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Properties

Label 16.0.119...000.1
Degree $16$
Signature $[0, 8]$
Discriminant $1.193\times 10^{20}$
Root discriminant \(17.98\)
Ramified primes $2,5,11,19$
Class number $2$
Class group [2]
Galois group $(C_2^2\times D_4^2):C_4$ (as 16T1086)

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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 + 4*x^14 + 24*x^12 + 58*x^10 + 159*x^8 + 226*x^6 + 311*x^4 + 207*x^2 + 121)
 
Copy content gp:K = bnfinit(y^16 + 4*y^14 + 24*y^12 + 58*y^10 + 159*y^8 + 226*y^6 + 311*y^4 + 207*y^2 + 121, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 4*x^14 + 24*x^12 + 58*x^10 + 159*x^8 + 226*x^6 + 311*x^4 + 207*x^2 + 121);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 + 4*x^14 + 24*x^12 + 58*x^10 + 159*x^8 + 226*x^6 + 311*x^4 + 207*x^2 + 121)
 

\( x^{16} + 4x^{14} + 24x^{12} + 58x^{10} + 159x^{8} + 226x^{6} + 311x^{4} + 207x^{2} + 121 \) 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:   \(119251860062500000000\) \(\medspace = 2^{8}\cdot 5^{12}\cdot 11^{4}\cdot 19^{4}\) 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:  \(17.98\)
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:  $2^{3/2}5^{3/4}11^{1/2}19^{1/2}\approx 136.72427835574135$
Ramified primes:   \(2\), \(5\), \(11\), \(19\) 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:  8.0.682515625.4

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{6}+\frac{1}{3}a^{4}+\frac{1}{3}$, $\frac{1}{3}a^{9}-\frac{1}{3}a^{7}+\frac{1}{3}a^{5}+\frac{1}{3}a$, $\frac{1}{3}a^{10}+\frac{1}{3}a^{4}+\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{3}a^{11}+\frac{1}{3}a^{5}+\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{6}a^{12}-\frac{1}{6}a^{11}-\frac{1}{2}a^{7}+\frac{1}{6}a^{6}+\frac{1}{3}a^{5}+\frac{1}{6}a^{4}-\frac{1}{6}a^{3}+\frac{1}{6}a^{2}-\frac{1}{6}a-\frac{1}{2}$, $\frac{1}{6}a^{13}-\frac{1}{6}a^{11}-\frac{1}{6}a^{8}-\frac{1}{3}a^{7}+\frac{1}{6}a^{6}-\frac{1}{2}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a-\frac{1}{6}$, $\frac{1}{834}a^{14}-\frac{11}{139}a^{12}-\frac{1}{6}a^{11}-\frac{41}{417}a^{10}-\frac{1}{6}a^{9}-\frac{20}{417}a^{8}-\frac{1}{3}a^{7}+\frac{53}{139}a^{6}-\frac{1}{3}a^{5}+\frac{67}{834}a^{4}-\frac{1}{6}a^{3}+\frac{347}{834}a^{2}-\frac{1}{3}a+\frac{127}{278}$, $\frac{1}{9174}a^{15}-\frac{205}{9174}a^{13}-\frac{736}{4587}a^{11}-\frac{1}{6}a^{10}-\frac{192}{1529}a^{9}-\frac{1}{6}a^{8}-\frac{3713}{9174}a^{7}-\frac{1}{3}a^{6}+\frac{683}{1529}a^{5}-\frac{1}{3}a^{4}-\frac{753}{1529}a^{3}-\frac{1}{6}a^{2}+\frac{689}{1529}a-\frac{1}{3}$ 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:  $C_{2}$, which has order $2$
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 $2$
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:   \( \frac{4}{417} a^{14} + \frac{14}{417} a^{12} + \frac{89}{417} a^{10} + \frac{118}{417} a^{8} + \frac{146}{139} a^{6} - \frac{10}{417} a^{4} + \frac{415}{417} a^{2} - \frac{283}{417} \)  (order $10$) 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:   $\frac{4}{417}a^{14}+\frac{14}{417}a^{12}+\frac{89}{417}a^{10}+\frac{257}{417}a^{8}+\frac{716}{417}a^{6}+\frac{460}{139}a^{4}+\frac{1666}{417}a^{2}+\frac{508}{139}$, $\frac{93}{3058}a^{15}+\frac{2}{417}a^{14}+\frac{907}{9174}a^{13}+\frac{7}{417}a^{12}+\frac{2600}{4587}a^{11}+\frac{89}{834}a^{10}+\frac{1476}{1529}a^{9}+\frac{257}{834}a^{8}+\frac{22141}{9174}a^{7}+\frac{358}{417}a^{6}+\frac{5941}{4587}a^{5}+\frac{230}{139}a^{4}+\frac{5803}{4587}a^{3}+\frac{2083}{834}a^{2}-\frac{8914}{4587}a+\frac{254}{139}$, $\frac{301}{9174}a^{15}+\frac{17}{834}a^{14}+\frac{164}{1529}a^{13}-\frac{5}{417}a^{12}+\frac{3227}{4587}a^{11}+\frac{45}{278}a^{10}+\frac{12563}{9174}a^{9}-\frac{67}{139}a^{8}+\frac{11751}{3058}a^{7}-\frac{5}{278}a^{6}+\frac{12097}{3058}a^{5}-\frac{2197}{834}a^{4}+\frac{42175}{9174}a^{3}-\frac{317}{417}a^{2}+\frac{5980}{4587}a-\frac{1279}{417}$, $\frac{245}{9174}a^{15}-\frac{25}{417}a^{14}+\frac{116}{4587}a^{13}-\frac{175}{834}a^{12}+\frac{1631}{4587}a^{11}-\frac{1043}{834}a^{10}-\frac{452}{4587}a^{9}-\frac{1085}{417}a^{8}+\frac{1031}{1529}a^{7}-\frac{2807}{417}a^{6}-\frac{25007}{9174}a^{5}-\frac{6547}{834}a^{4}-\frac{22849}{9174}a^{3}-\frac{1131}{139}a^{2}-\frac{43709}{9174}a-\frac{1880}{417}$, $\frac{8}{417}a^{15}+\frac{7}{834}a^{14}+\frac{28}{417}a^{13}-\frac{15}{278}a^{12}+\frac{178}{417}a^{11}-\frac{3}{139}a^{10}+\frac{889}{834}a^{9}-\frac{418}{417}a^{8}+\frac{723}{278}a^{7}-\frac{971}{834}a^{6}+\frac{1648}{417}a^{5}-\frac{2059}{417}a^{4}+\frac{1247}{417}a^{3}-\frac{545}{139}a^{2}+\frac{953}{834}a-\frac{598}{139}$, $\frac{245}{9174}a^{15}+\frac{25}{417}a^{14}+\frac{116}{4587}a^{13}+\frac{175}{834}a^{12}+\frac{1631}{4587}a^{11}+\frac{1043}{834}a^{10}-\frac{452}{4587}a^{9}+\frac{1085}{417}a^{8}+\frac{1031}{1529}a^{7}+\frac{2807}{417}a^{6}-\frac{25007}{9174}a^{5}+\frac{6547}{834}a^{4}-\frac{22849}{9174}a^{3}+\frac{1131}{139}a^{2}-\frac{43709}{9174}a+\frac{1880}{417}$, $\frac{23}{1529}a^{15}+\frac{43}{834}a^{14}+\frac{761}{9174}a^{13}+\frac{27}{278}a^{12}+\frac{1093}{3058}a^{11}+\frac{322}{417}a^{10}+\frac{3581}{3058}a^{9}+\frac{643}{834}a^{8}+\frac{21227}{9174}a^{7}+\frac{1277}{417}a^{6}+\frac{44135}{9174}a^{5}+\frac{179}{139}a^{4}+\frac{21580}{4587}a^{3}+\frac{1414}{417}a^{2}+\frac{14329}{3058}a-\frac{19}{834}$ 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:  \( 7775.62649708 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 7775.62649708 \cdot 2}{10\cdot\sqrt{119251860062500000000}}\cr\approx \mathstrut & 0.345916878017 \end{aligned}\]

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 + 4*x^14 + 24*x^12 + 58*x^10 + 159*x^8 + 226*x^6 + 311*x^4 + 207*x^2 + 121) 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 + 4*x^14 + 24*x^12 + 58*x^10 + 159*x^8 + 226*x^6 + 311*x^4 + 207*x^2 + 121, 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 + 4*x^14 + 24*x^12 + 58*x^10 + 159*x^8 + 226*x^6 + 311*x^4 + 207*x^2 + 121); 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 + 4*x^14 + 24*x^12 + 58*x^10 + 159*x^8 + 226*x^6 + 311*x^4 + 207*x^2 + 121); 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

$(C_2^2\times D_4^2):C_4$ (as 16T1086):

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); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A solvable group of order 1024
The 97 conjugacy class representatives for $(C_2^2\times D_4^2):C_4$
Character table for $(C_2^2\times D_4^2):C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 4.0.5225.1, 4.4.26125.1, 8.0.682515625.4

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 siblings: data not computed
Degree 32 siblings: data not computed
Minimal sibling: 16.0.698896000000000000.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.4.0.1}{4} }^{4}$ R ${\href{/padicField/7.8.0.1}{8} }^{2}$ R ${\href{/padicField/13.8.0.1}{8} }^{2}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ R ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{8}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.8.0.1}{8} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}$

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
\(2\) Copy content Toggle raw display 2.4.2.8a3.2$x^{8} + 2 x^{6} + 4 x^{5} + 2 x^{4} + 2 x^{3} + 5 x^{2} + 8 x + 3$$2$$4$$8$$C_8:C_2$$$[2, 2]^{4}$$
2.8.1.0a1.1$x^{8} + x^{4} + x^{3} + x^{2} + 1$$1$$8$$0$$C_8$$$[\ ]^{8}$$
\(5\) Copy content Toggle raw display 5.4.4.12a1.4$x^{16} + 16 x^{14} + 16 x^{13} + 104 x^{12} + 192 x^{11} + 448 x^{10} + 864 x^{9} + 1432 x^{8} + 2048 x^{7} + 2624 x^{6} + 2752 x^{5} + 2208 x^{4} + 1280 x^{3} + 512 x^{2} + 128 x + 21$$4$$4$$12$$C_4^2$$$[\ ]_{4}^{4}$$
\(11\) Copy content Toggle raw display $\Q_{11}$$x + 9$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{11}$$x + 9$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{11}$$x + 9$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{11}$$x + 9$$1$$1$$0$Trivial$$[\ ]$$
11.2.1.0a1.1$x^{2} + 7 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
11.2.1.0a1.1$x^{2} + 7 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
11.1.2.1a1.1$x^{2} + 11$$2$$1$$1$$C_2$$$[\ ]_{2}$$
11.1.2.1a1.1$x^{2} + 11$$2$$1$$1$$C_2$$$[\ ]_{2}$$
11.2.2.2a1.2$x^{4} + 14 x^{3} + 53 x^{2} + 28 x + 15$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
\(19\) Copy content Toggle raw display 19.2.1.0a1.1$x^{2} + 18 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
19.2.1.0a1.1$x^{2} + 18 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
19.2.1.0a1.1$x^{2} + 18 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
19.2.1.0a1.1$x^{2} + 18 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
19.4.2.4a1.2$x^{8} + 4 x^{6} + 22 x^{5} + 8 x^{4} + 44 x^{3} + 129 x^{2} + 44 x + 23$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$

Spectrum of ring of integers

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