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Properties

Label 16.4.289217423320703125.1
Degree $16$
Signature $[4, 6]$
Discriminant $2.892\times 10^{17}$
Root discriminant \(12.34\)
Ramified primes $5,29,61,131$
Class number $1$
Class group trivial
Galois group $C_4^4.C_2\wr D_4$ (as 16T1823)

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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 - x^15 - 5*x^14 + 12*x^13 - 2*x^12 - 22*x^11 + 28*x^10 - 4*x^9 - 13*x^8 - 4*x^7 + 28*x^6 - 22*x^5 - 2*x^4 + 12*x^3 - 5*x^2 - x + 1)
 
Copy content gp:K = bnfinit(y^16 - y^15 - 5*y^14 + 12*y^13 - 2*y^12 - 22*y^11 + 28*y^10 - 4*y^9 - 13*y^8 - 4*y^7 + 28*y^6 - 22*y^5 - 2*y^4 + 12*y^3 - 5*y^2 - y + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - x^15 - 5*x^14 + 12*x^13 - 2*x^12 - 22*x^11 + 28*x^10 - 4*x^9 - 13*x^8 - 4*x^7 + 28*x^6 - 22*x^5 - 2*x^4 + 12*x^3 - 5*x^2 - x + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - x^15 - 5*x^14 + 12*x^13 - 2*x^12 - 22*x^11 + 28*x^10 - 4*x^9 - 13*x^8 - 4*x^7 + 28*x^6 - 22*x^5 - 2*x^4 + 12*x^3 - 5*x^2 - x + 1)
 

\( x^{16} - x^{15} - 5 x^{14} + 12 x^{13} - 2 x^{12} - 22 x^{11} + 28 x^{10} - 4 x^{9} - 13 x^{8} + \cdots + 1 \) 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:  $[4, 6]$
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:   \(289217423320703125\) \(\medspace = 5^{8}\cdot 29^{4}\cdot 61\cdot 131^{2}\) 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:  \(12.34\)
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:  $5^{1/2}29^{1/2}61^{1/2}131^{1/2}\approx 1076.4269599002062$
Ramified primes:   \(5\), \(29\), \(61\), \(131\) 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:  \(\Q(\sqrt{61}) \)
$\Aut(K/\Q)$:   $C_2$
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.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$ 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:  Yes
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:  $9$
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:   $12a^{15}-8a^{14}-67a^{13}+120a^{12}+39a^{11}-270a^{10}+209a^{9}+83a^{8}-135a^{7}-138a^{6}+293a^{5}-108a^{4}-105a^{3}+93a^{2}-a-16$, $13a^{15}+3a^{14}-66a^{13}+71a^{12}+84a^{11}-195a^{10}+82a^{9}+100a^{8}-41a^{7}-148a^{6}+177a^{5}-13a^{4}-76a^{3}+38a^{2}+8a-5$, $11a^{15}-3a^{14}-58a^{13}+88a^{12}+45a^{11}-207a^{10}+146a^{9}+66a^{8}-83a^{7}-120a^{6}+217a^{5}-74a^{4}-73a^{3}+62a^{2}-a-10$, $a^{14}-5a^{12}+7a^{11}+5a^{10}-17a^{9}+11a^{8}+7a^{7}-6a^{6}-10a^{5}+18a^{4}-4a^{3}-6a^{2}+5a$, $7a^{15}-a^{14}-39a^{13}+51a^{12}+46a^{11}-137a^{10}+64a^{9}+78a^{8}-57a^{7}-90a^{6}+128a^{5}-9a^{4}-74a^{3}+39a^{2}+8a-9$, $5a^{15}+7a^{14}-22a^{13}-a^{12}+54a^{11}-31a^{10}-38a^{9}+51a^{8}+28a^{7}-55a^{6}+2a^{5}+49a^{4}-19a^{3}-10a^{2}+8a+3$, $5a^{15}-5a^{14}-30a^{13}+56a^{12}+13a^{11}-124a^{10}+101a^{9}+32a^{8}-69a^{7}-58a^{6}+136a^{5}-57a^{4}-50a^{3}+46a^{2}-3a-9$, $4a^{15}-7a^{14}-24a^{13}+63a^{12}-6a^{11}-127a^{10}+131a^{9}+21a^{8}-90a^{7}-43a^{6}+156a^{5}-81a^{4}-51a^{3}+59a^{2}-5a-11$, $a^{15}-4a^{13}+7a^{12}-a^{11}-12a^{10}+19a^{9}-8a^{8}-3a^{7}-2a^{6}+17a^{5}-19a^{4}+5a^{3}+6a^{2}-4a$ 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:  \( 181.198740019 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 181.198740019 \cdot 1}{2\cdot\sqrt{289217423320703125}}\cr\approx \mathstrut & 0.165848727732 \end{aligned}\]

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 - x^15 - 5*x^14 + 12*x^13 - 2*x^12 - 22*x^11 + 28*x^10 - 4*x^9 - 13*x^8 - 4*x^7 + 28*x^6 - 22*x^5 - 2*x^4 + 12*x^3 - 5*x^2 - x + 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 - x^15 - 5*x^14 + 12*x^13 - 2*x^12 - 22*x^11 + 28*x^10 - 4*x^9 - 13*x^8 - 4*x^7 + 28*x^6 - 22*x^5 - 2*x^4 + 12*x^3 - 5*x^2 - x + 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 - x^15 - 5*x^14 + 12*x^13 - 2*x^12 - 22*x^11 + 28*x^10 - 4*x^9 - 13*x^8 - 4*x^7 + 28*x^6 - 22*x^5 - 2*x^4 + 12*x^3 - 5*x^2 - x + 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 - x^15 - 5*x^14 + 12*x^13 - 2*x^12 - 22*x^11 + 28*x^10 - 4*x^9 - 13*x^8 - 4*x^7 + 28*x^6 - 22*x^5 - 2*x^4 + 12*x^3 - 5*x^2 - x + 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

$C_4^4.C_2\wr D_4$ (as 16T1823):

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 32768
The 230 conjugacy class representatives for $C_4^4.C_2\wr D_4$
Character table for $C_4^4.C_2\wr D_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 8.6.68856875.1

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: 16.2.134673762004296875.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $16$ ${\href{/padicField/3.8.0.1}{8} }^{2}$ R ${\href{/padicField/7.4.0.1}{4} }^{3}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{6}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ $16$ ${\href{/padicField/19.4.0.1}{4} }^{3}{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ R ${\href{/padicField/31.4.0.1}{4} }^{3}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ $16$ ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.4.0.1}{4} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{6}$ ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.4.0.1}{4} }^{2}$

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
\(5\) Copy content Toggle raw display 5.4.2.4a1.2$x^{8} + 8 x^{6} + 8 x^{5} + 20 x^{4} + 32 x^{3} + 32 x^{2} + 16 x + 9$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
5.4.2.4a1.2$x^{8} + 8 x^{6} + 8 x^{5} + 20 x^{4} + 32 x^{3} + 32 x^{2} + 16 x + 9$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
\(29\) Copy content Toggle raw display 29.8.1.0a1.1$x^{8} + 3 x^{4} + 24 x^{3} + 26 x^{2} + 23 x + 2$$1$$8$$0$$C_8$$$[\ ]^{8}$$
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$$$[\ ]_{2}^{4}$$
\(61\) Copy content Toggle raw display $\Q_{61}$$x + 59$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{61}$$x + 59$$1$$1$$0$Trivial$$[\ ]$$
61.2.1.0a1.1$x^{2} + 60 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
61.2.1.0a1.1$x^{2} + 60 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
61.2.1.0a1.1$x^{2} + 60 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
61.2.1.0a1.1$x^{2} + 60 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
61.1.2.1a1.2$x^{2} + 122$$2$$1$$1$$C_2$$$[\ ]_{2}$$
61.4.1.0a1.1$x^{4} + 3 x^{2} + 40 x + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
\(131\) Copy content Toggle raw display 131.1.2.1a1.1$x^{2} + 131$$2$$1$$1$$C_2$$$[\ ]_{2}$$
131.1.2.1a1.1$x^{2} + 131$$2$$1$$1$$C_2$$$[\ ]_{2}$$
131.4.1.0a1.1$x^{4} + 9 x^{2} + 109 x + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
131.8.1.0a1.1$x^{8} + 3 x^{4} + 72 x^{3} + 116 x^{2} + 104 x + 2$$1$$8$$0$$C_8$$$[\ ]^{8}$$

Spectrum of ring of integers

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