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Properties

Label 16.14.585...000.2
Degree $16$
Signature $[14, 1]$
Discriminant $-5.855\times 10^{23}$
Root discriminant \(30.58\)
Ramified primes $2,5,89$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^7.C_2\wr D_4$ (as 16T1778)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 18*x^14 + 112*x^12 - 244*x^10 - 169*x^8 + 1472*x^6 - 2014*x^4 + 890*x^2 - 89)
 
gp: K = bnfinit(y^16 - 18*y^14 + 112*y^12 - 244*y^10 - 169*y^8 + 1472*y^6 - 2014*y^4 + 890*y^2 - 89, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 18*x^14 + 112*x^12 - 244*x^10 - 169*x^8 + 1472*x^6 - 2014*x^4 + 890*x^2 - 89);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 18*x^14 + 112*x^12 - 244*x^10 - 169*x^8 + 1472*x^6 - 2014*x^4 + 890*x^2 - 89)
 

\( x^{16} - 18x^{14} + 112x^{12} - 244x^{10} - 169x^{8} + 1472x^{6} - 2014x^{4} + 890x^{2} - 89 \) 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:  $[14, 1]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-585531072079462400000000\) \(\medspace = -\,2^{28}\cdot 5^{8}\cdot 89^{5}\) 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:  \(30.58\)
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:  not computed
Ramified primes:   \(2\), \(5\), \(89\) 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:  \(\Q(\sqrt{-89}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{8}a^{12}-\frac{1}{4}a^{11}-\frac{1}{8}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{7}-\frac{1}{8}a^{6}-\frac{1}{4}a^{5}+\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{3}{8}a^{2}+\frac{1}{4}a-\frac{3}{8}$, $\frac{1}{8}a^{13}-\frac{1}{8}a^{11}-\frac{1}{4}a^{8}-\frac{1}{8}a^{7}+\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{3}{8}a^{3}-\frac{3}{8}a+\frac{1}{4}$, $\frac{1}{170008}a^{14}+\frac{9281}{170008}a^{12}-\frac{19341}{85004}a^{10}-\frac{1}{4}a^{9}+\frac{11515}{170008}a^{8}-\frac{3403}{21251}a^{6}-\frac{1}{4}a^{5}-\frac{33843}{170008}a^{4}+\frac{19239}{170008}a^{2}+\frac{1}{4}a-\frac{3909}{85004}$, $\frac{1}{170008}a^{15}+\frac{9281}{170008}a^{13}-\frac{19341}{85004}a^{11}-\frac{1}{4}a^{10}+\frac{11515}{170008}a^{9}-\frac{3403}{21251}a^{7}-\frac{1}{4}a^{6}-\frac{33843}{170008}a^{5}+\frac{19239}{170008}a^{3}+\frac{1}{4}a^{2}-\frac{3909}{85004}a$ 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:  $14$
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:   $\frac{25}{538}a^{14}-\frac{391}{538}a^{12}+\frac{944}{269}a^{10}-\frac{919}{269}a^{8}-\frac{3781}{269}a^{6}+\frac{9111}{269}a^{4}-\frac{12371}{538}a^{2}+\frac{2265}{538}$, $\frac{23}{85004}a^{15}-\frac{2837}{170008}a^{14}+\frac{953}{85004}a^{13}+\frac{21147}{85004}a^{12}-\frac{18395}{85004}a^{11}-\frac{190589}{170008}a^{10}+\frac{94837}{85004}a^{9}+\frac{143489}{170008}a^{8}-\frac{94877}{85004}a^{7}+\frac{794641}{170008}a^{6}-\frac{438373}{85004}a^{5}-\frac{1699507}{170008}a^{4}+\frac{465635}{42502}a^{3}+\frac{150354}{21251}a^{2}-\frac{55579}{21251}a-\frac{155135}{170008}$, $\frac{4015}{170008}a^{15}+\frac{3973}{85004}a^{14}-\frac{53541}{170008}a^{13}-\frac{60825}{85004}a^{12}+\frac{82043}{85004}a^{11}+\frac{279909}{85004}a^{10}+\frac{330565}{170008}a^{9}-\frac{195567}{85004}a^{8}-\frac{401073}{42502}a^{7}-\frac{1289673}{85004}a^{6}-\frac{43253}{170008}a^{5}+\frac{2440703}{85004}a^{4}+\frac{3376109}{170008}a^{3}-\frac{255837}{21251}a^{2}-\frac{776433}{85004}a+\frac{25275}{42502}$, $\frac{12205}{85004}a^{14}-\frac{46121}{21251}a^{12}+\frac{827195}{85004}a^{10}-\frac{481061}{85004}a^{8}-\frac{3919719}{85004}a^{6}+\frac{6823443}{85004}a^{4}-\frac{1248961}{42502}a^{2}+\frac{147057}{85004}$, $\frac{4259}{85004}a^{14}-\frac{31417}{42502}a^{12}+\frac{267377}{85004}a^{10}-\frac{89927}{85004}a^{8}-\frac{1340373}{85004}a^{6}+\frac{1942037}{85004}a^{4}-\frac{225613}{42502}a^{2}-\frac{124051}{85004}$, $\frac{1081}{170008}a^{15}+\frac{1767}{42502}a^{14}-\frac{9481}{85004}a^{13}-\frac{13748}{21251}a^{12}+\frac{112981}{170008}a^{11}+\frac{260507}{85004}a^{10}-\frac{217881}{170008}a^{9}-\frac{214165}{85004}a^{8}-\frac{251521}{170008}a^{7}-\frac{1153893}{85004}a^{6}+\frac{1370003}{170008}a^{5}+\frac{2358327}{85004}a^{4}-\frac{416237}{42502}a^{3}-\frac{1138877}{85004}a^{2}+\frac{877931}{170008}a+\frac{231249}{85004}$, $\frac{1081}{170008}a^{15}-\frac{1767}{42502}a^{14}-\frac{9481}{85004}a^{13}+\frac{13748}{21251}a^{12}+\frac{112981}{170008}a^{11}-\frac{260507}{85004}a^{10}-\frac{217881}{170008}a^{9}+\frac{214165}{85004}a^{8}-\frac{251521}{170008}a^{7}+\frac{1153893}{85004}a^{6}+\frac{1370003}{170008}a^{5}-\frac{2358327}{85004}a^{4}-\frac{416237}{42502}a^{3}+\frac{1138877}{85004}a^{2}+\frac{877931}{170008}a-\frac{231249}{85004}$, $\frac{768}{21251}a^{14}-\frac{12528}{21251}a^{12}+\frac{64875}{21251}a^{10}-\frac{81900}{21251}a^{8}-\frac{230809}{21251}a^{6}+\frac{678581}{21251}a^{4}-\frac{546419}{21251}a^{2}+\frac{158566}{21251}$, $\frac{4393}{85004}a^{15}+\frac{637}{21251}a^{14}-\frac{72989}{85004}a^{13}-\frac{72503}{170008}a^{12}+\frac{396739}{85004}a^{11}+\frac{277067}{170008}a^{10}-\frac{152066}{21251}a^{9}+\frac{28171}{42502}a^{8}-\frac{1205711}{85004}a^{7}-\frac{1855813}{170008}a^{6}+\frac{2284331}{42502}a^{5}+\frac{663375}{85004}a^{4}-\frac{2145501}{42502}a^{3}+\frac{1711161}{170008}a^{2}+\frac{1080943}{85004}a-\frac{844943}{170008}$, $\frac{724}{21251}a^{14}-\frac{47241}{85004}a^{12}+\frac{245961}{85004}a^{10}-\frac{78536}{21251}a^{8}-\frac{913287}{85004}a^{6}+\frac{1338955}{42502}a^{4}-\frac{1810313}{85004}a^{2}+\frac{76391}{85004}$, $\frac{3249}{42502}a^{15}-\frac{439}{85004}a^{14}-\frac{196059}{170008}a^{13}+\frac{5833}{85004}a^{12}+\frac{873675}{170008}a^{11}-\frac{9701}{42502}a^{10}-\frac{117031}{42502}a^{9}-\frac{9299}{42502}a^{8}-\frac{4245405}{170008}a^{7}+\frac{33945}{21251}a^{6}+\frac{3542451}{85004}a^{5}-\frac{20594}{21251}a^{4}-\frac{2198075}{170008}a^{3}-\frac{115529}{85004}a^{2}-\frac{129203}{170008}a+\frac{95695}{85004}$, $\frac{1081}{170008}a^{15}-\frac{1871}{21251}a^{14}-\frac{9481}{85004}a^{13}+\frac{58385}{42502}a^{12}+\frac{112981}{170008}a^{11}-\frac{558811}{85004}a^{10}-\frac{217881}{170008}a^{9}+\frac{504569}{85004}a^{8}-\frac{251521}{170008}a^{7}+\frac{2348689}{85004}a^{6}+\frac{1370003}{170008}a^{5}-\frac{5237403}{85004}a^{4}-\frac{416237}{42502}a^{3}+\frac{3093495}{85004}a^{2}+\frac{877931}{170008}a-\frac{419111}{85004}$, $\frac{10891}{85004}a^{15}-\frac{11705}{85004}a^{14}-\frac{342037}{170008}a^{13}+\frac{363281}{170008}a^{12}+\frac{1667153}{170008}a^{11}-\frac{1723323}{170008}a^{10}-\frac{421163}{42502}a^{9}+\frac{183638}{21251}a^{8}-\frac{6656827}{170008}a^{7}+\frac{7497953}{170008}a^{6}+\frac{8111113}{85004}a^{5}-\frac{7934199}{85004}a^{4}-\frac{10780079}{170008}a^{3}+\frac{8317845}{170008}a^{2}+\frac{2032683}{170008}a-\frac{1035531}{170008}$, $\frac{104}{21251}a^{15}+\frac{65}{1076}a^{14}-\frac{3393}{42502}a^{13}-\frac{2087}{2152}a^{12}+\frac{37797}{85004}a^{11}+\frac{10517}{2152}a^{10}-\frac{76239}{85004}a^{9}-\frac{5801}{1076}a^{8}-\frac{40903}{85004}a^{7}-\frac{42927}{2152}a^{6}+\frac{520749}{85004}a^{5}+\frac{13539}{269}a^{4}-\frac{815741}{85004}a^{3}-\frac{62231}{2152}a^{2}+\frac{41617}{85004}a+\frac{3439}{2152}$ 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:  \( 3528147.74875 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{14}\cdot(2\pi)^{1}\cdot 3528147.74875 \cdot 1}{2\cdot\sqrt{585531072079462400000000}}\cr\approx \mathstrut & 0.237323968855 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 18*x^14 + 112*x^12 - 244*x^10 - 169*x^8 + 1472*x^6 - 2014*x^4 + 890*x^2 - 89)
 
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 - 18*x^14 + 112*x^12 - 244*x^10 - 169*x^8 + 1472*x^6 - 2014*x^4 + 890*x^2 - 89, 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 - 18*x^14 + 112*x^12 - 244*x^10 - 169*x^8 + 1472*x^6 - 2014*x^4 + 890*x^2 - 89);
 
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 - 18*x^14 + 112*x^12 - 244*x^10 - 169*x^8 + 1472*x^6 - 2014*x^4 + 890*x^2 - 89);
 
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

$C_2^7.C_2\wr D_4$ (as 16T1778):

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 16384
The 148 conjugacy class representatives for $C_2^7.C_2\wr D_4$
Character table for $C_2^7.C_2\wr D_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.2225.1, 8.8.5069440000.1

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: 16.4.814254147110502400000000.3

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.4.0.1}{4} }^{4}$ R ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.4.0.1}{4} }^{2}$ ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.4.0.1}{4} }^{2}$ ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}$ ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.8.0.1}{8} }^{2}$ ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.4.0.1}{4} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }^{3}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.4.0.1}{4} }^{2}$ ${\href{/padicField/53.2.0.1}{2} }^{8}$ ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$

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

# 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 2.8.16.47$x^{8} - 6 x^{6} + 8 x^{5} + 40 x^{4} - 24 x^{3} + 76 x^{2} + 16 x + 52$$4$$2$$16$$(((C_4 \times C_2): C_2):C_2):C_2$$[2, 2, 3, 3]^{4}$
2.8.12.3$x^{8} - 12 x^{7} + 44 x^{6} + 152 x^{5} - 512 x^{4} + 560 x^{3} + 4432 x^{2} + 9376 x + 21744$$2$$4$$12$$C_2^2:C_4$$[2, 3]^{4}$
\(5\) 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}$
\(89\) Copy content Toggle raw display 89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.0.1$x^{2} + 82 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} + 82 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.4.3.4$x^{4} + 1157$$4$$1$$3$$C_4$$[\ ]_{4}$
89.4.0.1$x^{4} + 4 x^{2} + 72 x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$