Properties

Label 16.0.382...776.74
Degree $16$
Signature $(0, 8)$
Discriminant $3.829\times 10^{22}$
Root discriminant \(25.79\)
Ramified primes $2,3$
Class number $1$
Class group trivial
Galois group $D_8:C_2$ (as 16T38)

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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 - 18*x^12 + 333*x^8 - 1296*x^6 + 1890*x^4 - 648*x^2 + 81)
 
Copy content gp:K = bnfinit(y^16 - 18*y^12 + 333*y^8 - 1296*y^6 + 1890*y^4 - 648*y^2 + 81, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 18*x^12 + 333*x^8 - 1296*x^6 + 1890*x^4 - 648*x^2 + 81);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 18*x^12 + 333*x^8 - 1296*x^6 + 1890*x^4 - 648*x^2 + 81)
 

\( x^{16} - 18x^{12} + 333x^{8} - 1296x^{6} + 1890x^{4} - 648x^{2} + 81 \) 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:  $(0, 8)$
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:   \(38294359833110460235776\) \(\medspace = 2^{56}\cdot 3^{12}\) 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:  \(25.79\)
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:  $2^{29/8}3^{3/4}\approx 28.12384367863563$
Ramified primes:   \(2\), \(3\) 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: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.
Maximal CM subfield:  \(\Q(\sqrt{2}, \sqrt{-3})\)

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3}a^{4}$, $\frac{1}{3}a^{5}$, $\frac{1}{3}a^{6}$, $\frac{1}{3}a^{7}$, $\frac{1}{9}a^{8}$, $\frac{1}{9}a^{9}$, $\frac{1}{9}a^{10}$, $\frac{1}{9}a^{11}$, $\frac{1}{135}a^{12}-\frac{2}{45}a^{10}+\frac{2}{45}a^{8}-\frac{1}{15}a^{6}+\frac{1}{5}a^{2}-\frac{1}{5}$, $\frac{1}{135}a^{13}-\frac{2}{45}a^{11}+\frac{2}{45}a^{9}-\frac{1}{15}a^{7}+\frac{1}{5}a^{3}-\frac{1}{5}a$, $\frac{1}{1066905}a^{14}+\frac{1891}{1066905}a^{12}+\frac{11773}{355635}a^{10}-\frac{15814}{355635}a^{8}+\frac{263}{118545}a^{6}+\frac{343}{5645}a^{4}-\frac{552}{5645}a^{2}+\frac{19233}{39515}$, $\frac{1}{1066905}a^{15}+\frac{1891}{1066905}a^{13}+\frac{11773}{355635}a^{11}-\frac{15814}{355635}a^{9}+\frac{263}{118545}a^{7}+\frac{343}{5645}a^{5}-\frac{552}{5645}a^{3}+\frac{19233}{39515}a$ 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:  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:  $7$
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:   \( -\frac{1664}{213381} a^{14} - \frac{410}{71127} a^{12} + \frac{3056}{23709} a^{10} + \frac{601}{7903} a^{8} - \frac{58288}{23709} a^{6} + \frac{9554}{1129} a^{4} - \frac{11768}{1129} a^{2} + \frac{19244}{7903} \)  (order $6$) 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:   $\frac{1108}{118545}a^{14}+\frac{494}{1066905}a^{12}-\frac{61933}{355635}a^{10}-\frac{961}{39515}a^{8}+\frac{378184}{118545}a^{6}-\frac{198997}{16935}a^{4}+\frac{88522}{5645}a^{2}-\frac{85461}{39515}$, $\frac{892}{1066905}a^{14}+\frac{3433}{1066905}a^{12}-\frac{1571}{355635}a^{10}-\frac{7136}{355635}a^{8}+\frac{4243}{23709}a^{6}-\frac{7912}{16935}a^{4}+\frac{989}{5645}a^{2}-\frac{1896}{7903}$, $\frac{2123}{1066905}a^{14}-\frac{131}{1066905}a^{12}-\frac{3671}{118545}a^{10}+\frac{1345}{71127}a^{8}+\frac{68363}{118545}a^{6}-\frac{16951}{5645}a^{4}+\frac{27102}{5645}a^{2}-\frac{42657}{39515}$, $\frac{50}{3387}a^{15}+\frac{10817}{1066905}a^{14}-\frac{226}{50805}a^{13}+\frac{9886}{1066905}a^{12}+\frac{12896}{50805}a^{11}-\frac{18574}{118545}a^{10}+\frac{2194}{50805}a^{9}-\frac{2081}{23709}a^{8}-\frac{81442}{16935}a^{7}+\frac{123744}{39515}a^{6}+\frac{20325}{1129}a^{5}-\frac{60629}{5645}a^{4}-\frac{142683}{5645}a^{3}+\frac{78198}{5645}a^{2}+\frac{42093}{5645}a-\frac{145368}{39515}$, $\frac{28}{1129}a^{15}-\frac{5081}{213381}a^{14}-\frac{281}{152415}a^{13}-\frac{7409}{355635}a^{12}+\frac{2618}{5645}a^{11}+\frac{138224}{355635}a^{10}+\frac{4273}{50805}a^{9}+\frac{100501}{355635}a^{8}-\frac{142874}{16935}a^{7}-\frac{293571}{39515}a^{6}+\frac{104696}{3387}a^{5}+\frac{84725}{3387}a^{4}-\frac{228011}{5645}a^{3}-\frac{161196}{5645}a^{2}+\frac{17021}{5645}a+\frac{138767}{39515}$, $\frac{28234}{355635}a^{15}-\frac{2414}{118545}a^{14}-\frac{2885}{213381}a^{13}+\frac{43346}{1066905}a^{12}-\frac{106679}{71127}a^{11}+\frac{174263}{355635}a^{10}+\frac{17848}{355635}a^{9}-\frac{142312}{355635}a^{8}+\frac{3222493}{118545}a^{7}-\frac{192847}{23709}a^{6}-\frac{594754}{5645}a^{5}+\frac{613451}{16935}a^{4}+\frac{168853}{1129}a^{3}-\frac{337057}{5645}a^{2}-\frac{1323237}{39515}a+\frac{123119}{7903}$, $\frac{613}{355635}a^{15}+\frac{15374}{1066905}a^{14}-\frac{4696}{213381}a^{13}+\frac{8903}{213381}a^{12}-\frac{7057}{71127}a^{11}-\frac{10345}{71127}a^{10}+\frac{72221}{355635}a^{9}-\frac{154604}{355635}a^{8}+\frac{50577}{39515}a^{7}+\frac{143897}{39515}a^{6}-\frac{128579}{16935}a^{5}-\frac{138569}{16935}a^{4}+\frac{16452}{1129}a^{3}+\frac{2307}{1129}a^{2}-\frac{51759}{39515}a-\frac{18409}{39515}$ 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:  \( 212526.1837160089 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 212526.1837160089 \cdot 1}{6\cdot\sqrt{38294359833110460235776}}\cr\approx \mathstrut & 0.439675972985297 \end{aligned}\]

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 - 18*x^12 + 333*x^8 - 1296*x^6 + 1890*x^4 - 648*x^2 + 81) 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 - 18*x^12 + 333*x^8 - 1296*x^6 + 1890*x^4 - 648*x^2 + 81, 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 - 18*x^12 + 333*x^8 - 1296*x^6 + 1890*x^4 - 648*x^2 + 81); 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 - 18*x^12 + 333*x^8 - 1296*x^6 + 1890*x^4 - 648*x^2 + 81); 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

$D_8:C_2$ (as 16T38):

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 32
The 11 conjugacy class representatives for $D_8:C_2$
Character table for $D_8:C_2$

Intermediate fields

\(\Q(\sqrt{-6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{3 +3 \sqrt{2}})\), \(\Q(\sqrt{1 + \sqrt{2}})\), \(\Q(\sqrt{2}, \sqrt{-3})\), 8.2.195689447424.4, 8.2.195689447424.3, 8.0.84934656.2

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

Galois closure: deg 32
Degree 8 siblings: 8.2.195689447424.3, 8.2.195689447424.4
Degree 16 siblings: 16.0.9573589958277615058944.66, 16.4.153177439332441840943104.85, 16.0.153177439332441840943104.89
Minimal sibling: 8.2.195689447424.3

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 ${\href{/padicField/5.4.0.1}{4} }^{4}$ ${\href{/padicField/7.2.0.1}{2} }^{6}{,}\,{\href{/padicField/7.1.0.1}{1} }^{4}$ ${\href{/padicField/11.8.0.1}{8} }^{2}$ ${\href{/padicField/13.2.0.1}{2} }^{8}$ ${\href{/padicField/17.2.0.1}{2} }^{8}$ ${\href{/padicField/19.8.0.1}{8} }^{2}$ ${\href{/padicField/23.2.0.1}{2} }^{8}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.2.0.1}{2} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ ${\href{/padicField/37.2.0.1}{2} }^{8}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.8.0.1}{8} }^{2}$

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 2.2.8.56a1.32$x^{16} + 8 x^{15} + 36 x^{14} + 112 x^{13} + 270 x^{12} + 528 x^{11} + 876 x^{10} + 1256 x^{9} + 1587 x^{8} + 1760 x^{7} + 1724 x^{6} + 1464 x^{5} + 1090 x^{4} + 680 x^{3} + 376 x^{2} + 152 x + 55$$8$$2$$56$16T38$$[2, 3, \frac{7}{2}, \frac{9}{2}]^{2}$$
\(3\) 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$$$[\ ]_{4}^{4}$$

Spectrum of ring of integers

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