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Properties

Label 16.4.484...304.54
Degree $16$
Signature $[4, 6]$
Discriminant $4.841\times 10^{23}$
Root discriminant \(30.22\)
Ramified primes $2,3$
Class number $1$
Class group trivial
Galois group $C_2^4.(C_4\times D_4)$ (as 16T867)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^14 - 12*x^12 + 120*x^10 - 218*x^8 - 120*x^6 - 12*x^4 + 8*x^2 + 1)
 
gp: K = bnfinit(y^16 - 8*y^14 - 12*y^12 + 120*y^10 - 218*y^8 - 120*y^6 - 12*y^4 + 8*y^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^14 - 12*x^12 + 120*x^10 - 218*x^8 - 120*x^6 - 12*x^4 + 8*x^2 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^14 - 12*x^12 + 120*x^10 - 218*x^8 - 120*x^6 - 12*x^4 + 8*x^2 + 1)
 

\( x^{16} - 8x^{14} - 12x^{12} + 120x^{10} - 218x^{8} - 120x^{6} - 12x^{4} + 8x^{2} + 1 \) 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:   \(484116351470433472610304\) \(\medspace = 2^{66}\cdot 3^{8}\) 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:  \(30.22\)
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:  $2^{35/8}3^{1/2}\approx 35.93907196672871$
Ramified primes:   \(2\), \(3\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{4}+\frac{1}{4}a^{2}-\frac{1}{4}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{8}a^{8}-\frac{1}{4}a^{4}-\frac{3}{8}$, $\frac{1}{16}a^{9}-\frac{1}{16}a^{8}-\frac{1}{8}a^{5}+\frac{1}{8}a^{4}+\frac{5}{16}a-\frac{5}{16}$, $\frac{1}{16}a^{10}-\frac{1}{16}a^{8}-\frac{1}{8}a^{6}+\frac{1}{8}a^{4}+\frac{5}{16}a^{2}-\frac{5}{16}$, $\frac{1}{16}a^{11}-\frac{1}{16}a^{8}-\frac{1}{8}a^{7}+\frac{1}{8}a^{4}+\frac{5}{16}a^{3}-\frac{5}{16}$, $\frac{1}{32}a^{12}-\frac{1}{32}a^{8}+\frac{3}{32}a^{4}+\frac{5}{32}$, $\frac{1}{64}a^{13}-\frac{1}{64}a^{12}-\frac{1}{32}a^{11}-\frac{1}{32}a^{10}+\frac{1}{64}a^{9}-\frac{1}{64}a^{8}+\frac{1}{16}a^{7}+\frac{1}{16}a^{6}-\frac{1}{64}a^{5}+\frac{1}{64}a^{4}-\frac{5}{32}a^{3}-\frac{5}{32}a^{2}+\frac{15}{64}a-\frac{15}{64}$, $\frac{1}{320}a^{14}-\frac{1}{320}a^{12}-\frac{9}{320}a^{10}-\frac{3}{320}a^{8}-\frac{29}{320}a^{6}-\frac{43}{320}a^{4}-\frac{83}{320}a^{2}-\frac{73}{320}$, $\frac{1}{320}a^{15}-\frac{1}{320}a^{13}-\frac{9}{320}a^{11}-\frac{3}{320}a^{9}-\frac{29}{320}a^{7}+\frac{37}{320}a^{5}-\frac{1}{4}a^{4}+\frac{77}{320}a^{3}-\frac{1}{2}a^{2}-\frac{153}{320}a+\frac{1}{4}$ 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:   $a^{15}-8a^{13}-12a^{11}+120a^{9}-218a^{7}-120a^{5}-12a^{3}+8a$, $\frac{61}{160}a^{14}-\frac{129}{40}a^{12}-\frac{499}{160}a^{10}+\frac{3791}{80}a^{8}-\frac{16749}{160}a^{6}-\frac{4}{5}a^{4}+\frac{467}{160}a^{2}+\frac{71}{80}$, $\frac{39}{160}a^{14}-\frac{157}{80}a^{12}-\frac{451}{160}a^{10}+\frac{2349}{80}a^{8}-\frac{8771}{160}a^{6}-\frac{2071}{80}a^{4}-\frac{217}{160}a^{2}-\frac{1}{80}$, $\frac{71}{64}a^{15}+\frac{577}{64}a^{13}+\frac{1}{16}a^{12}+\frac{779}{64}a^{11}-\frac{1}{2}a^{10}-\frac{8621}{64}a^{9}-\frac{3}{4}a^{8}+\frac{16579}{64}a^{7}+\frac{15}{2}a^{6}+\frac{6443}{64}a^{5}-\frac{219}{16}a^{4}-\frac{143}{64}a^{3}-\frac{15}{2}a^{2}-\frac{231}{64}a+\frac{1}{8}$, $\frac{57}{160}a^{15}-\frac{241}{80}a^{13}-\frac{463}{160}a^{11}+\frac{1761}{40}a^{9}-\frac{15673}{160}a^{7}+\frac{227}{80}a^{5}-\frac{801}{160}a^{3}-\frac{11}{10}a$, $\frac{47}{320}a^{15}+\frac{1}{16}a^{14}+\frac{377}{320}a^{13}-\frac{1}{2}a^{12}+\frac{563}{320}a^{11}-\frac{3}{4}a^{10}-\frac{5709}{320}a^{9}+\frac{15}{2}a^{8}+\frac{10283}{320}a^{7}-\frac{219}{16}a^{6}+\frac{6291}{320}a^{5}-\frac{29}{4}a^{4}-\frac{919}{320}a^{3}+\frac{13}{8}a^{2}-\frac{519}{320}a-\frac{1}{4}$, $\frac{47}{320}a^{15}+\frac{1}{16}a^{14}-\frac{377}{320}a^{13}-\frac{1}{2}a^{12}-\frac{563}{320}a^{11}-\frac{3}{4}a^{10}+\frac{5709}{320}a^{9}+\frac{15}{2}a^{8}-\frac{10283}{320}a^{7}-\frac{219}{16}a^{6}-\frac{6291}{320}a^{5}-\frac{29}{4}a^{4}+\frac{919}{320}a^{3}+\frac{13}{8}a^{2}+\frac{519}{320}a-\frac{1}{4}$, $\frac{3}{5}a^{15}+\frac{113}{320}a^{14}-\frac{771}{160}a^{13}-\frac{943}{320}a^{12}-\frac{567}{80}a^{11}-\frac{1037}{320}a^{10}+\frac{11607}{160}a^{9}+\frac{13971}{320}a^{8}-\frac{5311}{40}a^{7}-\frac{29397}{320}a^{6}-\frac{11593}{160}a^{5}-\frac{4149}{320}a^{4}+\frac{481}{80}a^{3}+\frac{1801}{320}a^{2}+\frac{477}{160}a+\frac{361}{320}$, $\frac{83}{320}a^{15}-\frac{119}{320}a^{14}+\frac{623}{320}a^{13}+\frac{1009}{320}a^{12}+\frac{1307}{320}a^{11}+\frac{971}{320}a^{10}-\frac{9331}{320}a^{9}-\frac{14953}{320}a^{8}+\frac{13367}{320}a^{7}+\frac{32771}{320}a^{6}+\frac{16869}{320}a^{5}+\frac{1667}{320}a^{4}+\frac{9769}{320}a^{3}-\frac{4863}{320}a^{2}+\frac{1159}{320}a-\frac{1603}{320}$ 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:  \( 2702089.013366066 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 2702089.013366066 \cdot 1}{2\cdot\sqrt{484116351470433472610304}}\cr\approx \mathstrut & 1.91158658010628 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^14 - 12*x^12 + 120*x^10 - 218*x^8 - 120*x^6 - 12*x^4 + 8*x^2 + 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 - 8*x^14 - 12*x^12 + 120*x^10 - 218*x^8 - 120*x^6 - 12*x^4 + 8*x^2 + 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 - 8*x^14 - 12*x^12 + 120*x^10 - 218*x^8 - 120*x^6 - 12*x^4 + 8*x^2 + 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 - 8*x^14 - 12*x^12 + 120*x^10 - 218*x^8 - 120*x^6 - 12*x^4 + 8*x^2 + 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

$C_2^4.(C_4\times D_4)$ (as 16T867):

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 41 conjugacy class representatives for $C_2^4.(C_4\times D_4)$
Character table for $C_2^4.(C_4\times D_4)$

Intermediate fields

\(\Q(\sqrt{6}) \), 4.2.18432.2, 8.4.21743271936.1

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: 16.4.484116351470433472610304.79, 16.4.484116351470433472610304.60, 16.4.484116351470433472610304.120, 16.0.7564317991725523009536.41, 16.0.30257271966902092038144.58, 16.0.30257271966902092038144.59, 16.0.7564317991725523009536.42, 16.0.121029087867608368152576.152, 16.0.121029087867608368152576.153, 16.0.1494186269970473680896.24, 16.0.1494186269970473680896.33
Degree 32 siblings: deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32
Minimal sibling: 16.0.1494186269970473680896.24

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 ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{4}$ ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}$

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 2.16.66.994$x^{16} + 24 x^{12} + 16 x^{11} + 8 x^{10} + 16 x^{8} + 16 x^{4} + 16 x^{3} + 26$$16$$1$$66$16T867$[2, 3, 3, 7/2, 4, 4, 17/4, 19/4]^{2}$
\(3\) Copy content Toggle raw display 3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$