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Properties

Label 16.0.331...344.2
Degree $16$
Signature $[0, 8]$
Discriminant $3.315\times 10^{18}$
Root discriminant \(14.37\)
Ramified primes $2,3,7$
Class number $1$
Class group trivial
Galois group $D_8:C_2$ (as 16T45)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 38*x^14 - 124*x^13 + 304*x^12 - 588*x^11 + 944*x^10 - 1312*x^9 + 1591*x^8 - 1604*x^7 + 1208*x^6 - 552*x^5 + 52*x^4 + 64*x^3 + 14*x^2 - 40*x + 13)
 
gp: K = bnfinit(y^16 - 8*y^15 + 38*y^14 - 124*y^13 + 304*y^12 - 588*y^11 + 944*y^10 - 1312*y^9 + 1591*y^8 - 1604*y^7 + 1208*y^6 - 552*y^5 + 52*y^4 + 64*y^3 + 14*y^2 - 40*y + 13, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^15 + 38*x^14 - 124*x^13 + 304*x^12 - 588*x^11 + 944*x^10 - 1312*x^9 + 1591*x^8 - 1604*x^7 + 1208*x^6 - 552*x^5 + 52*x^4 + 64*x^3 + 14*x^2 - 40*x + 13);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^15 + 38*x^14 - 124*x^13 + 304*x^12 - 588*x^11 + 944*x^10 - 1312*x^9 + 1591*x^8 - 1604*x^7 + 1208*x^6 - 552*x^5 + 52*x^4 + 64*x^3 + 14*x^2 - 40*x + 13)
 

\( x^{16} - 8 x^{15} + 38 x^{14} - 124 x^{13} + 304 x^{12} - 588 x^{11} + 944 x^{10} - 1312 x^{9} + \cdots + 13 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(3315264163198009344\) \(\medspace = 2^{32}\cdot 3^{8}\cdot 7^{6}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(14.37\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2^{2}3^{1/2}7^{1/2}\approx 18.33030277982336$
Ramified primes:   \(2\), \(3\), \(7\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $4$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{9}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}$, $\frac{1}{9}a^{12}+\frac{1}{9}a^{11}-\frac{1}{9}a^{7}-\frac{1}{9}a^{6}+\frac{4}{9}a^{5}-\frac{1}{3}a^{3}+\frac{2}{9}a-\frac{2}{9}$, $\frac{1}{9}a^{13}-\frac{1}{9}a^{11}-\frac{1}{9}a^{8}-\frac{4}{9}a^{6}-\frac{4}{9}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{2}{9}a^{2}-\frac{4}{9}a+\frac{2}{9}$, $\frac{1}{9}a^{14}+\frac{1}{9}a^{11}-\frac{1}{9}a^{9}+\frac{4}{9}a^{7}+\frac{4}{9}a^{6}+\frac{1}{9}a^{5}+\frac{1}{3}a^{4}-\frac{1}{9}a^{3}-\frac{4}{9}a^{2}+\frac{4}{9}a-\frac{2}{9}$, $\frac{1}{621387}a^{15}+\frac{1216}{207129}a^{14}+\frac{11827}{621387}a^{13}+\frac{18470}{621387}a^{12}+\frac{7957}{69043}a^{11}-\frac{63559}{621387}a^{10}+\frac{9326}{207129}a^{9}+\frac{11191}{207129}a^{8}+\frac{2241}{5311}a^{7}-\frac{140470}{621387}a^{6}-\frac{9369}{69043}a^{5}-\frac{139219}{621387}a^{4}+\frac{138470}{621387}a^{3}-\frac{38326}{207129}a^{2}-\frac{301942}{621387}a+\frac{7813}{15933}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $3$

Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $7$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( \frac{37858}{69043} a^{15} - \frac{2374735}{621387} a^{14} + \frac{10449892}{621387} a^{13} - \frac{31203460}{621387} a^{12} + \frac{7815332}{69043} a^{11} - \frac{41699059}{207129} a^{10} + \frac{187335808}{621387} a^{9} - \frac{245570485}{621387} a^{8} + \frac{7113958}{15933} a^{7} - \frac{246473863}{621387} a^{6} + \frac{47920946}{207129} a^{5} - \frac{3400518}{69043} a^{4} - \frac{17190944}{621387} a^{3} + \frac{1524422}{207129} a^{2} + \frac{8629874}{621387} a - \frac{93697}{15933} \)  (order $12$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{125870}{621387}a^{15}-\frac{997835}{621387}a^{14}+\frac{1551416}{207129}a^{13}-\frac{14841241}{621387}a^{12}+\frac{35369461}{621387}a^{11}-\frac{66094985}{621387}a^{10}+\frac{102650543}{621387}a^{9}-\frac{138650692}{621387}a^{8}+\frac{4191515}{15933}a^{7}-\frac{17384960}{69043}a^{6}+\frac{105096265}{621387}a^{5}-\frac{35386931}{621387}a^{4}-\frac{1632691}{207129}a^{3}+\frac{3826864}{621387}a^{2}+\frac{1716155}{207129}a-\frac{232565}{47799}$, $\frac{77780}{621387}a^{15}-\frac{508591}{621387}a^{14}+\frac{2115461}{621387}a^{13}-\frac{1972933}{207129}a^{12}+\frac{12442855}{621387}a^{11}-\frac{20576948}{621387}a^{10}+\frac{29305798}{621387}a^{9}-\frac{4092925}{69043}a^{8}+\frac{332553}{5311}a^{7}-\frac{29528426}{621387}a^{6}+\frac{11920672}{621387}a^{5}+\frac{664558}{621387}a^{4}-\frac{2513647}{621387}a^{3}-\frac{539480}{621387}a^{2}+\frac{1308550}{621387}a-\frac{42794}{47799}$, $\frac{116}{5499}a^{15}-\frac{866}{5499}a^{14}+\frac{4514}{5499}a^{13}-\frac{15532}{5499}a^{12}+\frac{40889}{5499}a^{11}-\frac{83003}{5499}a^{10}+\frac{135458}{5499}a^{9}-\frac{61930}{1833}a^{8}+\frac{5731}{141}a^{7}-\frac{228275}{5499}a^{6}+\frac{163400}{5499}a^{5}-\frac{41000}{5499}a^{4}-\frac{15091}{1833}a^{3}+\frac{33065}{5499}a^{2}+\frac{8857}{5499}a-\frac{637}{423}$, $\frac{46226}{207129}a^{15}-\frac{315773}{207129}a^{14}+\frac{4101778}{621387}a^{13}-\frac{1332004}{69043}a^{12}+\frac{26376920}{621387}a^{11}-\frac{15211972}{207129}a^{10}+\frac{22229398}{207129}a^{9}-\frac{85829617}{621387}a^{8}+\frac{2429204}{15933}a^{7}-\frac{79844176}{621387}a^{6}+\frac{40528298}{621387}a^{5}-\frac{1282238}{207129}a^{4}-\frac{1857428}{207129}a^{3}-\frac{1854739}{621387}a^{2}+\frac{3109598}{621387}a-\frac{54865}{47799}$, $\frac{346154}{621387}a^{15}-\frac{791380}{207129}a^{14}+\frac{3494722}{207129}a^{13}-\frac{31193899}{621387}a^{12}+\frac{23519557}{207129}a^{11}-\frac{125479880}{621387}a^{10}+\frac{20939261}{69043}a^{9}-\frac{246925372}{621387}a^{8}+\frac{21514526}{47799}a^{7}-\frac{248998258}{621387}a^{6}+\frac{48726311}{207129}a^{5}-\frac{31649936}{621387}a^{4}-\frac{17797733}{621387}a^{3}+\frac{6301823}{621387}a^{2}+\frac{6802277}{621387}a-\frac{29457}{5311}$, $\frac{143708}{621387}a^{15}-\frac{1170446}{621387}a^{14}+\frac{1865156}{207129}a^{13}-\frac{18229684}{621387}a^{12}+\frac{44344159}{621387}a^{11}-\frac{84263633}{621387}a^{10}+\frac{131953535}{621387}a^{9}-\frac{178768483}{621387}a^{8}+\frac{5431472}{15933}a^{7}-\frac{68470477}{207129}a^{6}+\frac{139118221}{621387}a^{5}-\frac{43169645}{621387}a^{4}-\frac{1486203}{69043}a^{3}+\frac{9640270}{621387}a^{2}+\frac{2315539}{207129}a-\frac{368933}{47799}$, $\frac{96920}{621387}a^{15}-\frac{488344}{621387}a^{14}+\frac{582343}{207129}a^{13}-\frac{3831869}{621387}a^{12}+\frac{5914954}{621387}a^{11}-\frac{6335690}{621387}a^{10}+\frac{5142160}{621387}a^{9}-\frac{2241763}{621387}a^{8}-\frac{427993}{47799}a^{7}+\frac{17911778}{621387}a^{6}-\frac{24074546}{621387}a^{5}+\frac{14605126}{621387}a^{4}+\frac{486404}{621387}a^{3}-\frac{1147208}{207129}a^{2}-\frac{964477}{621387}a+\frac{113515}{47799}$ Copy content Toggle raw display
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 3545.42621346 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 3545.42621346 \cdot 1}{12\cdot\sqrt{3315264163198009344}}\cr\approx \mathstrut & 0.394155038598 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 38*x^14 - 124*x^13 + 304*x^12 - 588*x^11 + 944*x^10 - 1312*x^9 + 1591*x^8 - 1604*x^7 + 1208*x^6 - 552*x^5 + 52*x^4 + 64*x^3 + 14*x^2 - 40*x + 13)
 
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))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^16 - 8*x^15 + 38*x^14 - 124*x^13 + 304*x^12 - 588*x^11 + 944*x^10 - 1312*x^9 + 1591*x^8 - 1604*x^7 + 1208*x^6 - 552*x^5 + 52*x^4 + 64*x^3 + 14*x^2 - 40*x + 13, 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)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^16 - 8*x^15 + 38*x^14 - 124*x^13 + 304*x^12 - 588*x^11 + 944*x^10 - 1312*x^9 + 1591*x^8 - 1604*x^7 + 1208*x^6 - 552*x^5 + 52*x^4 + 64*x^3 + 14*x^2 - 40*x + 13);
 
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)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^15 + 38*x^14 - 124*x^13 + 304*x^12 - 588*x^11 + 944*x^10 - 1312*x^9 + 1591*x^8 - 1604*x^7 + 1208*x^6 - 552*x^5 + 52*x^4 + 64*x^3 + 14*x^2 - 40*x + 13);
 
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):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), 4.0.1008.2, 4.0.1008.1, \(\Q(\zeta_{12})\), 8.0.16257024.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Sibling fields

Galois closure: deg 32
Degree 8 siblings: 8.2.4248502272.1, 8.2.4248502272.2
Degree 16 siblings: 16.4.162447943996702457856.1, 16.0.162447943996702457856.17, 16.0.18049771555189161984.1
Minimal sibling: 8.2.4248502272.1

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 ${\href{/padicField/5.8.0.1}{8} }^{2}$ R ${\href{/padicField/11.8.0.1}{8} }^{2}$ ${\href{/padicField/13.2.0.1}{2} }^{6}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ ${\href{/padicField/17.8.0.1}{8} }^{2}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.8.0.1}{8} }^{2}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.8.0.1}{8} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.2.0.1}{2} }^{8}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ 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 Deg $16$$8$$2$$32$
\(3\) Copy content Toggle raw display 3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(7\) Copy content Toggle raw display 7.4.2.2$x^{4} - 42 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.0.1$x^{4} + 5 x^{2} + 4 x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
7.8.4.1$x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$