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Properties

Label 16.4.290...000.1
Degree $16$
Signature $[4, 6]$
Discriminant $2.900\times 10^{20}$
Root discriminant \(19.01\)
Ramified primes $2,3,5,29$
Class number $1$
Class group trivial
Galois group $D_4^2:C_4$ (as 16T610)

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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 + 11*x^14 + 39*x^12 + 42*x^10 - 21*x^8 - 46*x^6 - 4*x^4 + 8*x^2 + 1)
 
Copy content gp:K = bnfinit(y^16 + 11*y^14 + 39*y^12 + 42*y^10 - 21*y^8 - 46*y^6 - 4*y^4 + 8*y^2 + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 11*x^14 + 39*x^12 + 42*x^10 - 21*x^8 - 46*x^6 - 4*x^4 + 8*x^2 + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 + 11*x^14 + 39*x^12 + 42*x^10 - 21*x^8 - 46*x^6 - 4*x^4 + 8*x^2 + 1)
 

\( x^{16} + 11x^{14} + 39x^{12} + 42x^{10} - 21x^{8} - 46x^{6} - 4x^{4} + 8x^{2} + 1 \) 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:  $[4, 6]$
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:   \(290029415062500000000\) \(\medspace = 2^{8}\cdot 3^{8}\cdot 5^{12}\cdot 29^{4}\) 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:  \(19.01\)
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:  $2^{3/2}3^{1/2}5^{3/4}29^{1/2}\approx 88.21290473450735$
Ramified primes:   \(2\), \(3\), \(5\), \(29\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{1142}a^{14}-\frac{68}{571}a^{12}+\frac{23}{571}a^{10}-\frac{1}{2}a^{9}-\frac{439}{1142}a^{8}-\frac{11}{1142}a^{6}-\frac{71}{571}a^{4}-\frac{1}{2}a^{3}+\frac{157}{571}a^{2}-\frac{1}{2}a-\frac{235}{571}$, $\frac{1}{1142}a^{15}-\frac{68}{571}a^{13}+\frac{23}{571}a^{11}-\frac{1}{2}a^{10}-\frac{439}{1142}a^{9}-\frac{11}{1142}a^{7}-\frac{71}{571}a^{5}-\frac{1}{2}a^{4}+\frac{157}{571}a^{3}-\frac{1}{2}a^{2}-\frac{235}{571}a$ 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 $1$
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:  $9$
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:   $\frac{70}{571}a^{14}+\frac{758}{571}a^{12}+\frac{2649}{571}a^{10}+\frac{2959}{571}a^{8}-\frac{770}{571}a^{6}-\frac{3088}{571}a^{4}-\frac{1431}{571}a^{2}+\frac{789}{571}$, $\frac{370}{571}a^{15}+\frac{3925}{571}a^{13}+\frac{13023}{571}a^{11}+\frac{11725}{571}a^{9}-\frac{8638}{571}a^{7}-\frac{11428}{571}a^{5}+\frac{267}{571}a^{3}+\frac{1397}{571}a$, $a$, $\frac{148}{571}a^{14}+\frac{1570}{571}a^{12}+\frac{5095}{571}a^{10}+\frac{3548}{571}a^{8}-\frac{6767}{571}a^{6}-\frac{6170}{571}a^{4}+\frac{3076}{571}a^{2}+\frac{673}{571}$, $\frac{373}{1142}a^{15}+\frac{153}{1142}a^{14}+\frac{2044}{571}a^{13}+\frac{1461}{1142}a^{12}+\frac{14303}{1142}a^{11}+\frac{1806}{571}a^{10}+\frac{14405}{1142}a^{9}-\frac{751}{571}a^{8}-\frac{9813}{1142}a^{7}-\frac{5695}{571}a^{6}-\frac{15851}{1142}a^{5}-\frac{5167}{1142}a^{4}+\frac{2351}{1142}a^{3}+\frac{4075}{1142}a^{2}+\frac{1992}{571}a+\frac{1749}{1142}$, $\frac{76}{571}a^{15}-\frac{567}{1142}a^{14}+\frac{1597}{1142}a^{13}-\frac{5683}{1142}a^{12}+\frac{5279}{1142}a^{11}-\frac{16375}{1142}a^{10}+\frac{2609}{571}a^{9}-\frac{5753}{1142}a^{8}-\frac{1101}{1142}a^{7}+\frac{12540}{571}a^{6}-\frac{1085}{571}a^{5}+\frac{5426}{571}a^{4}-\frac{1949}{1142}a^{3}-\frac{9593}{1142}a^{2}-\frac{889}{571}a-\frac{940}{571}$, $\frac{153}{1142}a^{15}+\frac{435}{1142}a^{14}+\frac{1461}{1142}a^{13}+\frac{2396}{571}a^{12}+\frac{1806}{571}a^{11}+\frac{17155}{1142}a^{10}-\frac{751}{571}a^{9}+\frac{9867}{571}a^{8}-\frac{5695}{571}a^{7}-\frac{4785}{1142}a^{6}-\frac{5167}{1142}a^{5}-\frac{15519}{1142}a^{4}+\frac{4075}{1142}a^{3}-\frac{796}{571}a^{2}+\frac{1749}{1142}a+\frac{1681}{1142}$, $\frac{300}{571}a^{14}+\frac{3167}{571}a^{12}+\frac{10374}{571}a^{10}+\frac{8766}{571}a^{8}-\frac{7868}{571}a^{6}-\frac{8340}{571}a^{4}+\frac{1698}{571}a^{2}+\frac{37}{571}$, $\frac{518}{571}a^{15}+\frac{5495}{571}a^{13}+\frac{18118}{571}a^{11}+\frac{15273}{571}a^{9}-\frac{15405}{571}a^{7}-\frac{17598}{571}a^{5}+\frac{3343}{571}a^{3}+\frac{2070}{571}a$ 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:  \( 6845.02782643 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 6845.02782643 \cdot 1}{2\cdot\sqrt{290029415062500000000}}\cr\approx \mathstrut & 0.197844231318 \end{aligned}\]

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 + 11*x^14 + 39*x^12 + 42*x^10 - 21*x^8 - 46*x^6 - 4*x^4 + 8*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 + 11*x^14 + 39*x^12 + 42*x^10 - 21*x^8 - 46*x^6 - 4*x^4 + 8*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 + 11*x^14 + 39*x^12 + 42*x^10 - 21*x^8 - 46*x^6 - 4*x^4 + 8*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 = polynomial_ring(QQ); K, a = number_field(x^16 + 11*x^14 + 39*x^12 + 42*x^10 - 21*x^8 - 46*x^6 - 4*x^4 + 8*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

$D_4^2:C_4$ (as 16T610):

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 256
The 40 conjugacy class representatives for $D_4^2:C_4$
Character table for $D_4^2:C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.32625.1, \(\Q(\zeta_{15})^+\), 4.4.725.1, 8.8.1064390625.1

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(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: 16.0.36665447040000000000.3

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 R ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.8.0.1}{8} }^{2}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.8.0.1}{8} }^{2}$ R ${\href{/padicField/31.2.0.1}{2} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}$

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$$$[\ ]^{8}$$
\(3\) Copy content Toggle raw display 3.8.2.8a1.2$x^{16} + 4 x^{13} + 2 x^{12} + 8 x^{10} + 8 x^{9} + 5 x^{8} + 8 x^{7} + 12 x^{6} + 12 x^{5} + 8 x^{4} + 8 x^{3} + 12 x^{2} + 8 x + 7$$2$$8$$8$$C_8\times C_2$$$[\ ]_{2}^{8}$$
\(5\) 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$$$[\ ]_{4}^{4}$$
\(29\) Copy content Toggle raw display 29.2.1.0a1.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
29.1.2.1a1.2$x^{2} + 58$$2$$1$$1$$C_2$$$[\ ]_{2}$$
29.2.1.0a1.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
29.1.2.1a1.2$x^{2} + 58$$2$$1$$1$$C_2$$$[\ ]_{2}$$
29.1.2.1a1.2$x^{2} + 58$$2$$1$$1$$C_2$$$[\ ]_{2}$$
29.2.1.0a1.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
29.1.2.1a1.2$x^{2} + 58$$2$$1$$1$$C_2$$$[\ ]_{2}$$
29.2.1.0a1.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$

Spectrum of ring of integers

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