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Properties

Label 16.4.350749278894882816.1
Degree $16$
Signature $[4, 6]$
Discriminant $3.507\times 10^{17}$
Root discriminant \(12.49\)
Ramified primes $2,3,13$
Class number $1$
Class group trivial
Galois group $C_4^2:D_4$ (as 16T382)

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

\( x^{16} - 2x^{15} + x^{14} + 2x^{13} - 13x^{12} + 12x^{10} + 7x^{8} + 12x^{6} - 13x^{4} + 2x^{3} + x^{2} - 2x + 1 \) 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:   \(350749278894882816\) \(\medspace = 2^{16}\cdot 3^{8}\cdot 13^{8}\) 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:  \(12.49\)
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:  $2^{3/2}3^{3/4}13^{1/2}\approx 23.246501929569742$
Ramified primes:   \(2\), \(3\), \(13\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{5}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{9}-\frac{1}{3}a^{8}-\frac{1}{3}a^{6}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{9}-\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{9}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{12387}a^{14}+\frac{440}{4129}a^{13}-\frac{509}{4129}a^{12}-\frac{931}{12387}a^{11}+\frac{1174}{12387}a^{10}-\frac{3674}{12387}a^{9}-\frac{2486}{12387}a^{8}+\frac{3866}{12387}a^{7}+\frac{1643}{12387}a^{6}+\frac{1528}{4129}a^{5}-\frac{985}{4129}a^{4}-\frac{5060}{12387}a^{3}-\frac{5656}{12387}a^{2}+\frac{440}{4129}a+\frac{4130}{12387}$, $\frac{1}{12387}a^{15}-\frac{1489}{12387}a^{13}-\frac{81}{4129}a^{12}-\frac{116}{4129}a^{11}-\frac{850}{12387}a^{10}-\frac{4381}{12387}a^{9}-\frac{1298}{12387}a^{8}+\frac{2032}{4129}a^{7}+\frac{1183}{4129}a^{6}-\frac{721}{12387}a^{5}-\frac{2236}{12387}a^{4}-\frac{3049}{12387}a^{3}-\frac{707}{4129}a^{2}+\frac{13}{4129}a+\frac{2809}{12387}$ 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:   $\frac{2902}{12387}a^{15}-\frac{1552}{4129}a^{14}+\frac{1382}{4129}a^{13}+\frac{149}{4129}a^{12}-\frac{10679}{4129}a^{11}-\frac{9293}{12387}a^{10}-\frac{9032}{12387}a^{9}+\frac{8339}{12387}a^{8}+\frac{16573}{4129}a^{7}+\frac{15088}{12387}a^{6}+\frac{70951}{12387}a^{5}+\frac{19084}{12387}a^{4}+\frac{7793}{12387}a^{3}+\frac{257}{4129}a^{2}-\frac{20918}{12387}a-\frac{3584}{12387}$, $a$, $\frac{2999}{4129}a^{15}-\frac{15413}{12387}a^{14}+\frac{4624}{12387}a^{13}+\frac{6322}{4129}a^{12}-\frac{37143}{4129}a^{11}-\frac{10343}{4129}a^{10}+\frac{96800}{12387}a^{9}+\frac{31336}{12387}a^{8}+\frac{77597}{12387}a^{7}+\frac{16910}{12387}a^{6}+\frac{113561}{12387}a^{5}+\frac{26515}{12387}a^{4}-\frac{36344}{4129}a^{3}-\frac{10435}{12387}a^{2}-\frac{5788}{12387}a-\frac{8191}{12387}$, $\frac{9031}{12387}a^{15}-\frac{13723}{12387}a^{14}+\frac{557}{12387}a^{13}+\frac{23086}{12387}a^{12}-\frac{111115}{12387}a^{11}-\frac{53660}{12387}a^{10}+\frac{105704}{12387}a^{9}+\frac{38734}{12387}a^{8}+\frac{71615}{12387}a^{7}+\frac{12111}{4129}a^{6}+\frac{38259}{4129}a^{5}+\frac{24124}{4129}a^{4}-\frac{122101}{12387}a^{3}-\frac{28877}{12387}a^{2}+\frac{4936}{12387}a-\frac{8869}{4129}$, $\frac{641}{4129}a^{15}-\frac{145}{4129}a^{14}-\frac{2219}{12387}a^{13}+\frac{2893}{12387}a^{12}-\frac{5493}{4129}a^{11}-\frac{13150}{4129}a^{10}-\frac{9494}{12387}a^{9}+\frac{30506}{12387}a^{8}+\frac{10734}{4129}a^{7}+\frac{32134}{12387}a^{6}+\frac{46547}{12387}a^{5}+\frac{49315}{12387}a^{4}+\frac{20941}{12387}a^{3}-\frac{24529}{12387}a^{2}-\frac{11981}{12387}a-\frac{3589}{12387}$, $\frac{2235}{4129}a^{15}-\frac{9334}{12387}a^{14}+\frac{233}{12387}a^{13}+\frac{4579}{4129}a^{12}-\frac{25456}{4129}a^{11}-\frac{50528}{12387}a^{10}+\frac{58678}{12387}a^{9}+\frac{53863}{12387}a^{8}+\frac{21620}{4129}a^{7}+\frac{24736}{12387}a^{6}+\frac{31121}{4129}a^{5}+\frac{37430}{12387}a^{4}-\frac{68521}{12387}a^{3}-\frac{14203}{4129}a^{2}-\frac{2705}{12387}a-\frac{3229}{12387}$, $\frac{5952}{4129}a^{15}-\frac{30572}{12387}a^{14}+\frac{3037}{4129}a^{13}+\frac{38702}{12387}a^{12}-\frac{221374}{12387}a^{11}-\frac{21197}{4129}a^{10}+\frac{195302}{12387}a^{9}+\frac{17385}{4129}a^{8}+\frac{134879}{12387}a^{7}+\frac{35702}{12387}a^{6}+\frac{78580}{4129}a^{5}+\frac{77605}{12387}a^{4}-\frac{70441}{4129}a^{3}-\frac{2892}{4129}a^{2}+\frac{9764}{4129}a-\frac{10720}{4129}$, $\frac{10280}{12387}a^{15}-\frac{6139}{4129}a^{14}+\frac{8653}{12387}a^{13}+\frac{16661}{12387}a^{12}-\frac{122977}{12387}a^{11}-\frac{9308}{4129}a^{10}+\frac{30432}{4129}a^{9}+\frac{36830}{12387}a^{8}+\frac{25241}{4129}a^{7}+\frac{10444}{12387}a^{6}+\frac{150454}{12387}a^{5}+\frac{6198}{4129}a^{4}-\frac{76313}{12387}a^{3}-\frac{19136}{12387}a^{2}+\frac{18035}{12387}a-\frac{3955}{4129}$, $\frac{6665}{12387}a^{15}-\frac{7867}{12387}a^{14}-\frac{2203}{12387}a^{13}+\frac{17137}{12387}a^{12}-\frac{26014}{4129}a^{11}-\frac{65611}{12387}a^{10}+\frac{59087}{12387}a^{9}+\frac{34535}{12387}a^{8}+\frac{62807}{12387}a^{7}+\frac{23885}{4129}a^{6}+\frac{96016}{12387}a^{5}+\frac{25929}{4129}a^{4}-\frac{16288}{4129}a^{3}-\frac{26024}{12387}a^{2}-\frac{1442}{4129}a-\frac{27413}{12387}$ 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:  \( 218.035986284 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 218.035986284 \cdot 1}{2\cdot\sqrt{350749278894882816}}\cr\approx \mathstrut & 0.181217019171 \end{aligned}\]

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 - 2*x^15 + x^14 + 2*x^13 - 13*x^12 + 12*x^10 + 7*x^8 + 12*x^6 - 13*x^4 + 2*x^3 + x^2 - 2*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 - 2*x^15 + x^14 + 2*x^13 - 13*x^12 + 12*x^10 + 7*x^8 + 12*x^6 - 13*x^4 + 2*x^3 + x^2 - 2*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 - 2*x^15 + x^14 + 2*x^13 - 13*x^12 + 12*x^10 + 7*x^8 + 12*x^6 - 13*x^4 + 2*x^3 + x^2 - 2*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 - 2*x^15 + x^14 + 2*x^13 - 13*x^12 + 12*x^10 + 7*x^8 + 12*x^6 - 13*x^4 + 2*x^3 + x^2 - 2*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

$C_4^2:D_4$ (as 16T382):

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 128
The 26 conjugacy class representatives for $C_4^2:D_4$
Character table for $C_4^2:D_4$

Intermediate fields

\(\Q(\sqrt{13}) \), 4.2.507.1, 4.2.2704.1, 4.4.8112.1, 8.4.65804544.1

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: 16.0.168110601127133184.3

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 ${\href{/padicField/5.8.0.1}{8} }^{2}$ ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.8.0.1}{8} }^{2}$ R ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.2.0.1}{2} }^{8}$ ${\href{/padicField/41.8.0.1}{8} }^{2}$ ${\href{/padicField/43.2.0.1}{2} }^{8}$ ${\href{/padicField/47.8.0.1}{8} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ ${\href{/padicField/59.8.0.1}{8} }^{2}$

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 2.8.2.16a33.1$x^{16} + 2 x^{15} + 2 x^{14} + 4 x^{12} + 4 x^{11} + 8 x^{10} + 6 x^{9} + 7 x^{8} + 6 x^{7} + 9 x^{6} + 6 x^{5} + 9 x^{4} + 4 x^{3} + 4 x^{2} + 2 x + 3$$2$$8$$16$$C_2^2 : C_8$$$[2, 2]^{8}$$
\(3\) Copy content Toggle raw display 3.2.2.2a1.1$x^{4} + 4 x^{3} + 8 x^{2} + 11 x + 4$$2$$2$$2$$C_4$$$[\ ]_{2}^{2}$$
3.4.1.0a1.1$x^{4} + 2 x^{3} + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
3.2.4.6a1.3$x^{8} + 8 x^{7} + 32 x^{6} + 80 x^{5} + 136 x^{4} + 160 x^{3} + 128 x^{2} + 67 x + 19$$4$$2$$6$$Q_8$$$[\ ]_{4}^{2}$$
\(13\) Copy content Toggle raw display 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$$$[\ ]_{2}^{4}$$
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$$$[\ ]_{2}^{4}$$

Spectrum of ring of integers

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