminant $9.803\times 10^{24}$
Root discriminant \(36.47\)
Ramified primes $2,3$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $(C_2^3\times C_4):C_4$ (as 16T243)

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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 - 24*x^14 + 252*x^12 - 1512*x^10 + 6210*x^8 - 20088*x^6 + 44388*x^4 - 44712*x^2 + 3969)
 
Copy content gp:K = bnfinit(y^16 - 24*y^14 + 252*y^12 - 1512*y^10 + 6210*y^8 - 20088*y^6 + 44388*y^4 - 44712*y^2 + 3969, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 24*x^14 + 252*x^12 - 1512*x^10 + 6210*x^8 - 20088*x^6 + 44388*x^4 - 44712*x^2 + 3969);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 24*x^14 + 252*x^12 - 1512*x^10 + 6210*x^8 - 20088*x^6 + 44388*x^4 - 44712*x^2 + 3969)
 

\( x^{16} - 24x^{14} + 252x^{12} - 1512x^{10} + 6210x^{8} - 20088x^{6} + 44388x^{4} - 44712x^{2} + 3969 \) 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:   \(9803356117276277820358656\) \(\medspace = 2^{64}\cdot 3^{12}\) 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:  \(36.47\)
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:  $2^{137/32}3^{3/4}\approx 44.32263892833827$
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: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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3}a^{4}$, $\frac{1}{3}a^{5}$, $\frac{1}{3}a^{6}$, $\frac{1}{3}a^{7}$, $\frac{1}{144}a^{8}-\frac{1}{6}a^{7}+\frac{1}{12}a^{6}-\frac{1}{6}a^{5}+\frac{1}{24}a^{4}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a-\frac{1}{16}$, $\frac{1}{144}a^{9}+\frac{1}{12}a^{7}-\frac{1}{6}a^{6}+\frac{1}{24}a^{5}-\frac{1}{6}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{16}a-\frac{1}{2}$, $\frac{1}{144}a^{10}-\frac{1}{6}a^{7}+\frac{1}{24}a^{6}-\frac{1}{6}a^{5}+\frac{1}{12}a^{4}-\frac{1}{2}a^{3}-\frac{1}{16}a^{2}-\frac{1}{2}a-\frac{1}{4}$, $\frac{1}{144}a^{11}+\frac{1}{24}a^{7}-\frac{1}{6}a^{6}+\frac{1}{12}a^{5}-\frac{1}{6}a^{4}-\frac{1}{16}a^{3}-\frac{1}{2}a^{2}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{38448}a^{12}-\frac{1}{2136}a^{10}+\frac{11}{4272}a^{8}+\frac{175}{1068}a^{6}-\frac{137}{1424}a^{4}+\frac{19}{712}a^{2}-\frac{151}{1424}$, $\frac{1}{38448}a^{13}-\frac{1}{2136}a^{11}+\frac{11}{4272}a^{9}+\frac{175}{1068}a^{7}-\frac{137}{1424}a^{5}+\frac{19}{712}a^{3}-\frac{151}{1424}a$, $\frac{1}{38448}a^{14}+\frac{7}{6408}a^{10}+\frac{1}{534}a^{8}-\frac{1}{6}a^{7}+\frac{263}{4272}a^{6}-\frac{1}{6}a^{5}+\frac{137}{1068}a^{4}-\frac{1}{2}a^{3}-\frac{67}{356}a^{2}-\frac{1}{2}a-\frac{101}{356}$, $\frac{1}{269136}a^{15}-\frac{1}{89712}a^{13}+\frac{5}{2136}a^{11}+\frac{1}{6408}a^{9}+\frac{1049}{9968}a^{7}+\frac{1247}{29904}a^{5}-\frac{51}{2492}a^{3}+\frac{131}{623}a$ 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:   $\frac{1}{9612}a^{12}+\frac{1}{534}a^{10}-\frac{11}{1068}a^{8}+\frac{1}{89}a^{6}+\frac{55}{1068}a^{4}-\frac{19}{178}a^{2}+\frac{863}{356}$, $\frac{7}{19224}a^{12}-\frac{7}{1068}a^{10}+\frac{40}{801}a^{8}-\frac{55}{267}a^{6}+\frac{1573}{2136}a^{4}-\frac{757}{356}a^{2}+\frac{139}{356}$, $\frac{31}{38448}a^{14}+\frac{221}{12816}a^{12}-\frac{2009}{12816}a^{10}+\frac{10099}{12816}a^{8}-\frac{45}{16}a^{6}+\frac{11849}{1424}a^{4}-\frac{17317}{1424}a^{2}+\frac{699}{1424}$, $\frac{73}{38448}a^{14}-\frac{65}{1602}a^{12}+\frac{4775}{12816}a^{10}-\frac{6113}{3204}a^{8}+\frac{9851}{1424}a^{6}-\frac{3673}{178}a^{4}+\frac{46379}{1424}a^{2}-\frac{2595}{356}$, $\frac{37}{269136}a^{15}+\frac{11}{9612}a^{14}+\frac{839}{269136}a^{13}-\frac{155}{6408}a^{12}-\frac{133}{4272}a^{11}+\frac{469}{2136}a^{10}+\frac{1145}{6408}a^{9}-\frac{14237}{12816}a^{8}-\frac{21757}{29904}a^{7}+\frac{4259}{1068}a^{6}+\frac{70817}{29904}a^{5}-\frac{1039}{89}a^{4}-\frac{48893}{9968}a^{3}+\frac{12123}{712}a^{2}+\frac{2300}{623}a-\frac{3815}{1424}$, $\frac{97}{67284}a^{15}+\frac{1}{9612}a^{14}-\frac{4663}{134568}a^{13}-\frac{7}{9612}a^{12}+\frac{4663}{12816}a^{11}-\frac{11}{1068}a^{10}-\frac{1163}{534}a^{9}+\frac{703}{4272}a^{8}+\frac{1495}{168}a^{7}-\frac{247}{267}a^{6}-\frac{428255}{14952}a^{5}+\frac{7295}{2136}a^{4}+\frac{625559}{9968}a^{3}-\frac{23}{2}a^{2}-\frac{37316}{623}a+\frac{24595}{1424}$, $\frac{5}{14952}a^{15}+\frac{1}{9612}a^{14}+\frac{1045}{134568}a^{13}-\frac{17}{4806}a^{12}-\frac{967}{12816}a^{11}+\frac{43}{1068}a^{10}+\frac{433}{1068}a^{9}-\frac{1019}{4272}a^{8}-\frac{2791}{1869}a^{7}+\frac{469}{534}a^{6}+\frac{69521}{14952}a^{5}-\frac{6289}{2136}a^{4}-\frac{80067}{9968}a^{3}+\frac{911}{178}a^{2}+\frac{7489}{2492}a-\frac{1639}{1424}$, $\frac{5}{14952}a^{15}+\frac{1}{9612}a^{14}-\frac{1045}{134568}a^{13}-\frac{35}{9612}a^{12}+\frac{967}{12816}a^{11}+\frac{15}{356}a^{10}-\frac{433}{1068}a^{9}-\frac{1063}{4272}a^{8}+\frac{2791}{1869}a^{7}+\frac{475}{534}a^{6}-\frac{69521}{14952}a^{5}-\frac{6179}{2136}a^{4}+\frac{80067}{9968}a^{3}+\frac{446}{89}a^{2}-\frac{7489}{2492}a+\frac{389}{1424}$, $\frac{19}{33642}a^{15}+\frac{11}{9612}a^{14}-\frac{19}{1512}a^{13}-\frac{17}{712}a^{12}+\frac{775}{6408}a^{11}+\frac{457}{2136}a^{10}-\frac{8417}{12816}a^{9}-\frac{13663}{12816}a^{8}+\frac{4679}{1869}a^{7}+\frac{4045}{1068}a^{6}-\frac{4821}{623}a^{5}-\frac{986}{89}a^{4}+\frac{71569}{4984}a^{3}+\frac{11283}{712}a^{2}-\frac{92469}{9968}a-\frac{109}{1424}$, $\frac{5}{14952}a^{15}+\frac{11}{19224}a^{14}-\frac{145}{22428}a^{13}-\frac{467}{38448}a^{12}+\frac{667}{12816}a^{11}+\frac{473}{4272}a^{10}-\frac{1417}{6408}a^{9}-\frac{7229}{12816}a^{8}+\frac{855}{1246}a^{7}+\frac{545}{267}a^{6}-\frac{4469}{2492}a^{5}-\frac{8661}{1424}a^{4}+\frac{3655}{9968}a^{3}+\frac{13649}{1424}a^{2}+\frac{20897}{4984}a-\frac{2629}{1424}$, $\frac{17}{12816}a^{15}-\frac{11}{19224}a^{14}+\frac{121}{4272}a^{13}+\frac{457}{38448}a^{12}-\frac{833}{3204}a^{11}-\frac{151}{1424}a^{10}+\frac{1899}{1424}a^{9}+\frac{6721}{12816}a^{8}-\frac{20785}{4272}a^{7}-\frac{493}{267}a^{6}+\frac{701}{48}a^{5}+\frac{7539}{1424}a^{4}-\frac{16441}{712}a^{3}-\frac{10469}{1424}a^{2}+\frac{8355}{1424}a+\frac{1469}{1424}$ 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:  \( 11649698.970758608 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 11649698.970758608 \cdot 1}{2\cdot\sqrt{9803356117276277820358656}}\cr\approx \mathstrut & 0.742261195582682 \end{aligned}\] (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 - 24*x^14 + 252*x^12 - 1512*x^10 + 6210*x^8 - 20088*x^6 + 44388*x^4 - 44712*x^2 + 3969) 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 - 24*x^14 + 252*x^12 - 1512*x^10 + 6210*x^8 - 20088*x^6 + 44388*x^4 - 44712*x^2 + 3969, 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 - 24*x^14 + 252*x^12 - 1512*x^10 + 6210*x^8 - 20088*x^6 + 44388*x^4 - 44712*x^2 + 3969); 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 - 24*x^14 + 252*x^12 - 1512*x^10 + 6210*x^8 - 20088*x^6 + 44388*x^4 - 44712*x^2 + 3969); 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

$(C_2^3\times C_4):C_4$ (as 16T243):

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 $(C_2^3\times C_4):C_4$
Character table for $(C_2^3\times C_4):C_4$

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.18432.1, 4.2.9216.1, 4.2.2048.1, 8.4.5435817984.3

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: 16.8.612709757329767363772416.21, 16.0.612709757329767363772416.95, 16.0.9803356117276277820358656.257, 16.12.2450839029319069455089664.4, 16.4.2450839029319069455089664.110
Degree 32 siblings: deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, 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 ${\href{/padicField/5.4.0.1}{4} }^{4}$ ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{4}$ ${\href{/padicField/11.8.0.1}{8} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.8.0.1}{8} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{8}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.8.0.1}{8} }^{2}$

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 2.1.16.64l1.182$x^{16} + 8 x^{14} + 4 x^{12} + 8 x^{10} + 8 x^{8} + 16 x^{7} + 8 x^{6} + 16 x + 18$$16$$1$$64$16T243$$[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$$
\(3\) Copy content Toggle raw display 3.2.4.6a1.1$x^{8} + 8 x^{7} + 32 x^{6} + 80 x^{5} + 136 x^{4} + 160 x^{3} + 128 x^{2} + 67 x + 16$$4$$2$$6$$C_8:C_2$$$[\ ]_{4}^{4}$$
3.2.4.6a1.1$x^{8} + 8 x^{7} + 32 x^{6} + 80 x^{5} + 136 x^{4} + 160 x^{3} + 128 x^{2} + 67 x + 16$$4$$2$$6$$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)