Properties

Label 16.0.382...776.118
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 16T45)

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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 + 18*x^12 + 297*x^8 - 1296*x^6 + 2214*x^4 - 1944*x^2 + 729)
 
Copy content gp:K = bnfinit(y^16 + 18*y^12 + 297*y^8 - 1296*y^6 + 2214*y^4 - 1944*y^2 + 729, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 18*x^12 + 297*x^8 - 1296*x^6 + 2214*x^4 - 1944*x^2 + 729);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 + 18*x^12 + 297*x^8 - 1296*x^6 + 2214*x^4 - 1944*x^2 + 729)
 

\( x^{16} + 18x^{12} + 297x^{8} - 1296x^{6} + 2214x^{4} - 1944x^{2} + 729 \) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_4$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3}a^{4}$, $\frac{1}{3}a^{5}$, $\frac{1}{9}a^{6}+\frac{1}{3}a^{2}$, $\frac{1}{9}a^{7}+\frac{1}{3}a^{3}$, $\frac{1}{9}a^{8}$, $\frac{1}{9}a^{9}$, $\frac{1}{27}a^{10}-\frac{1}{3}a^{2}$, $\frac{1}{27}a^{11}-\frac{1}{3}a^{3}$, $\frac{1}{81}a^{12}-\frac{1}{27}a^{8}+\frac{1}{9}a^{4}$, $\frac{1}{81}a^{13}-\frac{1}{27}a^{9}+\frac{1}{9}a^{5}$, $\frac{1}{832437}a^{14}+\frac{3580}{832437}a^{12}+\frac{37}{30831}a^{10}+\frac{8}{277479}a^{8}-\frac{4123}{92493}a^{6}-\frac{12989}{92493}a^{4}+\frac{339}{10277}a^{2}+\frac{910}{10277}$, $\frac{1}{2497311}a^{15}+\frac{4619}{832437}a^{13}+\frac{37}{92493}a^{11}-\frac{13700}{277479}a^{9}-\frac{1600}{30831}a^{7}-\frac{904}{92493}a^{5}-\frac{9260}{92493}a^{3}+\frac{3729}{10277}a$ 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:   \( \frac{7552}{832437} a^{14} + \frac{850}{92493} a^{12} + \frac{16112}{92493} a^{10} + \frac{1857}{10277} a^{8} + \frac{89872}{30831} a^{6} - \frac{90094}{10277} a^{4} + \frac{363160}{30831} a^{2} - \frac{64655}{10277} \)  (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:   $\frac{475}{92493}a^{14}+\frac{2047}{832437}a^{12}+\frac{25903}{277479}a^{10}+\frac{13646}{277479}a^{8}+\frac{143108}{92493}a^{6}-\frac{535748}{92493}a^{4}+\frac{267706}{30831}a^{2}-\frac{45841}{10277}$, $\frac{5069}{277479}a^{14}-\frac{2705}{92493}a^{12}-\frac{33380}{92493}a^{10}-\frac{16943}{30831}a^{8}-\frac{556474}{92493}a^{6}+\frac{448690}{30831}a^{4}-\frac{378883}{30831}a^{2}+\frac{56134}{10277}$, $\frac{24092}{832437}a^{14}-\frac{11869}{277479}a^{12}-\frac{5953}{10277}a^{10}-\frac{77948}{92493}a^{8}-\frac{900265}{92493}a^{6}+\frac{721595}{30831}a^{4}-\frac{284729}{10277}a^{2}+\frac{140999}{10277}$, $\frac{215225}{2497311}a^{15}+\frac{80434}{832437}a^{14}-\frac{91727}{832437}a^{13}+\frac{105227}{832437}a^{12}-\frac{468692}{277479}a^{11}+\frac{526787}{277479}a^{10}-\frac{597913}{277479}a^{9}+\frac{684580}{277479}a^{8}-\frac{2615506}{92493}a^{7}+\frac{979451}{30831}a^{6}+\frac{6997873}{92493}a^{5}-\frac{7766818}{92493}a^{4}-\frac{8595851}{92493}a^{3}+\frac{3130943}{30831}a^{2}+\frac{481032}{10277}a-\frac{532258}{10277}$, $\frac{3266}{277479}a^{15}+\frac{85}{30831}a^{14}-\frac{1439}{832437}a^{13}-\frac{1753}{92493}a^{12}-\frac{18770}{92493}a^{11}+\frac{3737}{277479}a^{10}-\frac{6445}{277479}a^{9}-\frac{34988}{92493}a^{8}-\frac{310043}{92493}a^{7}+\frac{13109}{92493}a^{6}+\frac{1372972}{92493}a^{5}-\frac{307036}{30831}a^{4}-\frac{663881}{30831}a^{3}+\frac{202493}{10277}a^{2}+\frac{86472}{10277}a-\frac{121105}{10277}$, $\frac{17236}{832437}a^{15}-\frac{95}{3483}a^{14}+\frac{32603}{832437}a^{13}-\frac{56}{3483}a^{12}+\frac{118069}{277479}a^{11}-\frac{589}{1161}a^{10}+\frac{209827}{277479}a^{9}-\frac{373}{1161}a^{8}+\frac{659155}{92493}a^{7}-\frac{1090}{129}a^{6}-\frac{1319692}{92493}a^{5}+\frac{11551}{387}a^{4}+\frac{345868}{30831}a^{3}-\frac{5842}{129}a^{2}-\frac{69881}{10277}a+\frac{1141}{43}$, $\frac{20075}{832437}a^{15}-\frac{20044}{832437}a^{14}-\frac{7331}{277479}a^{13}-\frac{8020}{277479}a^{12}-\frac{128249}{277479}a^{11}-\frac{128203}{277479}a^{10}-\frac{46682}{92493}a^{9}-\frac{16675}{30831}a^{8}-\frac{710846}{92493}a^{7}-\frac{238445}{30831}a^{6}+\frac{707731}{30831}a^{5}+\frac{226569}{10277}a^{4}-\frac{848867}{30831}a^{3}-\frac{258744}{10277}a^{2}+\frac{158411}{10277}a+\frac{145513}{10277}$ 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:  \( 328226.7478372471 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 328226.7478372471 \cdot 1}{6\cdot\sqrt{38294359833110460235776}}\cr\approx \mathstrut & 0.679038282209887 \end{aligned}\]

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 + 297*x^8 - 1296*x^6 + 2214*x^4 - 1944*x^2 + 729) 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 + 297*x^8 - 1296*x^6 + 2214*x^4 - 1944*x^2 + 729, 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 + 297*x^8 - 1296*x^6 + 2214*x^4 - 1944*x^2 + 729); 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 + 297*x^8 - 1296*x^6 + 2214*x^4 - 1944*x^2 + 729); 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 16T45):

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

\(\Q(\sqrt{6}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{1 + \sqrt{-2}})\), \(\Q(\sqrt{-1 + \sqrt{-2}})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), 8.0.84934656.1

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

Galois closure: deg 32
Degree 8 siblings: 8.2.97844723712.2, 8.2.97844723712.3
Degree 16 siblings: 16.0.9573589958277615058944.78, 16.0.153177439332441840943104.158, 16.4.153177439332441840943104.86
Minimal sibling: 8.2.97844723712.2

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 ${\href{/padicField/5.8.0.1}{8} }^{2}$ ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.2.0.1}{2} }^{8}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.2.0.1}{2} }^{6}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ ${\href{/padicField/23.2.0.1}{2} }^{8}$ ${\href{/padicField/29.8.0.1}{8} }^{2}$ ${\href{/padicField/31.8.0.1}{8} }^{2}$ ${\href{/padicField/37.2.0.1}{2} }^{8}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.2.0.1}{2} }^{6}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.8.0.1}{8} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

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 2.2.8.56a1.846$x^{16} + 8 x^{15} + 44 x^{14} + 168 x^{13} + 494 x^{12} + 1144 x^{11} + 2164 x^{10} + 3400 x^{9} + 4511 x^{8} + 5080 x^{7} + 4876 x^{6} + 3960 x^{5} + 2710 x^{4} + 1520 x^{3} + 688 x^{2} + 224 x + 51$$8$$2$$56$16T45$$[2, 3, \frac{7}{2}, \frac{9}{2}]^{2}$$
\(3\) Copy content Toggle raw display 3.1.4.3a1.1$x^{4} + 3$$4$$1$$3$$D_{4}$$$[\ ]_{4}^{2}$$
3.1.4.3a1.1$x^{4} + 3$$4$$1$$3$$D_{4}$$$[\ ]_{4}^{2}$$
3.2.4.6a1.2$x^{8} + 8 x^{7} + 32 x^{6} + 80 x^{5} + 136 x^{4} + 160 x^{3} + 128 x^{2} + 64 x + 19$$4$$2$$6$$D_4$$$[\ ]_{4}^{2}$$

Spectrum of ring of integers

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