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Properties

Label 16.0.860...096.52
Degree $16$
Signature $[0, 8]$
Discriminant $8.607\times 10^{23}$
Root discriminant \(31.33\)
Ramified primes $2,3$
Class number $1$
Class group trivial
Galois group $(C_2^2\times C_4^2):D_8$ (as 16T1282)

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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 + 8*x^14 + 20*x^12 + 8*x^10 - 30*x^8 - 24*x^6 + 12*x^4 + 8*x^2 + 9)
 
Copy content gp:K = bnfinit(y^16 + 8*y^14 + 20*y^12 + 8*y^10 - 30*y^8 - 24*y^6 + 12*y^4 + 8*y^2 + 9, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 8*x^14 + 20*x^12 + 8*x^10 - 30*x^8 - 24*x^6 + 12*x^4 + 8*x^2 + 9);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 + 8*x^14 + 20*x^12 + 8*x^10 - 30*x^8 - 24*x^6 + 12*x^4 + 8*x^2 + 9)
 

\( x^{16} + 8x^{14} + 20x^{12} + 8x^{10} - 30x^{8} - 24x^{6} + 12x^{4} + 8x^{2} + 9 \) 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:   \(860651291502992840196096\) \(\medspace = 2^{70}\cdot 3^{6}\) 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:  \(31.33\)
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:  $2^{1269/256}3^{1/2}\approx 53.79918827448669$
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:  \(\Q\)
$\Aut(K/\Q)$:   $C_2$
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.
Maximal CM subfield:  \(\Q(\sqrt{-2}) \)

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{8}+\frac{1}{4}a^{2}-\frac{1}{4}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{8}+\frac{1}{4}a^{4}-\frac{1}{4}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{8}+\frac{1}{4}a^{6}-\frac{1}{4}$, $\frac{1}{12}a^{15}-\frac{1}{12}a^{13}-\frac{1}{12}a^{11}-\frac{1}{12}a^{9}-\frac{1}{4}a^{7}+\frac{1}{4}a^{5}+\frac{1}{4}a^{3}-\frac{1}{12}a$ 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:  No
Index:  Not computed
Inessential primes:  $3$

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:   \( -1 \)  (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:   $\frac{1}{12}a^{15}-\frac{2}{3}a^{13}-\frac{5}{3}a^{11}-\frac{11}{12}a^{9}+\frac{5}{4}a^{7}+a^{5}+a^{3}+\frac{1}{12}a$, $\frac{1}{12}a^{15}+\frac{5}{12}a^{13}+\frac{1}{6}a^{11}-\frac{4}{3}a^{9}-\frac{1}{4}a^{7}+\frac{7}{4}a^{5}-\frac{1}{2}a^{3}-\frac{1}{3}a$, $\frac{1}{12}a^{15}-\frac{1}{4}a^{14}+\frac{2}{3}a^{13}-\frac{5}{4}a^{12}+\frac{7}{6}a^{11}-\frac{1}{4}a^{10}-\frac{19}{12}a^{9}+\frac{19}{4}a^{8}-\frac{13}{4}a^{7}+\frac{3}{4}a^{6}+3a^{5}-\frac{25}{4}a^{4}+\frac{3}{2}a^{3}+\frac{7}{4}a^{2}-\frac{19}{12}a+\frac{3}{4}$, $\frac{1}{4}a^{14}-\frac{3}{2}a^{12}-\frac{7}{4}a^{10}+3a^{8}+\frac{15}{4}a^{6}-\frac{3}{2}a^{4}+\frac{5}{4}a^{2}-1$, $\frac{1}{12}a^{15}-\frac{1}{4}a^{14}+\frac{2}{3}a^{13}-\frac{5}{4}a^{12}+\frac{5}{3}a^{11}-\frac{1}{4}a^{10}+\frac{5}{12}a^{9}+\frac{21}{4}a^{8}-\frac{13}{4}a^{7}+\frac{11}{4}a^{6}-\frac{21}{4}a^{4}+6a^{3}-\frac{1}{4}a^{2}-\frac{31}{12}a+\frac{9}{4}$, $\frac{1}{6}a^{15}+\frac{1}{4}a^{14}+\frac{4}{3}a^{13}+\frac{3}{2}a^{12}+\frac{10}{3}a^{11}+\frac{7}{4}a^{10}+\frac{5}{6}a^{9}-\frac{7}{2}a^{8}-\frac{15}{2}a^{7}-\frac{23}{4}a^{6}-6a^{5}+\frac{3}{2}a^{4}+5a^{3}+\frac{11}{4}a^{2}+\frac{5}{6}a-\frac{3}{2}$, $\frac{2}{3}a^{15}+\frac{1}{2}a^{14}-\frac{61}{12}a^{13}+4a^{12}-\frac{151}{12}a^{11}+\frac{19}{2}a^{10}-\frac{35}{6}a^{9}+\frac{3}{2}a^{8}+19a^{7}-\frac{37}{2}a^{6}+\frac{73}{4}a^{5}-9a^{4}-\frac{37}{4}a^{3}+\frac{27}{2}a^{2}-\frac{65}{6}a+\frac{3}{2}$ 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:  \( 1788185.768620906 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 1788185.768620906 \cdot 1}{2\cdot\sqrt{860651291502992840196096}}\cr\approx \mathstrut & 2.34103535723568 \end{aligned}\]

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 + 8*x^14 + 20*x^12 + 8*x^10 - 30*x^8 - 24*x^6 + 12*x^4 + 8*x^2 + 9) 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 + 8*x^14 + 20*x^12 + 8*x^10 - 30*x^8 - 24*x^6 + 12*x^4 + 8*x^2 + 9, 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 + 8*x^14 + 20*x^12 + 8*x^10 - 30*x^8 - 24*x^6 + 12*x^4 + 8*x^2 + 9); 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 + 8*x^14 + 20*x^12 + 8*x^10 - 30*x^8 - 24*x^6 + 12*x^4 + 8*x^2 + 9); 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

$(C_2^2\times C_4^2):D_8$ (as 16T1282):

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 1024
The 43 conjugacy class representatives for $(C_2^2\times C_4^2):D_8$
Character table for $(C_2^2\times C_4^2):D_8$

Intermediate fields

\(\Q(\sqrt{-2}) \), 4.0.3072.2, 8.0.7247757312.4

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: 16.0.645488468627244630147072.35, 16.0.645488468627244630147072.48, 16.0.71720940958582736683008.18, 16.0.71720940958582736683008.23, 16.0.3442605166011971360784384.51, 16.0.645488468627244630147072.13, 16.0.645488468627244630147072.14, 16.0.71720940958582736683008.3, 16.0.71720940958582736683008.13, 16.0.7745861623526935561764864.258, 16.0.7745861623526935561764864.281, 16.0.860651291502992840196096.87, 16.0.382511685112441262309376.45, 16.0.30983446494107742247059456.113, 16.0.30983446494107742247059456.114, 16.0.860651291502992840196096.100, 16.0.382511685112441262309376.48, 16.0.3442605166011971360784384.16, 16.0.3442605166011971360784384.23, 16.0.3442605166011971360784384.134, 16.0.3442605166011971360784384.151, 16.0.3442605166011971360784384.160, 16.0.30983446494107742247059456.178, 16.0.7745861623526935561764864.160, 16.0.382511685112441262309376.59, 16.0.30983446494107742247059456.186, 16.0.7745861623526935561764864.190, 16.0.382511685112441262309376.63, 16.0.3442605166011971360784384.30, 16.0.860651291502992840196096.37, 16.0.3442605166011971360784384.45
Degree 32 siblings: deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32
Minimal sibling: 16.0.71720940958582736683008.3

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 ${\href{/padicField/5.8.0.1}{8} }^{2}$ ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.8.0.1}{8} }^{2}$ ${\href{/padicField/17.4.0.1}{4} }^{3}{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{5}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ ${\href{/padicField/29.8.0.1}{8} }^{2}$ ${\href{/padicField/31.8.0.1}{8} }^{2}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{5}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}$ ${\href{/padicField/53.8.0.1}{8} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{6}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$

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 2.1.16.70c1.2523$x^{16} + 16 x^{15} + 8 x^{14} + 16 x^{13} + 4 x^{12} + 8 x^{10} + 12 x^{8} + 16 x^{7} + 2$$16$$1$$70$16T1282$$[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{9}{2}, \frac{19}{4}, \frac{39}{8}, \frac{21}{4}]^{2}$$
\(3\) Copy content Toggle raw display $\Q_{3}$$x + 1$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$$[\ ]$$
3.2.2.2a1.2$x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
3.4.2.4a1.2$x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{4} + 8 x^{3} + 7$$2$$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)