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Properties

Label 16.8.463...000.2
Degree $16$
Signature $[8, 4]$
Discriminant $4.635\times 10^{18}$
Root discriminant \(14.68\)
Ramified primes $2,5,29$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^6:D_4$ (as 16T969)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 8*x^14 + 26*x^13 - 19*x^12 - 4*x^11 + 4*x^10 - 16*x^9 + 53*x^8 - 56*x^7 + 6*x^6 + 54*x^5 - 53*x^4 + 6*x^3 + 16*x^2 - 8*x + 1)
 
gp: K = bnfinit(y^16 - 2*y^15 - 8*y^14 + 26*y^13 - 19*y^12 - 4*y^11 + 4*y^10 - 16*y^9 + 53*y^8 - 56*y^7 + 6*y^6 + 54*y^5 - 53*y^4 + 6*y^3 + 16*y^2 - 8*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 - 8*x^14 + 26*x^13 - 19*x^12 - 4*x^11 + 4*x^10 - 16*x^9 + 53*x^8 - 56*x^7 + 6*x^6 + 54*x^5 - 53*x^4 + 6*x^3 + 16*x^2 - 8*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 2*x^15 - 8*x^14 + 26*x^13 - 19*x^12 - 4*x^11 + 4*x^10 - 16*x^9 + 53*x^8 - 56*x^7 + 6*x^6 + 54*x^5 - 53*x^4 + 6*x^3 + 16*x^2 - 8*x + 1)
 

\( x^{16} - 2 x^{15} - 8 x^{14} + 26 x^{13} - 19 x^{12} - 4 x^{11} + 4 x^{10} - 16 x^{9} + 53 x^{8} + \cdots + 1 \) 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:  $[8, 4]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(4635236761600000000\) \(\medspace = 2^{24}\cdot 5^{8}\cdot 29^{4}\) 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:  \(14.68\)
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:  $2^{15/8}5^{1/2}29^{1/2}\approx 44.16876366153594$
Ramified primes:   \(2\), \(5\), \(29\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{488525633}a^{15}-\frac{233924844}{488525633}a^{14}-\frac{201724859}{488525633}a^{13}-\frac{165534725}{488525633}a^{12}+\frac{46132093}{488525633}a^{11}+\frac{243676117}{488525633}a^{10}+\frac{127657761}{488525633}a^{9}+\frac{236196375}{488525633}a^{8}+\frac{48363370}{488525633}a^{7}+\frac{169328324}{488525633}a^{6}-\frac{21435671}{488525633}a^{5}-\frac{86489123}{488525633}a^{4}-\frac{18142882}{488525633}a^{3}-\frac{28143547}{488525633}a^{2}-\frac{212447606}{488525633}a-\frac{54943864}{488525633}$ Copy content Toggle raw display

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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:  $11$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) 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:   $\frac{1187130660}{488525633}a^{15}-\frac{1883960630}{488525633}a^{14}-\frac{10227734176}{488525633}a^{13}+\frac{26513267611}{488525633}a^{12}-\frac{11867792429}{488525633}a^{11}-\frac{8228726863}{488525633}a^{10}-\frac{757926955}{488525633}a^{9}-\frac{17898485169}{488525633}a^{8}+\frac{54661579967}{488525633}a^{7}-\frac{43412585943}{488525633}a^{6}-\frac{8921553243}{488525633}a^{5}+\frac{56058239103}{488525633}a^{4}-\frac{36572370495}{488525633}a^{3}-\frac{8084426311}{488525633}a^{2}+\frac{13851431998}{488525633}a-\frac{2653019536}{488525633}$, $\frac{574099201}{488525633}a^{15}-\frac{685896729}{488525633}a^{14}-\frac{5243863321}{488525633}a^{13}+\frac{10920048019}{488525633}a^{12}-\frac{1373918135}{488525633}a^{11}-\frac{6052563002}{488525633}a^{10}-\frac{188297043}{488525633}a^{9}-\frac{9932968390}{488525633}a^{8}+\frac{23096894333}{488525633}a^{7}-\frac{12706121048}{488525633}a^{6}-\frac{11780206994}{488525633}a^{5}+\frac{27878635537}{488525633}a^{4}-\frac{10822561380}{488525633}a^{3}-\frac{8113299764}{488525633}a^{2}+\frac{6290421118}{488525633}a-\frac{317697569}{488525633}$, $\frac{590022463}{488525633}a^{15}-\frac{527051747}{488525633}a^{14}-\frac{5434428621}{488525633}a^{13}+\frac{9412716128}{488525633}a^{12}+\frac{461468781}{488525633}a^{11}-\frac{3704108747}{488525633}a^{10}-\frac{2457666378}{488525633}a^{9}-\frac{10834952203}{488525633}a^{8}+\frac{19274528036}{488525633}a^{7}-\frac{8479881504}{488525633}a^{6}-\frac{10099690659}{488525633}a^{5}+\frac{21495395658}{488525633}a^{4}-\frac{3816918355}{488525633}a^{3}-\frac{6455395821}{488525633}a^{2}+\frac{3233171705}{488525633}a+\frac{40908773}{488525633}$, $\frac{372198470}{488525633}a^{15}-\frac{315475256}{488525633}a^{14}-\frac{3475767850}{488525633}a^{13}+\frac{5723645384}{488525633}a^{12}+\frac{960089100}{488525633}a^{11}-\frac{2095029462}{488525633}a^{10}-\frac{3186984714}{488525633}a^{9}-\frac{6208518624}{488525633}a^{8}+\frac{11649919472}{488525633}a^{7}-\frac{3007240232}{488525633}a^{6}-\frac{6656334510}{488525633}a^{5}+\frac{11087823205}{488525633}a^{4}+\frac{604516853}{488525633}a^{3}-\frac{5809320632}{488525633}a^{2}+\frac{1022608304}{488525633}a+\frac{894532488}{488525633}$, $a$, $\frac{1650765199}{488525633}a^{15}-\frac{2268454923}{488525633}a^{14}-\frac{14633444535}{488525633}a^{13}+\frac{33760619868}{488525633}a^{12}-\frac{10142180038}{488525633}a^{11}-\frac{13018326209}{488525633}a^{10}-\frac{1851380838}{488525633}a^{9}-\frac{27196436296}{488525633}a^{8}+\frac{70642527548}{488525633}a^{7}-\frac{48415540404}{488525633}a^{6}-\frac{20293153035}{488525633}a^{5}+\frac{75622189134}{488525633}a^{4}-\frac{38860004750}{488525633}a^{3}-\frac{15023983270}{488525633}a^{2}+\frac{16582788132}{488525633}a-\frac{2134761197}{488525633}$, $\frac{1084981418}{488525633}a^{15}-\frac{1389334355}{488525633}a^{14}-\frac{9788330911}{488525633}a^{13}+\frac{21442178469}{488525633}a^{12}-\frac{4556711534}{488525633}a^{11}-\frac{10621227790}{488525633}a^{10}+\frac{1071534997}{488525633}a^{9}-\frac{20497137468}{488525633}a^{8}+\frac{46185251621}{488525633}a^{7}-\frac{29133947142}{488525633}a^{6}-\frac{17520144077}{488525633}a^{5}+\frac{53980869409}{488525633}a^{4}-\frac{26258249484}{488525633}a^{3}-\frac{10470622825}{488525633}a^{2}+\frac{11935819370}{488525633}a-\frac{2226049716}{488525633}$, $\frac{947630537}{488525633}a^{15}-\frac{1223322198}{488525633}a^{14}-\frac{8630284690}{488525633}a^{13}+\frac{18753809673}{488525633}a^{12}-\frac{3236986207}{488525633}a^{11}-\frac{9475446697}{488525633}a^{10}-\frac{790200668}{488525633}a^{9}-\frac{17132023555}{488525633}a^{8}+\frac{40077788572}{488525633}a^{7}-\frac{22589910696}{488525633}a^{6}-\frac{16426381686}{488525633}a^{5}+\frac{46253596995}{488525633}a^{4}-\frac{20000283499}{488525633}a^{3}-\frac{11127652395}{488525633}a^{2}+\frac{9188968948}{488525633}a-\frac{1750486972}{488525633}$, $\frac{282152457}{488525633}a^{15}-\frac{355775087}{488525633}a^{14}-\frac{2568341775}{488525633}a^{13}+\frac{5540322672}{488525633}a^{12}-\frac{907033994}{488525633}a^{11}-\frac{3016017704}{488525633}a^{10}-\frac{32514253}{488525633}a^{9}-\frac{5238265761}{488525633}a^{8}+\frac{12518862384}{488525633}a^{7}-\frac{7067797552}{488525633}a^{6}-\frac{5011377033}{488525633}a^{5}+\frac{12839078185}{488525633}a^{4}-\frac{6248548099}{488525633}a^{3}-\frac{2698659765}{488525633}a^{2}+\frac{2897808102}{488525633}a-\frac{122174692}{488525633}$, $\frac{462875086}{488525633}a^{15}-\frac{737538209}{488525633}a^{14}-\frac{4029300133}{488525633}a^{13}+\frac{10301192159}{488525633}a^{12}-\frac{4188824106}{488525633}a^{11}-\frac{3044441553}{488525633}a^{10}-\frac{1785572867}{488525633}a^{9}-\frac{6150775833}{488525633}a^{8}+\frac{20959484406}{488525633}a^{7}-\frac{15609162404}{488525633}a^{6}-\frac{3310210160}{488525633}a^{5}+\frac{20250946001}{488525633}a^{4}-\frac{10791885235}{488525633}a^{3}-\frac{5257749652}{488525633}a^{2}+\frac{3296367274}{488525633}a-\frac{104863597}{488525633}$, $\frac{528232762}{488525633}a^{15}-\frac{1004670875}{488525633}a^{14}-\frac{4264319208}{488525633}a^{13}+\frac{13187463605}{488525633}a^{12}-\frac{9096539032}{488525633}a^{11}-\frac{1540709580}{488525633}a^{10}-\frac{220155777}{488525633}a^{9}-\frac{6347864278}{488525633}a^{8}+\frac{25649230663}{488525633}a^{7}-\frac{26423518911}{488525633}a^{6}+\frac{1928892846}{488525633}a^{5}+\frac{24714459018}{488525633}a^{4}-\frac{21369058345}{488525633}a^{3}-\frac{485438494}{488525633}a^{2}+\frac{7103591808}{488525633}a-\frac{2392601258}{488525633}$ Copy content Toggle raw display (assuming GRH)
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:  \( 1635.12490583 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 1635.12490583 \cdot 1}{2\cdot\sqrt{4635236761600000000}}\cr\approx \mathstrut & 0.151511087566 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 8*x^14 + 26*x^13 - 19*x^12 - 4*x^11 + 4*x^10 - 16*x^9 + 53*x^8 - 56*x^7 + 6*x^6 + 54*x^5 - 53*x^4 + 6*x^3 + 16*x^2 - 8*x + 1)
 
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 - 2*x^15 - 8*x^14 + 26*x^13 - 19*x^12 - 4*x^11 + 4*x^10 - 16*x^9 + 53*x^8 - 56*x^7 + 6*x^6 + 54*x^5 - 53*x^4 + 6*x^3 + 16*x^2 - 8*x + 1, 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 - 2*x^15 - 8*x^14 + 26*x^13 - 19*x^12 - 4*x^11 + 4*x^10 - 16*x^9 + 53*x^8 - 56*x^7 + 6*x^6 + 54*x^5 - 53*x^4 + 6*x^3 + 16*x^2 - 8*x + 1);
 
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 - 2*x^15 - 8*x^14 + 26*x^13 - 19*x^12 - 4*x^11 + 4*x^10 - 16*x^9 + 53*x^8 - 56*x^7 + 6*x^6 + 54*x^5 - 53*x^4 + 6*x^3 + 16*x^2 - 8*x + 1);
 
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

$C_2^6:D_4$ (as 16T969):

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 512
The 44 conjugacy class representatives for $C_2^6:D_4$
Character table for $C_2^6:D_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 8.6.134560000.1, 8.4.134560000.3, 8.6.134560000.2

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

Degree 16 siblings: data not computed
Degree 32 siblings: data not computed
Minimal sibling: 16.4.4635236761600000000.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.8.0.1}{8} }^{2}$ R ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{4}$ ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ R ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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$$4$$4$$24$
\(5\) Copy content Toggle raw display 5.8.4.1$x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
\(29\) Copy content Toggle raw display 29.2.0.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.2.1$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$