Properties

Label 16.0.153...104.166
Degree $16$
Signature $(0, 8)$
Discriminant $1.532\times 10^{23}$
Root discriminant \(28.12\)
Ramified primes $2,3$
Class number $2$
Class group [2]
Galois group $D_8:C_2$ (as 16T38)

Related objects

Downloads

Learn more

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 + 36*x^12 - 144*x^10 + 522*x^8 - 2592*x^6 + 5292*x^4 - 3888*x^2 + 2025)
 
Copy content gp:K = bnfinit(y^16 + 36*y^12 - 144*y^10 + 522*y^8 - 2592*y^6 + 5292*y^4 - 3888*y^2 + 2025, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 36*x^12 - 144*x^10 + 522*x^8 - 2592*x^6 + 5292*x^4 - 3888*x^2 + 2025);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 + 36*x^12 - 144*x^10 + 522*x^8 - 2592*x^6 + 5292*x^4 - 3888*x^2 + 2025)
 

\( x^{16} + 36x^{12} - 144x^{10} + 522x^{8} - 2592x^{6} + 5292x^{4} - 3888x^{2} + 2025 \) 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:   \(153177439332441840943104\) \(\medspace = 2^{58}\cdot 3^{12}\) 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:  \(28.12\)
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:  \(\Q(i, \sqrt{6})\)

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3}a^{4}$, $\frac{1}{3}a^{5}$, $\frac{1}{3}a^{6}$, $\frac{1}{3}a^{7}$, $\frac{1}{72}a^{8}+\frac{1}{12}a^{4}-\frac{1}{8}$, $\frac{1}{72}a^{9}+\frac{1}{12}a^{5}-\frac{1}{8}a$, $\frac{1}{72}a^{10}+\frac{1}{12}a^{6}-\frac{1}{8}a^{2}$, $\frac{1}{360}a^{11}+\frac{1}{180}a^{9}-\frac{7}{60}a^{7}-\frac{1}{30}a^{5}-\frac{17}{40}a^{3}+\frac{7}{20}a$, $\frac{1}{1080}a^{12}-\frac{1}{360}a^{10}+\frac{1}{360}a^{8}-\frac{3}{20}a^{6}+\frac{13}{120}a^{4}-\frac{7}{40}a^{2}-\frac{3}{8}$, $\frac{1}{2160}a^{13}-\frac{1}{2160}a^{12}-\frac{1}{180}a^{10}+\frac{1}{240}a^{9}-\frac{1}{720}a^{8}-\frac{2}{15}a^{7}+\frac{1}{30}a^{6}+\frac{3}{80}a^{5}+\frac{9}{80}a^{4}+\frac{1}{5}a^{3}-\frac{7}{20}a^{2}+\frac{39}{80}a-\frac{5}{16}$, $\frac{1}{645840}a^{14}+\frac{11}{71760}a^{12}+\frac{289}{215280}a^{10}-\frac{1337}{215280}a^{8}-\frac{2203}{71760}a^{6}+\frac{77}{624}a^{4}+\frac{8361}{23920}a^{2}-\frac{725}{4784}$, $\frac{1}{645840}a^{15}+\frac{11}{71760}a^{13}+\frac{289}{215280}a^{11}-\frac{1337}{215280}a^{9}-\frac{2203}{71760}a^{7}+\frac{77}{624}a^{5}+\frac{8361}{23920}a^{3}-\frac{725}{4784}a$ 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:  $C_{2}$, which has order $2$
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:  $C_{2}$, which has order $2$
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:   \( \frac{1}{1242} a^{14} + \frac{5}{4968} a^{12} + \frac{13}{414} a^{10} - \frac{127}{1656} a^{8} + \frac{17}{46} a^{6} - \frac{41}{24} a^{4} + \frac{127}{46} a^{2} - \frac{263}{184} \)  (order $4$) 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:   $\frac{1}{1656}a^{14}+\frac{13}{24840}a^{12}+\frac{43}{2070}a^{10}-\frac{199}{2760}a^{8}+\frac{581}{2760}a^{6}-\frac{157}{120}a^{4}+\frac{757}{460}a^{2}-\frac{157}{184}$, $\frac{113}{645840}a^{14}+\frac{47}{71760}a^{12}+\frac{1561}{215280}a^{10}-\frac{77}{43056}a^{8}+\frac{4613}{71760}a^{6}-\frac{539}{3120}a^{4}+\frac{9521}{23920}a^{2}-\frac{597}{4784}$, $\frac{7}{645840}a^{14}+\frac{1291}{645840}a^{12}+\frac{95}{14352}a^{10}+\frac{18149}{215280}a^{8}-\frac{151}{4784}a^{6}+\frac{751}{1040}a^{4}-\frac{65259}{23920}a^{2}+\frac{2101}{4784}$, $\frac{67}{71760}a^{15}+\frac{283}{215280}a^{14}+\frac{1093}{645840}a^{13}+\frac{509}{215280}a^{12}+\frac{8023}{215280}a^{11}+\frac{2189}{43056}a^{10}-\frac{13861}{215280}a^{9}-\frac{21637}{215280}a^{8}+\frac{1857}{4784}a^{7}+\frac{6737}{14352}a^{6}-\frac{1097}{624}a^{5}-\frac{7859}{3120}a^{4}+\frac{41207}{23920}a^{3}+\frac{51657}{23920}a^{2}-\frac{38457}{23920}a-\frac{4369}{4784}$, $\frac{139}{64584}a^{15}-\frac{113}{80730}a^{14}-\frac{167}{161460}a^{13}-\frac{62}{40365}a^{12}-\frac{1433}{17940}a^{11}-\frac{1189}{21528}a^{10}+\frac{28627}{107640}a^{9}+\frac{1837}{13455}a^{8}-\frac{7733}{7176}a^{7}-\frac{2503}{3588}a^{6}+\frac{2003}{390}a^{5}+\frac{614}{195}a^{4}-\frac{14256}{1495}a^{3}-\frac{59911}{11960}a^{2}+\frac{90897}{11960}a+\frac{1943}{299}$, $\frac{283}{64584}a^{15}+\frac{15}{4784}a^{14}-\frac{73}{161460}a^{13}-\frac{151}{215280}a^{12}+\frac{5549}{35880}a^{11}+\frac{7901}{71760}a^{10}-\frac{69749}{107640}a^{9}-\frac{103733}{215280}a^{8}+\frac{26953}{11960}a^{7}+\frac{118409}{71760}a^{6}-\frac{2173}{195}a^{5}-\frac{26063}{3120}a^{4}+\frac{55491}{2392}a^{3}+\frac{418451}{23920}a^{2}-\frac{137271}{11960}a-\frac{62825}{4784}$, $\frac{275}{129168}a^{15}+\frac{209}{215280}a^{14}-\frac{379}{645840}a^{13}+\frac{479}{645840}a^{12}-\frac{5483}{71760}a^{11}+\frac{6587}{215280}a^{10}+\frac{13061}{43056}a^{9}-\frac{10931}{71760}a^{8}-\frac{23237}{23920}a^{7}+\frac{20431}{71760}a^{6}+\frac{3403}{624}a^{5}-\frac{2033}{1040}a^{4}-\frac{276101}{23920}a^{3}+\frac{28999}{4784}a^{2}+\frac{135193}{23920}a-\frac{22221}{4784}$ 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:  \( 322588.2352814882 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 322588.2352814882 \cdot 2}{4\cdot\sqrt{153177439332441840943104}}\cr\approx \mathstrut & 1.00105991935464 \end{aligned}\]

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 + 36*x^12 - 144*x^10 + 522*x^8 - 2592*x^6 + 5292*x^4 - 3888*x^2 + 2025) 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 + 36*x^12 - 144*x^10 + 522*x^8 - 2592*x^6 + 5292*x^4 - 3888*x^2 + 2025, 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 + 36*x^12 - 144*x^10 + 522*x^8 - 2592*x^6 + 5292*x^4 - 3888*x^2 + 2025); 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 + 36*x^12 - 144*x^10 + 522*x^8 - 2592*x^6 + 5292*x^4 - 3888*x^2 + 2025); 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

\(\Q(\sqrt{-6}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{1 + i})\), \(\Q(\sqrt{3 +3 i})\), \(\Q(i, \sqrt{6})\), 8.0.97844723712.6, 8.0.97844723712.2, 8.0.339738624.7

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: 8.0.97844723712.2, 8.0.97844723712.6
Degree 16 siblings: 16.0.9573589958277615058944.70, 16.0.153177439332441840943104.169, 16.4.153177439332441840943104.88
Minimal sibling: 8.0.97844723712.2

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 ${\href{/padicField/5.2.0.1}{2} }^{6}{,}\,{\href{/padicField/5.1.0.1}{1} }^{4}$ ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.8.0.1}{8} }^{2}$ ${\href{/padicField/13.2.0.1}{2} }^{8}$ ${\href{/padicField/17.2.0.1}{2} }^{8}$ ${\href{/padicField/19.8.0.1}{8} }^{2}$ ${\href{/padicField/23.2.0.1}{2} }^{8}$ ${\href{/padicField/29.2.0.1}{2} }^{6}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.2.0.1}{2} }^{8}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.2.0.1}{2} }^{6}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ ${\href{/padicField/59.8.0.1}{8} }^{2}$

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 2.1.16.58m1.769$x^{16} + 8 x^{15} + 4 x^{14} + 4 x^{12} + 8 x^{11} + 26 x^{8} + 16 x^{7} + 8 x^{6} + 16 x^{5} + 4 x^{4} + 8 x^{2} + 10$$16$$1$$58$16T38$$[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)