Properties

Label 16.4.989068630116335616.1
Degree $16$
Signature $(4, 6)$
Discriminant $9.891\times 10^{17}$
Root discriminant \(13.33\)
Ramified primes $2,3,73$
Class number $1$
Class group trivial
Galois group $D_4^2:C_2^2$ (as 16T696)

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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^15 + 3*x^14 + 8*x^13 - 18*x^12 + 12*x^11 + 3*x^10 + 10*x^9 - 36*x^8 + 12*x^7 + 21*x^6 - 18*x^5 + 8*x^4 - 8*x^3 + 12*x^2 - 6*x + 1)
 
Copy content gp:K = bnfinit(y^16 - 4*y^15 + 3*y^14 + 8*y^13 - 18*y^12 + 12*y^11 + 3*y^10 + 10*y^9 - 36*y^8 + 12*y^7 + 21*y^6 - 18*y^5 + 8*y^4 - 8*y^3 + 12*y^2 - 6*y + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^15 + 3*x^14 + 8*x^13 - 18*x^12 + 12*x^11 + 3*x^10 + 10*x^9 - 36*x^8 + 12*x^7 + 21*x^6 - 18*x^5 + 8*x^4 - 8*x^3 + 12*x^2 - 6*x + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 4*x^15 + 3*x^14 + 8*x^13 - 18*x^12 + 12*x^11 + 3*x^10 + 10*x^9 - 36*x^8 + 12*x^7 + 21*x^6 - 18*x^5 + 8*x^4 - 8*x^3 + 12*x^2 - 6*x + 1)
 

\( x^{16} - 4 x^{15} + 3 x^{14} + 8 x^{13} - 18 x^{12} + 12 x^{11} + 3 x^{10} + 10 x^{9} - 36 x^{8} + \cdots + 1 \) 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:   \(989068630116335616\) \(\medspace = 2^{16}\cdot 3^{12}\cdot 73^{4}\) 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:  \(13.33\)
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:  $2\cdot 3^{3/4}73^{1/2}\approx 38.95223366419873$
Ramified primes:   \(2\), \(3\), \(73\) 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:Automorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{318600011}a^{15}+\frac{23885367}{318600011}a^{14}-\frac{134031331}{318600011}a^{13}+\frac{105372496}{318600011}a^{12}+\frac{86058726}{318600011}a^{11}+\frac{91527525}{318600011}a^{10}-\frac{26943000}{318600011}a^{9}+\frac{115966031}{318600011}a^{8}+\frac{63999048}{318600011}a^{7}+\frac{63948886}{318600011}a^{6}-\frac{137989847}{318600011}a^{5}+\frac{88767493}{318600011}a^{4}-\frac{19558758}{318600011}a^{3}+\frac{102600250}{318600011}a^{2}-\frac{123468336}{318600011}a-\frac{100492339}{318600011}$ 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:   $a$, $\frac{83616340}{318600011}a^{15}-\frac{283658898}{318600011}a^{14}+\frac{48392135}{318600011}a^{13}+\frac{812780388}{318600011}a^{12}-\frac{1041063743}{318600011}a^{11}+\frac{10323848}{318600011}a^{10}+\frac{746962903}{318600011}a^{9}+\frac{1296159384}{318600011}a^{8}-\frac{2750717312}{318600011}a^{7}-\frac{832819511}{318600011}a^{6}+\frac{2396268782}{318600011}a^{5}+\frac{207569027}{318600011}a^{4}-\frac{235840729}{318600011}a^{3}-\frac{759915993}{318600011}a^{2}+\frac{786435707}{318600011}a-\frac{291103720}{318600011}$, $\frac{387808189}{318600011}a^{15}-\frac{1391142449}{318600011}a^{14}+\frac{594840962}{318600011}a^{13}+\frac{3364086402}{318600011}a^{12}-\frac{5653975005}{318600011}a^{11}+\frac{2263595877}{318600011}a^{10}+\frac{2309525975}{318600011}a^{9}+\frac{4735102586}{318600011}a^{8}-\frac{12241110646}{318600011}a^{7}+\frac{29384803}{318600011}a^{6}+\frac{8480556201}{318600011}a^{5}-\frac{3620550967}{318600011}a^{4}+\frac{940812468}{318600011}a^{3}-\frac{2582917351}{318600011}a^{2}+\frac{3533574055}{318600011}a-\frac{956823435}{318600011}$, $\frac{129537075}{318600011}a^{15}-\frac{419343490}{318600011}a^{14}+\frac{98945381}{318600011}a^{13}+\frac{1006820908}{318600011}a^{12}-\frac{1536222956}{318600011}a^{11}+\frac{698141370}{318600011}a^{10}+\frac{427174984}{318600011}a^{9}+\frac{1693252493}{318600011}a^{8}-\frac{2854925069}{318600011}a^{7}-\frac{616256797}{318600011}a^{6}+\frac{1400268807}{318600011}a^{5}-\frac{1125301116}{318600011}a^{4}+\frac{1042521845}{318600011}a^{3}-\frac{814801002}{318600011}a^{2}+\frac{779443053}{318600011}a+\frac{168073876}{318600011}$, $\frac{83886143}{318600011}a^{15}-\frac{262408694}{318600011}a^{14}+\frac{50242875}{318600011}a^{13}+\frac{674937102}{318600011}a^{12}-\frac{1070214423}{318600011}a^{11}+\frac{361498835}{318600011}a^{10}+\frac{622584879}{318600011}a^{9}+\frac{846541011}{318600011}a^{8}-\frac{1824565902}{318600011}a^{7}-\frac{59325725}{318600011}a^{6}+\frac{1688664024}{318600011}a^{5}-\frac{1449544041}{318600011}a^{4}-\frac{275443210}{318600011}a^{3}-\frac{466620978}{318600011}a^{2}+\frac{520328026}{318600011}a-\frac{246106826}{318600011}$, $\frac{85797899}{318600011}a^{15}-\frac{296710806}{318600011}a^{14}+\frac{182262433}{318600011}a^{13}+\frac{531244921}{318600011}a^{12}-\frac{1237111912}{318600011}a^{11}+\frac{981741447}{318600011}a^{10}-\frac{74144762}{318600011}a^{9}+\frac{1147247075}{318600011}a^{8}-\frac{2266981911}{318600011}a^{7}+\frac{1092357160}{318600011}a^{6}+\frac{854230736}{318600011}a^{5}-\frac{1639455472}{318600011}a^{4}+\frac{641888757}{318600011}a^{3}-\frac{875654161}{318600011}a^{2}+\frac{1015519602}{318600011}a-\frac{634311088}{318600011}$, $\frac{239995668}{318600011}a^{15}-\frac{919926475}{318600011}a^{14}+\frac{578523259}{318600011}a^{13}+\frac{2017420095}{318600011}a^{12}-\frac{4091092598}{318600011}a^{11}+\frac{2311955326}{318600011}a^{10}+\frac{1225528502}{318600011}a^{9}+\frac{2245047366}{318600011}a^{8}-\frac{8031578885}{318600011}a^{7}+\frac{1802338886}{318600011}a^{6}+\frac{5666615416}{318600011}a^{5}-\frac{4235516368}{318600011}a^{4}+\frac{1518205582}{318600011}a^{3}-\frac{1531039241}{318600011}a^{2}+\frac{2324702920}{318600011}a-\frac{1173897671}{318600011}$, $\frac{89354608}{318600011}a^{15}-\frac{415466709}{318600011}a^{14}+\frac{423671691}{318600011}a^{13}+\frac{766580944}{318600011}a^{12}-\frac{2068787142}{318600011}a^{11}+\frac{1501817191}{318600011}a^{10}+\frac{345176829}{318600011}a^{9}+\frac{592158509}{318600011}a^{8}-\frac{4196627972}{318600011}a^{7}+\frac{2034480500}{318600011}a^{6}+\frac{2652242306}{318600011}a^{5}-\frac{2002295517}{318600011}a^{4}+\frac{550683130}{318600011}a^{3}-\frac{998076135}{318600011}a^{2}+\frac{1443015195}{318600011}a-\frac{581323067}{318600011}$, $\frac{123029844}{318600011}a^{15}-\frac{278965961}{318600011}a^{14}-\frac{333912856}{318600011}a^{13}+\frac{1131576268}{318600011}a^{12}-\frac{394935896}{318600011}a^{11}-\frac{1100928571}{318600011}a^{10}+\frac{1077554750}{318600011}a^{9}+\frac{2404436327}{318600011}a^{8}-\frac{1439662507}{318600011}a^{7}-\frac{4129284253}{318600011}a^{6}+\frac{1134550416}{318600011}a^{5}+\frac{1582000397}{318600011}a^{4}+\frac{66506663}{318600011}a^{3}-\frac{177557636}{318600011}a^{2}-\frac{98358867}{318600011}a+\frac{509099685}{318600011}$ 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:  \( 390.937838278 \)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 
Unit signature rank:  \( 3 \)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 390.937838278 \cdot 1}{2\cdot\sqrt{989068630116335616}}\cr\approx \mathstrut & 0.19349230153 \end{aligned}\]

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^15 + 3*x^14 + 8*x^13 - 18*x^12 + 12*x^11 + 3*x^10 + 10*x^9 - 36*x^8 + 12*x^7 + 21*x^6 - 18*x^5 + 8*x^4 - 8*x^3 + 12*x^2 - 6*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 - 4*x^15 + 3*x^14 + 8*x^13 - 18*x^12 + 12*x^11 + 3*x^10 + 10*x^9 - 36*x^8 + 12*x^7 + 21*x^6 - 18*x^5 + 8*x^4 - 8*x^3 + 12*x^2 - 6*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 - 4*x^15 + 3*x^14 + 8*x^13 - 18*x^12 + 12*x^11 + 3*x^10 + 10*x^9 - 36*x^8 + 12*x^7 + 21*x^6 - 18*x^5 + 8*x^4 - 8*x^3 + 12*x^2 - 6*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 = polynomial_ring(QQ); K, a = number_field(x^16 - 4*x^15 + 3*x^14 + 8*x^13 - 18*x^12 + 12*x^11 + 3*x^10 + 10*x^9 - 36*x^8 + 12*x^7 + 21*x^6 - 18*x^5 + 8*x^4 - 8*x^3 + 12*x^2 - 6*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

$D_4^2:C_2^2$ (as 16T696):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
Copy content gp:polgalois(K.pol)
 
Copy content magma: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 25 conjugacy class representatives for $D_4^2:C_2^2$
Character table for $D_4^2:C_2^2$

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{11 +4 \sqrt{3}})\), 8.4.110502144.1

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: 16.0.72202009998492499968.2

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 ${\href{/padicField/5.4.0.1}{4} }^{4}$ ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{3}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.8.0.1}{8} }^{2}$ ${\href{/padicField/23.2.0.1}{2} }^{6}{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ ${\href{/padicField/41.8.0.1}{8} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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 2.8.2.16a1.1$x^{16} + 2 x^{12} + 2 x^{11} + 2 x^{10} + 5 x^{8} + 2 x^{7} + 3 x^{6} + 2 x^{5} + 5 x^{4} + 4 x^{3} + 4 x^{2} + 5$$2$$8$$16$$C_8\times C_2$$$[2]^{8}$$
\(3\) Copy content Toggle raw display 3.4.4.12a1.1$x^{16} + 8 x^{15} + 24 x^{14} + 32 x^{13} + 24 x^{12} + 48 x^{11} + 96 x^{10} + 64 x^{9} + 24 x^{8} + 96 x^{7} + 96 x^{6} + 32 x^{4} + 64 x^{3} + 3 x^{2} + 16$$4$$4$$12$$C_8: C_2$$$[\ ]_{4}^{4}$$
\(73\) Copy content Toggle raw display 73.2.1.0a1.1$x^{2} + 70 x + 5$$1$$2$$0$$C_2$$$[\ ]^{2}$$
73.2.1.0a1.1$x^{2} + 70 x + 5$$1$$2$$0$$C_2$$$[\ ]^{2}$$
73.2.1.0a1.1$x^{2} + 70 x + 5$$1$$2$$0$$C_2$$$[\ ]^{2}$$
73.2.1.0a1.1$x^{2} + 70 x + 5$$1$$2$$0$$C_2$$$[\ ]^{2}$$
73.2.2.2a1.2$x^{4} + 140 x^{3} + 4910 x^{2} + 700 x + 98$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
73.2.2.2a1.2$x^{4} + 140 x^{3} + 4910 x^{2} + 700 x + 98$$2$$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)