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Properties

Label 16.4.60371254156640625.1
Degree $16$
Signature $[4, 6]$
Discriminant $6.037\times 10^{16}$
Root discriminant \(11.19\)
Ramified primes $3,5,11,19$
Class number $1$
Class group trivial
Galois group $D_4^2:C_2^2$ (as 16T602)

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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 - 4*x^14 - 4*x^13 + 10*x^12 + 9*x^11 - 14*x^10 - 18*x^9 + 7*x^8 + 23*x^7 + 8*x^6 - 6*x^5 - 9*x^4 - 9*x^3 + 4*x + 1)
 
Copy content gp:K = bnfinit(y^16 - 4*y^14 - 4*y^13 + 10*y^12 + 9*y^11 - 14*y^10 - 18*y^9 + 7*y^8 + 23*y^7 + 8*y^6 - 6*y^5 - 9*y^4 - 9*y^3 + 4*y + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^14 - 4*x^13 + 10*x^12 + 9*x^11 - 14*x^10 - 18*x^9 + 7*x^8 + 23*x^7 + 8*x^6 - 6*x^5 - 9*x^4 - 9*x^3 + 4*x + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 4*x^14 - 4*x^13 + 10*x^12 + 9*x^11 - 14*x^10 - 18*x^9 + 7*x^8 + 23*x^7 + 8*x^6 - 6*x^5 - 9*x^4 - 9*x^3 + 4*x + 1)
 

\( x^{16} - 4 x^{14} - 4 x^{13} + 10 x^{12} + 9 x^{11} - 14 x^{10} - 18 x^{9} + 7 x^{8} + 23 x^{7} + \cdots + 1 \) 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:   \(60371254156640625\) \(\medspace = 3^{4}\cdot 5^{8}\cdot 11^{4}\cdot 19^{4}\) 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:  \(11.19\)
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:  $3^{1/2}5^{1/2}11^{1/2}19^{1/2}\approx 55.991070716677676$
Ramified primes:   \(3\), \(5\), \(11\), \(19\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_2^2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{8868583}a^{15}+\frac{2946792}{8868583}a^{14}-\frac{14240}{99647}a^{13}+\frac{2678006}{8868583}a^{12}+\frac{4314455}{8868583}a^{11}-\frac{4001605}{8868583}a^{10}+\frac{938784}{8868583}a^{9}-\frac{2520029}{8868583}a^{8}+\frac{255093}{8868583}a^{7}-\frac{3951984}{8868583}a^{6}-\frac{1491866}{8868583}a^{5}-\frac{120697}{8868583}a^{4}-\frac{3301401}{8868583}a^{3}+\frac{832160}{8868583}a^{2}+\frac{3756888}{8868583}a+\frac{52821}{8868583}$ 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:  $9$
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:   $\frac{5241478}{8868583}a^{15}-\frac{5319973}{8868583}a^{14}-\frac{153957}{99647}a^{13}-\frac{4736050}{8868583}a^{12}+\frac{50224126}{8868583}a^{11}-\frac{19354194}{8868583}a^{10}-\frac{46646134}{8868583}a^{9}-\frac{13020071}{8868583}a^{8}+\frac{45642957}{8868583}a^{7}+\frac{23434918}{8868583}a^{6}-\frac{11289520}{8868583}a^{5}+\frac{18566722}{8868583}a^{4}-\frac{10064155}{8868583}a^{3}-\frac{15895746}{8868583}a^{2}+\frac{3741758}{8868583}a-\frac{8183239}{8868583}$, $\frac{224064}{8868583}a^{15}-\frac{4870245}{8868583}a^{14}+\frac{25580}{99647}a^{13}+\frac{14147770}{8868583}a^{12}+\frac{11892371}{8868583}a^{11}-\frac{46224335}{8868583}a^{10}-\frac{6222001}{8868583}a^{9}+\frac{50376086}{8868583}a^{8}+\frac{43483432}{8868583}a^{7}-\frac{40279090}{8868583}a^{6}-\frac{52044486}{8868583}a^{5}-\frac{3543041}{8868583}a^{4}+\frac{12262949}{8868583}a^{3}+\frac{21746414}{8868583}a^{2}+\frac{12928804}{8868583}a-\frac{4273761}{8868583}$, $\frac{2863568}{8868583}a^{15}-\frac{2159065}{8868583}a^{14}-\frac{161215}{99647}a^{13}-\frac{2566109}{8868583}a^{12}+\frac{45326885}{8868583}a^{11}+\frac{15221668}{8868583}a^{10}-\frac{79480644}{8868583}a^{9}-\frac{32576534}{8868583}a^{8}+\frac{77393110}{8868583}a^{7}+\frac{86841234}{8868583}a^{6}-\frac{27832456}{8868583}a^{5}-\frac{59730301}{8868583}a^{4}-\frac{27546679}{8868583}a^{3}-\frac{23768054}{8868583}a^{2}+\frac{25305319}{8868583}a+\frac{20579429}{8868583}$, $\frac{1037116}{8868583}a^{15}-\frac{1781426}{8868583}a^{14}-\frac{49264}{99647}a^{13}+\frac{2126837}{8868583}a^{12}+\frac{17707794}{8868583}a^{11}-\frac{4207666}{8868583}a^{10}-\frac{27214877}{8868583}a^{9}+\frac{145153}{8868583}a^{8}+\frac{28938064}{8868583}a^{7}+\frac{20743970}{8868583}a^{6}-\frac{16239693}{8868583}a^{5}-\frac{14477973}{8868583}a^{4}-\frac{24223540}{8868583}a^{3}-\frac{8572668}{8868583}a^{2}+\frac{14268371}{8868583}a+\frac{9135628}{8868583}$, $\frac{17561319}{8868583}a^{15}-\frac{10965300}{8868583}a^{14}-\frac{665712}{99647}a^{13}-\frac{33038114}{8868583}a^{12}+\frac{182941759}{8868583}a^{11}+\frac{27990632}{8868583}a^{10}-\frac{233129244}{8868583}a^{9}-\frac{147713109}{8868583}a^{8}+\frac{177325703}{8868583}a^{7}+\frac{241818309}{8868583}a^{6}-\frac{10261804}{8868583}a^{5}-\frac{51525258}{8868583}a^{4}-\frac{102404529}{8868583}a^{3}-\frac{81037267}{8868583}a^{2}+\frac{45343615}{8868583}a+\frac{23597763}{8868583}$, $\frac{1750413}{8868583}a^{15}+\frac{2123551}{8868583}a^{14}-\frac{80893}{99647}a^{13}-\frac{14925500}{8868583}a^{12}+\frac{9660516}{8868583}a^{11}+\frac{34126365}{8868583}a^{10}-\frac{8894861}{8868583}a^{9}-\frac{53446603}{8868583}a^{8}-\frac{15050641}{8868583}a^{7}+\frac{41336519}{8868583}a^{6}+\frac{52440939}{8868583}a^{5}+\frac{15523531}{8868583}a^{4}-\frac{11071481}{8868583}a^{3}-\frac{32207087}{8868583}a^{2}-\frac{19648420}{8868583}a+\frac{3587298}{8868583}$, $a$, $\frac{6354633}{8868583}a^{15}-\frac{9602589}{8868583}a^{14}-\frac{234279}{99647}a^{13}+\frac{7623341}{8868583}a^{12}+\frac{85890495}{8868583}a^{11}-\frac{38258891}{8868583}a^{10}-\frac{117231917}{8868583}a^{9}+\frac{7849998}{8868583}a^{8}+\frac{138086708}{8868583}a^{7}+\frac{68939633}{8868583}a^{6}-\frac{91562915}{8868583}a^{5}-\frac{56687110}{8868583}a^{4}-\frac{26539353}{8868583}a^{3}-\frac{7456713}{8868583}a^{2}+\frac{48695497}{8868583}a+\frac{8808892}{8868583}$, $\frac{11301227}{8868583}a^{15}-\frac{6366601}{8868583}a^{14}-\frac{464303}{99647}a^{13}-\frac{21866527}{8868583}a^{12}+\frac{123809249}{8868583}a^{11}+\frac{31050500}{8868583}a^{10}-\frac{172309345}{8868583}a^{9}-\frac{102799586}{8868583}a^{8}+\frac{130994961}{8868583}a^{7}+\frac{181761122}{8868583}a^{6}-\frac{12812776}{8868583}a^{5}-\frac{53866985}{8868583}a^{4}-\frac{63838432}{8868583}a^{3}-\frac{64673204}{8868583}a^{2}+\frac{25321708}{8868583}a+\frac{15526803}{8868583}$ 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:  \( 73.3855457794 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 73.3855457794 \cdot 1}{2\cdot\sqrt{60371254156640625}}\cr\approx \mathstrut & 0.147016008228 \end{aligned}\]

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^13 + 10*x^12 + 9*x^11 - 14*x^10 - 18*x^9 + 7*x^8 + 23*x^7 + 8*x^6 - 6*x^5 - 9*x^4 - 9*x^3 + 4*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^14 - 4*x^13 + 10*x^12 + 9*x^11 - 14*x^10 - 18*x^9 + 7*x^8 + 23*x^7 + 8*x^6 - 6*x^5 - 9*x^4 - 9*x^3 + 4*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^14 - 4*x^13 + 10*x^12 + 9*x^11 - 14*x^10 - 18*x^9 + 7*x^8 + 23*x^7 + 8*x^6 - 6*x^5 - 9*x^4 - 9*x^3 + 4*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 - 4*x^14 - 4*x^13 + 10*x^12 + 9*x^11 - 14*x^10 - 18*x^9 + 7*x^8 + 23*x^7 + 8*x^6 - 6*x^5 - 9*x^4 - 9*x^3 + 4*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 16T602):

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 256
The 40 conjugacy class representatives for $D_4^2:C_2^2$
Character table for $D_4^2:C_2^2$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.275.1, 4.2.475.1, 4.4.5225.1, 8.2.12931875.2, 8.2.12931875.1, 8.4.27300625.1

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: 16.0.13545904672265625.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.8.0.1}{8} }^{2}$ R R ${\href{/padicField/7.4.0.1}{4} }^{4}$ R ${\href{/padicField/13.8.0.1}{8} }^{2}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ R ${\href{/padicField/23.2.0.1}{2} }^{8}$ ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.2.0.1}{2} }^{6}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$

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
\(3\) Copy content Toggle raw display 3.4.1.0a1.1$x^{4} + 2 x^{3} + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
3.4.1.0a1.1$x^{4} + 2 x^{3} + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
3.4.2.4a1.2$x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{4} + 8 x^{3} + 7$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
\(5\) Copy content Toggle raw display 5.4.2.4a1.2$x^{8} + 8 x^{6} + 8 x^{5} + 20 x^{4} + 32 x^{3} + 32 x^{2} + 16 x + 9$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
5.4.2.4a1.2$x^{8} + 8 x^{6} + 8 x^{5} + 20 x^{4} + 32 x^{3} + 32 x^{2} + 16 x + 9$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
\(11\) Copy content Toggle raw display 11.2.1.0a1.1$x^{2} + 7 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
11.2.1.0a1.1$x^{2} + 7 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
11.2.1.0a1.1$x^{2} + 7 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
11.2.1.0a1.1$x^{2} + 7 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
11.2.2.2a1.2$x^{4} + 14 x^{3} + 53 x^{2} + 28 x + 15$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
11.2.2.2a1.2$x^{4} + 14 x^{3} + 53 x^{2} + 28 x + 15$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
\(19\) Copy content Toggle raw display 19.2.1.0a1.1$x^{2} + 18 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
19.2.1.0a1.1$x^{2} + 18 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
19.2.1.0a1.1$x^{2} + 18 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
19.2.1.0a1.1$x^{2} + 18 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
19.2.2.2a1.2$x^{4} + 36 x^{3} + 328 x^{2} + 72 x + 23$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
19.2.2.2a1.2$x^{4} + 36 x^{3} + 328 x^{2} + 72 x + 23$$2$$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)