Properties

Label 16.4.206...984.1
Degree $16$
Signature $[4, 6]$
Discriminant $2.070\times 10^{18}$
Root discriminant \(13.96\)
Ramified primes $2,7$
Class number $1$
Class group trivial
Galois group $C_4\wr C_2$ (as 16T28)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 6*x^14 - 4*x^13 - 5*x^12 + 12*x^11 + 12*x^10 - 68*x^9 + 126*x^8 - 132*x^7 + 70*x^6 - 24*x^5 + 28*x^4 - 36*x^3 + 24*x^2 - 8*x + 1)
 
gp: K = bnfinit(y^16 - 4*y^15 + 6*y^14 - 4*y^13 - 5*y^12 + 12*y^11 + 12*y^10 - 68*y^9 + 126*y^8 - 132*y^7 + 70*y^6 - 24*y^5 + 28*y^4 - 36*y^3 + 24*y^2 - 8*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^15 + 6*x^14 - 4*x^13 - 5*x^12 + 12*x^11 + 12*x^10 - 68*x^9 + 126*x^8 - 132*x^7 + 70*x^6 - 24*x^5 + 28*x^4 - 36*x^3 + 24*x^2 - 8*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 4*x^15 + 6*x^14 - 4*x^13 - 5*x^12 + 12*x^11 + 12*x^10 - 68*x^9 + 126*x^8 - 132*x^7 + 70*x^6 - 24*x^5 + 28*x^4 - 36*x^3 + 24*x^2 - 8*x + 1)
 

\( x^{16} - 4 x^{15} + 6 x^{14} - 4 x^{13} - 5 x^{12} + 12 x^{11} + 12 x^{10} - 68 x^{9} + 126 x^{8} - 132 x^{7} + 70 x^{6} - 24 x^{5} + 28 x^{4} - 36 x^{3} + 24 x^{2} - 8 x + 1 \) 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:  $[4, 6]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(2069703095939497984\) \(\medspace = 2^{44}\cdot 7^{6}\) 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:  \(13.96\)
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:  $2^{11/4}7^{1/2}\approx 17.798422345016238$
Ramified primes:   \(2\), \(7\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{162122591}a^{15}+\frac{45424346}{162122591}a^{14}+\frac{49494767}{162122591}a^{13}+\frac{2043155}{162122591}a^{12}-\frac{34864797}{162122591}a^{11}+\frac{835662}{162122591}a^{10}+\frac{19712972}{162122591}a^{9}-\frac{11118121}{162122591}a^{8}+\frac{13623379}{162122591}a^{7}+\frac{19572739}{162122591}a^{6}-\frac{2662856}{9536623}a^{5}-\frac{21530626}{162122591}a^{4}+\frac{52787661}{162122591}a^{3}+\frac{26703603}{162122591}a^{2}+\frac{40710260}{162122591}a+\frac{31377500}{162122591}$ 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 $1$

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:  $9$
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:   $\frac{311497354}{162122591}a^{15}-\frac{1033906602}{162122591}a^{14}+\frac{1165489544}{162122591}a^{13}-\frac{470028008}{162122591}a^{12}-\frac{1838666762}{162122591}a^{11}+\frac{2469728748}{162122591}a^{10}+\frac{5409432464}{162122591}a^{9}-\frac{17399288206}{162122591}a^{8}+\frac{27382010314}{162122591}a^{7}-\frac{22859545403}{162122591}a^{6}+\frac{402173188}{9536623}a^{5}-\frac{3532601944}{162122591}a^{4}+\frac{6536830308}{162122591}a^{3}-\frac{6565540449}{162122591}a^{2}+\frac{3233734820}{162122591}a-\frac{604386706}{162122591}$, $\frac{177713584}{162122591}a^{15}-\frac{802331092}{162122591}a^{14}+\frac{1295069240}{162122591}a^{13}-\frac{814725266}{162122591}a^{12}-\frac{996409842}{162122591}a^{11}+\frac{2736573289}{162122591}a^{10}+\frac{1879308492}{162122591}a^{9}-\frac{14204316824}{162122591}a^{8}+\frac{26113970486}{162122591}a^{7}-\frac{27451339622}{162122591}a^{6}+\frac{775656064}{9536623}a^{5}-\frac{1881192114}{162122591}a^{4}+\frac{5312423196}{162122591}a^{3}-\frac{7933924910}{162122591}a^{2}+\frac{4731903908}{162122591}a-\frac{967323865}{162122591}$, $\frac{309954798}{162122591}a^{15}-\frac{1085422187}{162122591}a^{14}+\frac{1269587313}{162122591}a^{13}-\frac{487863148}{162122591}a^{12}-\frac{1863978060}{162122591}a^{11}+\frac{2775262899}{162122591}a^{10}+\frac{5407854356}{162122591}a^{9}-\frac{18510300593}{162122591}a^{8}+\frac{28832710102}{162122591}a^{7}-\frac{24598752858}{162122591}a^{6}+\frac{411487810}{9536623}a^{5}-\frac{2959661193}{162122591}a^{4}+\frac{7603165180}{162122591}a^{3}-\frac{7071261392}{162122591}a^{2}+\frac{3239288928}{162122591}a-\frac{617856247}{162122591}$, $\frac{91476756}{162122591}a^{15}-\frac{228787736}{162122591}a^{14}+\frac{103870930}{162122591}a^{13}+\frac{107841922}{162122591}a^{12}-\frac{604010281}{162122591}a^{11}+\frac{253254516}{162122591}a^{10}+\frac{2119793112}{162122591}a^{9}-\frac{3722058902}{162122591}a^{8}+\frac{3993146534}{162122591}a^{7}-\frac{746202208}{162122591}a^{6}-\frac{140231406}{9536623}a^{5}-\frac{336442844}{162122591}a^{4}+\frac{1536235886}{162122591}a^{3}-\frac{466777892}{162122591}a^{2}-\frac{292262216}{162122591}a+\frac{177713584}{162122591}$, $\frac{272000763}{162122591}a^{15}-\frac{893172328}{162122591}a^{14}+\frac{961050838}{162122591}a^{13}-\frac{310761635}{162122591}a^{12}-\frac{1641392537}{162122591}a^{11}+\frac{2072944059}{162122591}a^{10}+\frac{4932967058}{162122591}a^{9}-\frac{15095690477}{162122591}a^{8}+\frac{22749698818}{162122591}a^{7}-\frac{18178841118}{162122591}a^{6}+\frac{252671424}{9536623}a^{5}-\frac{2558897871}{162122591}a^{4}+\frac{6067696677}{162122591}a^{3}-\frac{4906796723}{162122591}a^{2}+\frac{2066044878}{162122591}a-\frac{244973002}{162122591}$, $\frac{311497354}{162122591}a^{15}-\frac{1033906602}{162122591}a^{14}+\frac{1165489544}{162122591}a^{13}-\frac{470028008}{162122591}a^{12}-\frac{1838666762}{162122591}a^{11}+\frac{2469728748}{162122591}a^{10}+\frac{5409432464}{162122591}a^{9}-\frac{17399288206}{162122591}a^{8}+\frac{27382010314}{162122591}a^{7}-\frac{22859545403}{162122591}a^{6}+\frac{402173188}{9536623}a^{5}-\frac{3532601944}{162122591}a^{4}+\frac{6536830308}{162122591}a^{3}-\frac{6565540449}{162122591}a^{2}+\frac{3395857411}{162122591}a-\frac{604386706}{162122591}$, $\frac{387525287}{162122591}a^{15}-\frac{1445117866}{162122591}a^{14}+\frac{1821846952}{162122591}a^{13}-\frac{730655679}{162122591}a^{12}-\frac{2411095003}{162122591}a^{11}+\frac{4009682087}{162122591}a^{10}+\frac{6440528662}{162122591}a^{9}-\frac{25204883387}{162122591}a^{8}+\frac{39750495998}{162122591}a^{7}-\frac{34917555532}{162122591}a^{6}+\frac{608692943}{9536623}a^{5}-\frac{1960310423}{162122591}a^{4}+\frac{10547431755}{162122591}a^{3}-\frac{10420449738}{162122591}a^{2}+\frac{4209270063}{162122591}a-\frac{782368941}{162122591}$, $\frac{182242031}{162122591}a^{15}-\frac{526666412}{162122591}a^{14}+\frac{431535634}{162122591}a^{13}-\frac{21250396}{162122591}a^{12}-\frac{1134422160}{162122591}a^{11}+\frac{938782580}{162122591}a^{10}+\frac{3735656920}{162122591}a^{9}-\frac{8694711403}{162122591}a^{8}+\frac{11763146479}{162122591}a^{7}-\frac{7116535024}{162122591}a^{6}-\frac{25659304}{9536623}a^{5}-\frac{1653682536}{162122591}a^{4}+\frac{3622557255}{162122591}a^{3}-\frac{2514508447}{162122591}a^{2}+\frac{309578932}{162122591}a+\frac{141130368}{162122591}$, $\frac{397142741}{162122591}a^{15}-\frac{1282904979}{162122591}a^{14}+\frac{1328739203}{162122591}a^{13}-\frac{363702691}{162122591}a^{12}-\frac{2457258318}{162122591}a^{11}+\frac{2912464855}{162122591}a^{10}+\frac{7424923231}{162122591}a^{9}-\frac{21700576269}{162122591}a^{8}+\frac{32061467999}{162122591}a^{7}-\frac{24354520347}{162122591}a^{6}+\frac{246569043}{9536623}a^{5}-\frac{2955818151}{162122591}a^{4}+\frac{8500621441}{162122591}a^{3}-\frac{7332753121}{162122591}a^{2}+\frac{2665450555}{162122591}a-\frac{219429157}{162122591}$ 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:  \( 586.589293907 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 586.589293907 \cdot 1}{2\cdot\sqrt{2069703095939497984}}\cr\approx \mathstrut & 0.200700892939 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 6*x^14 - 4*x^13 - 5*x^12 + 12*x^11 + 12*x^10 - 68*x^9 + 126*x^8 - 132*x^7 + 70*x^6 - 24*x^5 + 28*x^4 - 36*x^3 + 24*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 - 4*x^15 + 6*x^14 - 4*x^13 - 5*x^12 + 12*x^11 + 12*x^10 - 68*x^9 + 126*x^8 - 132*x^7 + 70*x^6 - 24*x^5 + 28*x^4 - 36*x^3 + 24*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 - 4*x^15 + 6*x^14 - 4*x^13 - 5*x^12 + 12*x^11 + 12*x^10 - 68*x^9 + 126*x^8 - 132*x^7 + 70*x^6 - 24*x^5 + 28*x^4 - 36*x^3 + 24*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 - 4*x^15 + 6*x^14 - 4*x^13 - 5*x^12 + 12*x^11 + 12*x^10 - 68*x^9 + 126*x^8 - 132*x^7 + 70*x^6 - 24*x^5 + 28*x^4 - 36*x^3 + 24*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

$C_4\wr C_2$ (as 16T28):

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 14 conjugacy class representatives for $C_4\wr C_2$
Character table for $C_4\wr C_2$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 4.2.448.1, 4.2.14336.1, 8.4.205520896.3

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: 8.0.4917248.1, 8.0.314703872.3
Degree 16 sibling: 16.0.99038527051792384.1
Minimal sibling: 8.0.4917248.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.8.0.1}{8} }^{2}$ ${\href{/padicField/5.8.0.1}{8} }^{2}$ R ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.8.0.1}{8} }^{2}$ ${\href{/padicField/17.2.0.1}{2} }^{6}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ ${\href{/padicField/19.8.0.1}{8} }^{2}$ ${\href{/padicField/23.2.0.1}{2} }^{8}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.2.0.1}{2} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.2.0.1}{2} }^{6}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.8.0.1}{8} }^{2}$

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

# 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$$44$
\(7\) Copy content Toggle raw display 7.4.0.1$x^{4} + 5 x^{2} + 4 x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.2.2$x^{4} - 42 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
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}$