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Properties

Label 16.0.56801432950182144.1
Degree $16$
Signature $[0, 8]$
Discriminant $5.680\times 10^{16}$
Root discriminant \(11.15\)
Ramified primes $2,3,7,139$
Class number $1$
Class group trivial
Galois group $C_2^2\wr C_2^2.D_4$ (as 16T1728)

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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 + 4*x^12 + 8*x^10 - 17*x^8 + 2*x^6 + 13*x^4 - 7*x^2 + 1)
 
Copy content gp:K = bnfinit(y^16 - 4*y^14 + 4*y^12 + 8*y^10 - 17*y^8 + 2*y^6 + 13*y^4 - 7*y^2 + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^14 + 4*x^12 + 8*x^10 - 17*x^8 + 2*x^6 + 13*x^4 - 7*x^2 + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 4*x^14 + 4*x^12 + 8*x^10 - 17*x^8 + 2*x^6 + 13*x^4 - 7*x^2 + 1)
 

\( x^{16} - 4x^{14} + 4x^{12} + 8x^{10} - 17x^{8} + 2x^{6} + 13x^{4} - 7x^{2} + 1 \) 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:  $[0, 8]$
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:   \(56801432950182144\) \(\medspace = 2^{8}\cdot 3^{14}\cdot 7^{4}\cdot 139^{2}\) 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:  \(11.15\)
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:  $2^{3/2}3^{7/8}7^{1/2}139^{1/2}\approx 230.71854887288097$
Ramified primes:   \(2\), \(3\), \(7\), \(139\) 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:  \(\Q(\sqrt{-3}) \)

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$ 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:   \( 4 a^{14} - 15 a^{12} + 13 a^{10} + 32 a^{8} - 55 a^{6} - 5 a^{4} + 43 a^{2} - 11 \)  (order $6$) 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:   $a$, $3a^{15}-5a^{14}-11a^{13}+\frac{37}{2}a^{12}+\frac{17}{2}a^{11}-\frac{29}{2}a^{10}+\frac{53}{2}a^{9}-44a^{8}-\frac{85}{2}a^{7}+71a^{6}-\frac{13}{2}a^{5}+12a^{4}+\frac{73}{2}a^{3}-\frac{121}{2}a^{2}-10a+\frac{31}{2}$, $\frac{1}{2}a^{15}-3a^{14}-2a^{13}+11a^{12}+\frac{3}{2}a^{11}-\frac{17}{2}a^{10}+\frac{11}{2}a^{9}-\frac{53}{2}a^{8}-\frac{19}{2}a^{7}+\frac{85}{2}a^{6}-2a^{5}+\frac{13}{2}a^{4}+\frac{19}{2}a^{3}-\frac{73}{2}a^{2}-3a+10$, $4a^{15}+\frac{1}{2}a^{14}-14a^{13}-\frac{3}{2}a^{12}+\frac{17}{2}a^{11}+39a^{9}+\frac{13}{2}a^{8}-54a^{7}-\frac{13}{2}a^{6}-17a^{5}-5a^{4}+51a^{3}+9a^{2}-\frac{23}{2}a-\frac{3}{2}$, $3a^{15}-11a^{13}-\frac{1}{2}a^{12}+\frac{17}{2}a^{11}+\frac{3}{2}a^{10}+\frac{53}{2}a^{9}-a^{8}-\frac{85}{2}a^{7}-3a^{6}-\frac{13}{2}a^{5}+3a^{4}+\frac{73}{2}a^{3}+\frac{1}{2}a^{2}-10a-\frac{3}{2}$, $\frac{5}{2}a^{15}-\frac{9}{2}a^{14}-9a^{13}+\frac{33}{2}a^{12}+6a^{11}-\frac{25}{2}a^{10}+\frac{49}{2}a^{9}-40a^{8}-\frac{73}{2}a^{7}+63a^{6}-9a^{5}+\frac{23}{2}a^{4}+\frac{69}{2}a^{3}-\frac{107}{2}a^{2}-\frac{17}{2}a+\frac{29}{2}$, $4a^{15}-\frac{9}{2}a^{14}-14a^{13}+\frac{33}{2}a^{12}+\frac{17}{2}a^{11}-13a^{10}+39a^{9}-\frac{77}{2}a^{8}-54a^{7}+\frac{123}{2}a^{6}-17a^{5}+10a^{4}+51a^{3}-51a^{2}-\frac{23}{2}a+\frac{25}{2}$ 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:  \( 130.436728236 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 130.436728236 \cdot 1}{6\cdot\sqrt{56801432950182144}}\cr\approx \mathstrut & 0.221568464151 \end{aligned}\]

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 + 4*x^12 + 8*x^10 - 17*x^8 + 2*x^6 + 13*x^4 - 7*x^2 + 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 - 4*x^14 + 4*x^12 + 8*x^10 - 17*x^8 + 2*x^6 + 13*x^4 - 7*x^2 + 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 - 4*x^14 + 4*x^12 + 8*x^10 - 17*x^8 + 2*x^6 + 13*x^4 - 7*x^2 + 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 - 4*x^14 + 4*x^12 + 8*x^10 - 17*x^8 + 2*x^6 + 13*x^4 - 7*x^2 + 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

$C_2^2\wr C_2^2.D_4$ (as 16T1728):

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 8192
The 152 conjugacy class representatives for $C_2^2\wr C_2^2.D_4$
Character table for $C_2^2\wr C_2^2.D_4$

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.189.1, 8.0.14895657.1

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: 16.0.34870903681646592.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ R ${\href{/padicField/11.8.0.1}{8} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }^{3}{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ ${\href{/padicField/17.8.0.1}{8} }^{2}$ ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{5}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.8.0.1}{8} }^{2}$ ${\href{/padicField/29.8.0.1}{8} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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$$$[\ ]^{8}$$
\(3\) Copy content Toggle raw display 3.2.8.14a1.2$x^{16} + 16 x^{15} + 128 x^{14} + 672 x^{13} + 2576 x^{12} + 7616 x^{11} + 17920 x^{10} + 34176 x^{9} + 53344 x^{8} + 68352 x^{7} + 71680 x^{6} + 60928 x^{5} + 41216 x^{4} + 21504 x^{3} + 8192 x^{2} + 2048 x + 259$$8$$2$$14$$QD_{16}$$$[\ ]_{8}^{2}$$
\(7\) Copy content Toggle raw display 7.4.1.0a1.1$x^{4} + 5 x^{2} + 4 x + 3$$1$$4$$0$$C_4$$$[\ ]^{4}$$
7.4.1.0a1.1$x^{4} + 5 x^{2} + 4 x + 3$$1$$4$$0$$C_4$$$[\ ]^{4}$$
7.4.2.4a1.2$x^{8} + 10 x^{6} + 8 x^{5} + 31 x^{4} + 40 x^{3} + 46 x^{2} + 24 x + 16$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
\(139\) Copy content Toggle raw display $\Q_{139}$$x + 137$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{139}$$x + 137$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{139}$$x + 137$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{139}$$x + 137$$1$$1$$0$Trivial$$[\ ]$$
139.2.2.2a1.2$x^{4} + 276 x^{3} + 19048 x^{2} + 552 x + 143$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
139.4.1.0a1.1$x^{4} + 7 x^{2} + 96 x + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
139.4.1.0a1.1$x^{4} + 7 x^{2} + 96 x + 2$$1$$4$$0$$C_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)