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Properties

Label 16.0.6884340550074368.1
Degree $16$
Signature $[0, 8]$
Discriminant $6.884\times 10^{15}$
Root discriminant \(9.77\)
Ramified primes $2,17$
Class number $1$
Class group trivial
Galois group $C_8\wr C_2$ (as 16T289)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 22*x^14 - 56*x^13 + 110*x^12 - 174*x^11 + 232*x^10 - 266*x^9 + 259*x^8 - 202*x^7 + 112*x^6 - 30*x^5 - 14*x^4 + 20*x^3 - 6*x^2 - 2*x + 1)
 
gp: K = bnfinit(y^16 - 6*y^15 + 22*y^14 - 56*y^13 + 110*y^12 - 174*y^11 + 232*y^10 - 266*y^9 + 259*y^8 - 202*y^7 + 112*y^6 - 30*y^5 - 14*y^4 + 20*y^3 - 6*y^2 - 2*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 6*x^15 + 22*x^14 - 56*x^13 + 110*x^12 - 174*x^11 + 232*x^10 - 266*x^9 + 259*x^8 - 202*x^7 + 112*x^6 - 30*x^5 - 14*x^4 + 20*x^3 - 6*x^2 - 2*x + 1);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 6*x^15 + 22*x^14 - 56*x^13 + 110*x^12 - 174*x^11 + 232*x^10 - 266*x^9 + 259*x^8 - 202*x^7 + 112*x^6 - 30*x^5 - 14*x^4 + 20*x^3 - 6*x^2 - 2*x + 1)
 

\( x^{16} - 6 x^{15} + 22 x^{14} - 56 x^{13} + 110 x^{12} - 174 x^{11} + 232 x^{10} - 266 x^{9} + 259 x^{8} + \cdots + 1 \) 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:  $[0, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(6884340550074368\) \(\medspace = 2^{24}\cdot 17^{7}\) 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:  \(9.77\)
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:  $2^{3/2}17^{7/8}\approx 33.74331915933294$
Ramified primes:   \(2\), \(17\) 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:  \(\Q(\sqrt{17}) \)
$\card{ \Aut(K/\Q) }$:  $8$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{12}-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{2}$, $\frac{1}{298}a^{15}+\frac{31}{298}a^{14}-\frac{23}{298}a^{13}-\frac{13}{298}a^{12}-\frac{73}{298}a^{11}-\frac{22}{149}a^{10}-\frac{55}{298}a^{9}-\frac{33}{149}a^{8}-\frac{97}{298}a^{7}-\frac{33}{149}a^{6}-\frac{95}{298}a^{5}-\frac{59}{149}a^{4}+\frac{45}{149}a^{3}-\frac{77}{298}a^{2}-\frac{12}{149}a-\frac{145}{298}$ 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:  $7$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -\frac{413}{2} a^{15} + 1123 a^{14} - 3912 a^{13} + \frac{18731}{2} a^{12} - \frac{34901}{2} a^{11} + \frac{52239}{2} a^{10} - \frac{66437}{2} a^{9} + 36242 a^{8} - \frac{66179}{2} a^{7} + \frac{46169}{2} a^{6} - \frac{20241}{2} a^{5} + 478 a^{4} + 3178 a^{3} - \frac{4703}{2} a^{2} - \frac{163}{2} a + \frac{737}{2} \)  (order $4$) 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:   $\frac{51471}{298}a^{15}-\frac{280011}{298}a^{14}+\frac{487812}{149}a^{13}-\frac{1168142}{149}a^{12}+\frac{4354177}{298}a^{11}-\frac{3259486}{149}a^{10}+\frac{8292839}{298}a^{9}-\frac{9050295}{298}a^{8}+\frac{8266525}{298}a^{7}-\frac{2885924}{149}a^{6}+\frac{2536117}{298}a^{5}-\frac{126541}{298}a^{4}-\frac{394860}{149}a^{3}+\frac{586001}{298}a^{2}+\frac{20021}{298}a-\frac{91967}{298}$, $\frac{35019}{149}a^{15}-\frac{190447}{149}a^{14}+\frac{1326959}{298}a^{13}-\frac{1588541}{149}a^{12}+\frac{5920527}{298}a^{11}-\frac{8863021}{298}a^{10}+\frac{5636897}{149}a^{9}-\frac{6151133}{149}a^{8}+\frac{11235165}{298}a^{7}-\frac{7842547}{298}a^{6}+\frac{1721911}{149}a^{5}-\frac{84806}{149}a^{4}-\frac{1074911}{298}a^{3}+\frac{796981}{298}a^{2}+\frac{27373}{298}a-\frac{62415}{149}$, $\frac{35235}{298}a^{15}-\frac{191797}{298}a^{14}+\frac{334210}{149}a^{13}-\frac{1601183}{298}a^{12}+\frac{2984949}{298}a^{11}-\frac{2235221}{149}a^{10}+\frac{5688491}{298}a^{9}-\frac{3105119}{149}a^{8}+\frac{5675079}{298}a^{7}-\frac{1983000}{149}a^{6}+\frac{1746687}{298}a^{5}-\frac{45909}{149}a^{4}-\frac{269773}{149}a^{3}+\frac{402197}{298}a^{2}+\frac{12749}{298}a-\frac{63041}{298}$, $\frac{24039}{298}a^{15}-\frac{65381}{149}a^{14}+\frac{455535}{298}a^{13}-\frac{545367}{149}a^{12}+\frac{1016292}{149}a^{11}-\frac{1521496}{149}a^{10}+\frac{1935476}{149}a^{9}-\frac{2112384}{149}a^{8}+\frac{1929509}{149}a^{7}-\frac{1347418}{149}a^{6}+\frac{592433}{149}a^{5}-\frac{30366}{149}a^{4}-\frac{367255}{298}a^{3}+\frac{136496}{149}a^{2}+\frac{9379}{298}a-\frac{21356}{149}$, $\frac{27396}{149}a^{15}-\frac{149024}{149}a^{14}+\frac{1038407}{298}a^{13}-\frac{2486439}{298}a^{12}+\frac{4633691}{298}a^{11}-\frac{3468436}{149}a^{10}+\frac{8823745}{298}a^{9}-\frac{4814360}{149}a^{8}+\frac{8793539}{298}a^{7}-\frac{3068974}{149}a^{6}+\frac{2694591}{298}a^{5}-\frac{65733}{149}a^{4}-\frac{842321}{298}a^{3}+\frac{312354}{149}a^{2}+\frac{10463}{149}a-\frac{97755}{298}$, $\frac{667}{298}a^{15}-\frac{3759}{298}a^{14}+\frac{13267}{298}a^{13}-\frac{16181}{149}a^{12}+\frac{30561}{149}a^{11}-\frac{92971}{298}a^{10}+\frac{119765}{298}a^{9}-\frac{66413}{149}a^{8}+\frac{61893}{149}a^{7}-\frac{89765}{298}a^{6}+\frac{43021}{298}a^{5}-\frac{3146}{149}a^{4}-\frac{9553}{298}a^{3}+\frac{4344}{149}a^{2}-\frac{256}{149}a-\frac{603}{149}$, $\frac{63209}{298}a^{15}-\frac{343763}{298}a^{14}+\frac{1197499}{298}a^{13}-\frac{2866891}{298}a^{12}+\frac{2670887}{149}a^{11}-\frac{7995451}{298}a^{10}+\frac{5084239}{149}a^{9}-\frac{5547018}{149}a^{8}+\frac{5064474}{149}a^{7}-\frac{7066417}{298}a^{6}+\frac{1549001}{149}a^{5}-\frac{73318}{149}a^{4}-\frac{972533}{298}a^{3}+\frac{359682}{149}a^{2}+\frac{25283}{298}a-\frac{112959}{298}$ 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:  \( 23.5886655901 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 23.5886655901 \cdot 1}{4\cdot\sqrt{6884340550074368}}\cr\approx \mathstrut & 0.172643867485 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 22*x^14 - 56*x^13 + 110*x^12 - 174*x^11 + 232*x^10 - 266*x^9 + 259*x^8 - 202*x^7 + 112*x^6 - 30*x^5 - 14*x^4 + 20*x^3 - 6*x^2 - 2*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 - 6*x^15 + 22*x^14 - 56*x^13 + 110*x^12 - 174*x^11 + 232*x^10 - 266*x^9 + 259*x^8 - 202*x^7 + 112*x^6 - 30*x^5 - 14*x^4 + 20*x^3 - 6*x^2 - 2*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 - 6*x^15 + 22*x^14 - 56*x^13 + 110*x^12 - 174*x^11 + 232*x^10 - 266*x^9 + 259*x^8 - 202*x^7 + 112*x^6 - 30*x^5 - 14*x^4 + 20*x^3 - 6*x^2 - 2*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 - 6*x^15 + 22*x^14 - 56*x^13 + 110*x^12 - 174*x^11 + 232*x^10 - 266*x^9 + 259*x^8 - 202*x^7 + 112*x^6 - 30*x^5 - 14*x^4 + 20*x^3 - 6*x^2 - 2*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

$C_8\wr C_2$ (as 16T289):

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 128
The 44 conjugacy class representatives for $C_8\wr C_2$
Character table for $C_8\wr C_2$

Intermediate fields

\(\Q(\sqrt{-1}) \), 4.0.272.1, 8.0.1257728.1

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: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R $16$ ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.4.0.1}{4} }^{2}$ $16$ $16$ ${\href{/padicField/13.8.0.1}{8} }^{2}$ R ${\href{/padicField/19.8.0.1}{8} }^{2}$ $16$ ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}$ $16$ ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.4.0.1}{4} }^{2}$ ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{8}$ ${\href{/padicField/59.8.0.1}{8} }^{2}$

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$$24$
\(17\) Copy content Toggle raw display 17.4.0.1$x^{4} + 7 x^{2} + 10 x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} + 7 x^{2} + 10 x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
17.8.7.2$x^{8} + 136$$8$$1$$7$$C_8$$[\ ]_{8}$