Properties

Label 16.0.957...944.78
Degree $16$
Signature $(0, 8)$
Discriminant $9.574\times 10^{21}$
Root discriminant \(23.65\)
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 - 12*x^14 + 90*x^12 - 432*x^10 + 1278*x^8 - 2160*x^6 + 1620*x^4 + 324)
 
Copy content gp:K = bnfinit(y^16 - 12*y^14 + 90*y^12 - 432*y^10 + 1278*y^8 - 2160*y^6 + 1620*y^4 + 324, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 12*x^14 + 90*x^12 - 432*x^10 + 1278*x^8 - 2160*x^6 + 1620*x^4 + 324);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 12*x^14 + 90*x^12 - 432*x^10 + 1278*x^8 - 2160*x^6 + 1620*x^4 + 324)
 

\( x^{16} - 12x^{14} + 90x^{12} - 432x^{10} + 1278x^{8} - 2160x^{6} + 1620x^{4} + 324 \) 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:   \(9573589958277615058944\) \(\medspace = 2^{54}\cdot 3^{12}\) 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:  \(23.65\)
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:  \(\Q(\zeta_{12})\)

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3}a^{4}$, $\frac{1}{3}a^{5}$, $\frac{1}{3}a^{6}$, $\frac{1}{3}a^{7}$, $\frac{1}{18}a^{8}$, $\frac{1}{18}a^{9}$, $\frac{1}{54}a^{10}+\frac{1}{3}a^{2}$, $\frac{1}{54}a^{11}+\frac{1}{3}a^{3}$, $\frac{1}{162}a^{12}+\frac{1}{54}a^{8}-\frac{1}{9}a^{6}+\frac{1}{9}a^{4}-\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{162}a^{13}+\frac{1}{54}a^{9}-\frac{1}{9}a^{7}+\frac{1}{9}a^{5}-\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{298566}a^{14}-\frac{433}{298566}a^{12}-\frac{71}{11058}a^{10}+\frac{820}{49761}a^{8}+\frac{1721}{16587}a^{6}+\frac{1481}{16587}a^{4}+\frac{1684}{5529}a^{2}-\frac{1252}{5529}$, $\frac{1}{298566}a^{15}-\frac{433}{298566}a^{13}-\frac{71}{11058}a^{11}+\frac{820}{49761}a^{9}+\frac{1721}{16587}a^{7}+\frac{1481}{16587}a^{5}+\frac{1684}{5529}a^{3}-\frac{1252}{5529}a$ 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:   \( -\frac{44}{149283} a^{14} + \frac{622}{149283} a^{12} - \frac{1372}{49761} a^{10} + \frac{12335}{99522} a^{8} - \frac{1336}{5529} a^{6} - \frac{1318}{16587} a^{4} + \frac{6620}{5529} a^{2} - \frac{4090}{5529} \)  (order $12$) 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:   $\frac{44}{149283}a^{14}+\frac{622}{149283}a^{12}-\frac{1372}{49761}a^{10}+\frac{12335}{99522}a^{8}-\frac{1336}{5529}a^{6}-\frac{1318}{16587}a^{4}+\frac{6620}{5529}a^{2}-\frac{9619}{5529}$, $\frac{241}{99522}a^{14}-\frac{7121}{298566}a^{12}+\frac{17183}{99522}a^{10}-\frac{73049}{99522}a^{8}+\frac{31589}{16587}a^{6}-\frac{44252}{16587}a^{4}+\frac{8524}{5529}a^{2}-\frac{2126}{5529}$, $\frac{256}{149283}a^{14}+\frac{6065}{298566}a^{12}-\frac{13787}{99522}a^{10}+\frac{61547}{99522}a^{8}-\frac{24157}{16587}a^{6}+\frac{21317}{16587}a^{4}+\frac{2159}{5529}a^{2}-\frac{340}{5529}$, $\frac{8}{149283}a^{15}+\frac{676}{149283}a^{14}-\frac{74}{49761}a^{13}-\frac{7043}{149283}a^{12}+\frac{1009}{99522}a^{11}+\frac{33613}{99522}a^{10}-\frac{221}{3686}a^{9}-\frac{147289}{99522}a^{8}+\frac{1952}{16587}a^{7}+\frac{21196}{5529}a^{6}+\frac{2545}{5529}a^{5}-\frac{91336}{16587}a^{4}-\frac{12200}{5529}a^{3}+\frac{15407}{5529}a^{2}+\frac{6063}{1843}a+\frac{6542}{5529}$, $\frac{5}{7857}a^{15}+\frac{86}{49761}a^{14}-\frac{53}{5238}a^{13}-\frac{2977}{149283}a^{12}+\frac{200}{2619}a^{11}+\frac{14917}{99522}a^{10}-\frac{709}{1746}a^{9}-\frac{69731}{99522}a^{8}+\frac{1108}{873}a^{7}+\frac{34727}{16587}a^{6}-\frac{219}{97}a^{5}-\frac{57782}{16587}a^{4}+\frac{544}{291}a^{3}+\frac{5826}{1843}a^{2}-\frac{67}{97}a-\frac{8354}{5529}$, $\frac{226}{149283}a^{15}-\frac{155}{49761}a^{14}-\frac{674}{49761}a^{13}+\frac{5987}{149283}a^{12}+\frac{541}{5529}a^{11}-\frac{14332}{49761}a^{10}-\frac{6692}{16587}a^{9}+\frac{135343}{99522}a^{8}+\frac{16733}{16587}a^{7}-\frac{59782}{16587}a^{6}-\frac{9695}{5529}a^{5}+\frac{73111}{16587}a^{4}+\frac{1846}{1843}a^{3}-\frac{471}{1843}a^{2}+\frac{580}{1843}a-\frac{416}{5529}$, $\frac{127}{49761}a^{15}+\frac{925}{298566}a^{14}-\frac{1544}{49761}a^{13}-\frac{13495}{298566}a^{12}+\frac{12718}{49761}a^{11}+\frac{5617}{16587}a^{10}-\frac{2455}{1843}a^{9}-\frac{169345}{99522}a^{8}+\frac{8715}{1843}a^{7}+\frac{82508}{16587}a^{6}-\frac{60617}{5529}a^{5}-\frac{117376}{16587}a^{4}+\frac{81572}{5529}a^{3}+\frac{4051}{5529}a^{2}-\frac{12070}{1843}a+\frac{25106}{5529}$ 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:  \( 188559.9528595578 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 188559.9528595578 \cdot 1}{12\cdot\sqrt{9573589958277615058944}}\cr\approx \mathstrut & 0.390094431142523 \end{aligned}\]

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 - 12*x^14 + 90*x^12 - 432*x^10 + 1278*x^8 - 2160*x^6 + 1620*x^4 + 324) 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 - 12*x^14 + 90*x^12 - 432*x^10 + 1278*x^8 - 2160*x^6 + 1620*x^4 + 324, 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 - 12*x^14 + 90*x^12 - 432*x^10 + 1278*x^8 - 2160*x^6 + 1620*x^4 + 324); 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 - 12*x^14 + 90*x^12 - 432*x^10 + 1278*x^8 - 2160*x^6 + 1620*x^4 + 324); 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

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-1 + \sqrt{3}})\), \(\Q(\sqrt{1 + \sqrt{3}})\), \(\Q(\zeta_{12})\), 8.2.97844723712.2, 8.2.97844723712.3, 8.0.21233664.1

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: 16.0.38294359833110460235776.118, 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 ${\href{/padicField/5.8.0.1}{8} }^{2}$ ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.2.0.1}{2} }^{6}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.2.0.1}{2} }^{8}$ ${\href{/padicField/29.8.0.1}{8} }^{2}$ ${\href{/padicField/31.8.0.1}{8} }^{2}$ ${\href{/padicField/37.2.0.1}{2} }^{6}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.2.0.1}{2} }^{8}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.8.0.1}{8} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

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 2.2.8.54a1.584$x^{16} + 8 x^{15} + 48 x^{14} + 200 x^{13} + 626 x^{12} + 1520 x^{11} + 2956 x^{10} + 4704 x^{9} + 6201 x^{8} + 6840 x^{7} + 6304 x^{6} + 4848 x^{5} + 3044 x^{4} + 1528 x^{3} + 584 x^{2} + 160 x + 33$$8$$2$$54$16T38$$[2, 3, \frac{7}{2}, \frac{9}{2}]^{2}$$
\(3\) Copy content Toggle raw display 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$$$[\ ]_{4}^{2}$$
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)