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Properties

Label 16.2.202...000.2
Degree $16$
Signature $[2, 7]$
Discriminant $-2.020\times 10^{18}$
Root discriminant \(13.93\)
Ramified primes $2,5,13,29$
Class number $1$
Class group trivial
Galois group $C_2^7.C_2\wr D_4$ (as 16T1778)

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Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^16 + 3*x^14 - 3*x^12 - 14*x^10 - 9*x^8 + 2*x^6 + 8*x^4 - 6*x^2 - 1)
 
Copy content gp:K = bnfinit(y^16 + 3*y^14 - 3*y^12 - 14*y^10 - 9*y^8 + 2*y^6 + 8*y^4 - 6*y^2 - 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 3*x^14 - 3*x^12 - 14*x^10 - 9*x^8 + 2*x^6 + 8*x^4 - 6*x^2 - 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 + 3*x^14 - 3*x^12 - 14*x^10 - 9*x^8 + 2*x^6 + 8*x^4 - 6*x^2 - 1)
 

\( x^{16} + 3x^{14} - 3x^{12} - 14x^{10} - 9x^{8} + 2x^{6} + 8x^{4} - 6x^{2} - 1 \) 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:  $[2, 7]$
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:   \(-2020065264100000000\) \(\medspace = -\,2^{8}\cdot 5^{8}\cdot 13^{4}\cdot 29^{4}\) 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:  \(13.93\)
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:  $2^{15/8}5^{1/2}13^{1/2}29^{1/2}\approx 159.2527421555184$
Ramified primes:   \(2\), \(5\), \(13\), \(29\) 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:  \(\Q(\sqrt{-1}) \)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{1142}a^{14}-\frac{65}{571}a^{12}-\frac{1}{2}a^{11}+\frac{157}{1142}a^{10}+\frac{116}{571}a^{8}+\frac{270}{571}a^{6}-\frac{443}{1142}a^{4}+\frac{57}{571}a^{2}-\frac{1}{2}a+\frac{249}{1142}$, $\frac{1}{1142}a^{15}-\frac{65}{571}a^{13}+\frac{157}{1142}a^{11}-\frac{1}{2}a^{10}-\frac{339}{1142}a^{9}-\frac{31}{1142}a^{7}+\frac{64}{571}a^{5}-\frac{457}{1142}a^{3}-\frac{161}{571}a-\frac{1}{2}$ 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:  $8$
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:   $a$, $\frac{146}{571}a^{14}+\frac{434}{571}a^{12}-\frac{489}{571}a^{10}-\frac{2101}{571}a^{8}-\frac{1100}{571}a^{6}+\frac{416}{571}a^{4}+\frac{1227}{571}a^{2}-\frac{190}{571}$, $\frac{133}{571}a^{14}+\frac{411}{571}a^{12}-\frac{246}{571}a^{10}-\frac{1691}{571}a^{8}-\frac{1839}{571}a^{6}-\frac{677}{571}a^{4}+\frac{887}{571}a^{2}-\frac{572}{571}$, $\frac{154}{571}a^{15}+\frac{536}{571}a^{13}-\frac{375}{571}a^{11}-\frac{2529}{571}a^{9}-\frac{1919}{571}a^{7}+\frac{298}{571}a^{5}+\frac{2139}{571}a^{3}-\frac{482}{571}a$, $\frac{1}{1142}a^{15}+\frac{21}{1142}a^{14}-\frac{65}{571}a^{13}+\frac{125}{1142}a^{12}-\frac{207}{571}a^{11}-\frac{129}{1142}a^{10}+\frac{116}{571}a^{9}-\frac{419}{571}a^{8}+\frac{841}{571}a^{7}-\frac{40}{571}a^{6}+\frac{1841}{1142}a^{5}+\frac{975}{1142}a^{4}+\frac{628}{571}a^{3}+\frac{681}{1142}a^{2}-\frac{161}{571}a-\frac{481}{1142}$, $\frac{1}{1142}a^{15}-\frac{21}{1142}a^{14}-\frac{65}{571}a^{13}-\frac{125}{1142}a^{12}-\frac{207}{571}a^{11}+\frac{129}{1142}a^{10}+\frac{116}{571}a^{9}+\frac{419}{571}a^{8}+\frac{841}{571}a^{7}+\frac{40}{571}a^{6}+\frac{1841}{1142}a^{5}-\frac{975}{1142}a^{4}+\frac{628}{571}a^{3}-\frac{681}{1142}a^{2}-\frac{161}{571}a+\frac{481}{1142}$, $\frac{1}{1142}a^{15}-\frac{287}{1142}a^{14}-\frac{65}{571}a^{13}-\frac{947}{1142}a^{12}-\frac{207}{571}a^{11}+\frac{621}{1142}a^{10}+\frac{116}{571}a^{9}+\frac{2110}{571}a^{8}+\frac{841}{571}a^{7}+\frac{1879}{571}a^{6}+\frac{1841}{1142}a^{5}+\frac{379}{1142}a^{4}+\frac{628}{571}a^{3}-\frac{2455}{1142}a^{2}-\frac{161}{571}a+\frac{1625}{1142}$, $\frac{149}{1142}a^{15}+\frac{51}{571}a^{14}+\frac{615}{1142}a^{13}+\frac{222}{571}a^{12}-\frac{9}{571}a^{11}+\frac{13}{571}a^{10}-\frac{2547}{1142}a^{9}-\frac{2031}{1142}a^{8}-\frac{3477}{1142}a^{7}-\frac{2591}{1142}a^{6}-\frac{742}{571}a^{5}-\frac{77}{1142}a^{4}+\frac{1070}{571}a^{3}+\frac{1921}{1142}a^{2}+\frac{557}{1142}a+\frac{845}{1142}$ 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:  \( 441.631865837 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{7}\cdot 441.631865837 \cdot 1}{2\cdot\sqrt{2020065264100000000}}\cr\approx \mathstrut & 0.240251869633 \end{aligned}\]

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 + 3*x^14 - 3*x^12 - 14*x^10 - 9*x^8 + 2*x^6 + 8*x^4 - 6*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 + 3*x^14 - 3*x^12 - 14*x^10 - 9*x^8 + 2*x^6 + 8*x^4 - 6*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 + 3*x^14 - 3*x^12 - 14*x^10 - 9*x^8 + 2*x^6 + 8*x^4 - 6*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 + 3*x^14 - 3*x^12 - 14*x^10 - 9*x^8 + 2*x^6 + 8*x^4 - 6*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

$C_2^7.C_2\wr D_4$ (as 16T1778):

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 16384
The 148 conjugacy class representatives for $C_2^7.C_2\wr D_4$
Character table for $C_2^7.C_2\wr D_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 8.4.88830625.1

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: 16.4.3059980518400000000.3

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.4.0.1}{4} }^{2}$ R ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.4.0.1}{4} }^{2}$ ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ R ${\href{/padicField/17.8.0.1}{8} }^{2}$ ${\href{/padicField/19.4.0.1}{4} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ ${\href{/padicField/23.4.0.1}{4} }^{3}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ R ${\href{/padicField/31.4.0.1}{4} }^{3}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.4.0.1}{4} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.4.0.1}{4} }^{2}$ ${\href{/padicField/53.8.0.1}{8} }^{2}$ ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.4.0.1}{4} }^{2}$

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
\(2\) Copy content Toggle raw display 2.4.1.0a1.1$x^{4} + x + 1$$1$$4$$0$$C_4$$$[\ ]^{4}$$
2.4.1.0a1.1$x^{4} + x + 1$$1$$4$$0$$C_4$$$[\ ]^{4}$$
2.4.2.8a4.2$x^{8} + 2 x^{7} + 2 x^{5} + 4 x^{4} + 2 x^{3} + x^{2} + 2 x + 7$$2$$4$$8$$((C_8 : C_2):C_2):C_2$$$[2, 2, 2, 2]^{4}$$
\(5\) Copy content Toggle raw display 5.8.2.8a1.2$x^{16} + 2 x^{12} + 6 x^{10} + 8 x^{9} + 5 x^{8} + 6 x^{6} + 8 x^{5} + 13 x^{4} + 24 x^{3} + 28 x^{2} + 16 x + 9$$2$$8$$8$$C_8\times C_2$$$[\ ]_{2}^{8}$$
\(13\) Copy content Toggle raw display 13.2.1.0a1.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
13.2.1.0a1.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
13.2.1.0a1.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
13.2.1.0a1.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
13.4.2.4a1.2$x^{8} + 6 x^{6} + 24 x^{5} + 13 x^{4} + 72 x^{3} + 156 x^{2} + 48 x + 17$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
\(29\) Copy content Toggle raw display 29.2.1.0a1.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
29.2.1.0a1.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
29.2.1.0a1.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
29.2.1.0a1.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
29.2.2.2a1.2$x^{4} + 48 x^{3} + 580 x^{2} + 96 x + 33$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
29.2.2.2a1.2$x^{4} + 48 x^{3} + 580 x^{2} + 96 x + 33$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$

Spectrum of ring of integers

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