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Properties

Label 16.4.311502095702824709.1
Degree $16$
Signature $[4, 6]$
Discriminant $3.115\times 10^{17}$
Root discriminant \(12.40\)
Ramified primes $23,563,5861$
Class number $1$
Class group trivial
Galois group $C_4^4.C_2\wr S_4$ (as 16T1879)

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

\( x^{16} - x^{15} - 6 x^{14} + 6 x^{13} + 11 x^{12} - 13 x^{11} - 2 x^{10} + 15 x^{9} - 9 x^{8} - 15 x^{7} + \cdots + 1 \) 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:   \(311502095702824709\) \(\medspace = 23^{2}\cdot 563^{4}\cdot 5861\) 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:  \(12.40\)
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:  $23^{1/2}563^{1/2}5861^{1/2}\approx 8711.721356884642$
Ramified primes:   \(23\), \(563\), \(5861\) 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:  \(\Q(\sqrt{5861}) \)
$\Aut(K/\Q)$:   $C_2$
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:   $a$, $a^{15}+2a^{14}-8a^{13}-11a^{12}+22a^{11}+14a^{10}-24a^{9}+17a^{8}+20a^{7}-29a^{6}-34a^{5}-17a^{4}+24a^{3}+17a^{2}-5a-3$, $10a^{15}+3a^{14}-66a^{13}-18a^{12}+144a^{11}+11a^{10}-99a^{9}+118a^{8}+53a^{7}-191a^{6}-194a^{5}-17a^{4}+162a^{3}+85a^{2}-36a-22$, $a^{15}-a^{14}-5a^{13}+5a^{12}+6a^{11}-8a^{10}+4a^{9}+7a^{8}-5a^{7}-8a^{6}-7a^{5}+5a^{4}+4a^{3}-a^{2}-2a$, $13a^{15}-3a^{14}-83a^{13}+18a^{12}+170a^{11}-59a^{10}-84a^{9}+164a^{8}-13a^{7}-221a^{6}-167a^{5}+53a^{4}+195a^{3}+46a^{2}-52a-18$, $11a^{15}+2a^{14}-73a^{13}-11a^{12}+159a^{11}-5a^{10}-104a^{9}+134a^{8}+40a^{7}-208a^{6}-197a^{5}+3a^{4}+184a^{3}+82a^{2}-44a-24$, $9a^{15}-6a^{14}-53a^{13}+33a^{12}+93a^{11}-66a^{10}-15a^{9}+91a^{8}-39a^{7}-117a^{6}-90a^{5}+62a^{4}+105a^{3}+12a^{2}-27a-11$, $a^{15}+2a^{14}-8a^{13}-11a^{12}+22a^{11}+14a^{10}-24a^{9}+17a^{8}+20a^{7}-29a^{6}-34a^{5}-17a^{4}+24a^{3}+18a^{2}-5a-4$, $9a^{15}-2a^{14}-57a^{13}+12a^{12}+114a^{11}-38a^{10}-51a^{9}+105a^{8}-7a^{7}-143a^{6}-125a^{5}+32a^{4}+131a^{3}+35a^{2}-32a-14$ 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:  \( 191.193924186 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 191.193924186 \cdot 1}{2\cdot\sqrt{311502095702824709}}\cr\approx \mathstrut & 0.168621441542 \end{aligned}\]

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

$C_4^4.C_2\wr S_4$ (as 16T1879):

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 98304
The 190 conjugacy class representatives for $C_4^4.C_2\wr S_4$
Character table for $C_4^4.C_2\wr S_4$

Intermediate fields

4.2.563.1, 8.2.7290287.1

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: 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 ${\href{/padicField/2.8.0.1}{8} }{,}\,{\href{/padicField/2.4.0.1}{4} }^{2}$ ${\href{/padicField/3.6.0.1}{6} }^{2}{,}\,{\href{/padicField/3.4.0.1}{4} }$ ${\href{/padicField/5.8.0.1}{8} }^{2}$ ${\href{/padicField/7.3.0.1}{3} }^{4}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.4.0.1}{4} }^{2}$ ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$ ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ R ${\href{/padicField/29.4.0.1}{4} }^{3}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{3}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.3.0.1}{3} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ $16$ ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$

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
\(23\) Copy content Toggle raw display 23.3.1.0a1.1$x^{3} + 2 x + 18$$1$$3$$0$$C_3$$$[\ ]^{3}$$
23.3.1.0a1.1$x^{3} + 2 x + 18$$1$$3$$0$$C_3$$$[\ ]^{3}$$
23.2.2.2a1.2$x^{4} + 42 x^{3} + 451 x^{2} + 210 x + 48$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
23.6.1.0a1.1$x^{6} + x^{4} + 9 x^{3} + 9 x^{2} + x + 5$$1$$6$$0$$C_6$$$[\ ]^{6}$$
\(563\) Copy content Toggle raw display $\Q_{563}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{563}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $4$$1$$4$$0$$C_4$$$[\ ]^{4}$$
Deg $4$$2$$2$$2$
\(5861\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $12$$1$$12$$0$$C_{12}$$$[\ ]^{12}$$

Spectrum of ring of integers

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