Properties

Label 16.0.184...016.2
Degree $16$
Signature $[0, 8]$
Discriminant $1.847\times 10^{18}$
Root discriminant \(13.86\)
Ramified primes $2,3$
Class number $1$
Class group trivial
Galois group $Q_8 : C_2$ (as 16T11)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 12*x^14 + 60*x^12 - 144*x^10 + 191*x^8 - 144*x^6 + 60*x^4 - 12*x^2 + 1)
 
gp: K = bnfinit(y^16 - 12*y^14 + 60*y^12 - 144*y^10 + 191*y^8 - 144*y^6 + 60*y^4 - 12*y^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 12*x^14 + 60*x^12 - 144*x^10 + 191*x^8 - 144*x^6 + 60*x^4 - 12*x^2 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 12*x^14 + 60*x^12 - 144*x^10 + 191*x^8 - 144*x^6 + 60*x^4 - 12*x^2 + 1)
 

\( x^{16} - 12x^{14} + 60x^{12} - 144x^{10} + 191x^{8} - 144x^{6} + 60x^{4} - 12x^{2} + 1 \) 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:  $[0, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(1846757322198614016\) \(\medspace = 2^{48}\cdot 3^{8}\) 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:  \(13.86\)
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:  $2^{3}3^{1/2}\approx 13.856406460551018$
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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $16$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5}a^{10}+\frac{1}{5}a^{8}+\frac{1}{5}a^{6}+\frac{1}{5}a^{4}+\frac{1}{5}a^{2}+\frac{1}{5}$, $\frac{1}{5}a^{11}+\frac{1}{5}a^{9}+\frac{1}{5}a^{7}+\frac{1}{5}a^{5}+\frac{1}{5}a^{3}+\frac{1}{5}a$, $\frac{1}{5}a^{12}-\frac{1}{5}$, $\frac{1}{5}a^{13}-\frac{1}{5}a$, $\frac{1}{25}a^{14}+\frac{2}{25}a^{12}-\frac{2}{25}a^{10}+\frac{8}{25}a^{8}+\frac{8}{25}a^{6}-\frac{2}{25}a^{4}+\frac{12}{25}a^{2}+\frac{11}{25}$, $\frac{1}{25}a^{15}+\frac{2}{25}a^{13}-\frac{2}{25}a^{11}+\frac{8}{25}a^{9}+\frac{8}{25}a^{7}-\frac{2}{25}a^{5}+\frac{12}{25}a^{3}+\frac{11}{25}a$ 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 $1$

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:   \( -\frac{36}{25} a^{14} + \frac{408}{25} a^{12} - \frac{1888}{25} a^{10} + \frac{3927}{25} a^{8} - \frac{4273}{25} a^{6} + \frac{2387}{25} a^{4} - \frac{617}{25} a^{2} + \frac{39}{25} \)  (order $24$) 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:   $\frac{56}{25}a^{15}-\frac{653}{25}a^{13}+\frac{3138}{25}a^{11}-\frac{7002}{25}a^{9}+\frac{8373}{25}a^{7}-\frac{5462}{25}a^{5}+\frac{1822}{25}a^{3}-\frac{219}{25}a$, $\frac{17}{25}a^{15}-\frac{196}{25}a^{13}+\frac{931}{25}a^{11}-\frac{2049}{25}a^{9}+\frac{2476}{25}a^{7}-\frac{1719}{25}a^{5}+\frac{669}{25}a^{3}-\frac{93}{25}a$, $\frac{8}{25}a^{15}-\frac{74}{25}a^{13}+\frac{229}{25}a^{11}+\frac{9}{25}a^{9}-\frac{841}{25}a^{7}+\frac{1179}{25}a^{5}-\frac{534}{25}a^{3}+\frac{48}{25}a$, $\frac{59}{25}a^{15}-\frac{4}{5}a^{14}-\frac{692}{25}a^{13}+\frac{47}{5}a^{12}+\frac{3352}{25}a^{11}-\frac{229}{5}a^{10}-\frac{7583}{25}a^{9}+\frac{526}{5}a^{8}+\frac{9192}{25}a^{7}-\frac{659}{5}a^{6}-\frac{5948}{25}a^{5}+\frac{441}{5}a^{4}+\frac{1828}{25}a^{3}-28a^{2}-\frac{121}{25}a+\frac{9}{5}$, $\frac{42}{25}a^{15}+\frac{39}{25}a^{14}-\frac{496}{25}a^{13}-\frac{457}{25}a^{12}+\frac{2421}{25}a^{11}+\frac{2207}{25}a^{10}-\frac{5534}{25}a^{9}-\frac{4953}{25}a^{8}+\frac{6716}{25}a^{7}+\frac{5897}{25}a^{6}-\frac{4229}{25}a^{5}-\frac{3743}{25}a^{4}+\frac{1159}{25}a^{3}+\frac{1153}{25}a^{2}-\frac{28}{25}a-\frac{101}{25}$, $\frac{17}{25}a^{15}-\frac{17}{25}a^{14}-\frac{196}{25}a^{13}+\frac{211}{25}a^{12}+\frac{931}{25}a^{11}-\frac{1081}{25}a^{10}-\frac{2049}{25}a^{9}+\frac{2624}{25}a^{8}+\frac{2476}{25}a^{7}-\frac{3226}{25}a^{6}-\frac{1719}{25}a^{5}+\frac{2019}{25}a^{4}+\frac{669}{25}a^{3}-\frac{544}{25}a^{2}-\frac{68}{25}a+\frac{53}{25}$, $\frac{28}{25}a^{15}-\frac{28}{25}a^{14}-\frac{324}{25}a^{13}+\frac{324}{25}a^{12}+\frac{1544}{25}a^{11}-\frac{1544}{25}a^{10}-\frac{3401}{25}a^{9}+\frac{3401}{25}a^{8}+\frac{4024}{25}a^{7}-\frac{4024}{25}a^{6}-\frac{2556}{25}a^{5}+\frac{2556}{25}a^{4}+\frac{836}{25}a^{3}-\frac{836}{25}a^{2}-\frac{87}{25}a+\frac{87}{25}$ 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:  \( 3719.97240803 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 3719.97240803 \cdot 1}{24\cdot\sqrt{1846757322198614016}}\cr\approx \mathstrut & 0.277052777863 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 12*x^14 + 60*x^12 - 144*x^10 + 191*x^8 - 144*x^6 + 60*x^4 - 12*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 - 12*x^14 + 60*x^12 - 144*x^10 + 191*x^8 - 144*x^6 + 60*x^4 - 12*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 - 12*x^14 + 60*x^12 - 144*x^10 + 191*x^8 - 144*x^6 + 60*x^4 - 12*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 - 12*x^14 + 60*x^12 - 144*x^10 + 191*x^8 - 144*x^6 + 60*x^4 - 12*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

$D_4:C_2$ (as 16T11):

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 16
The 10 conjugacy class representatives for $Q_8 : C_2$
Character table for $Q_8 : C_2$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-6}) \), \(\Q(\zeta_{12})\), \(\Q(\zeta_{8})\), \(\Q(i, \sqrt{6})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\zeta_{24})\), 8.0.150994944.1 x2, 8.0.1358954496.5 x2, 8.4.1358954496.1 x2

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 8 siblings: 8.4.1358954496.1, 8.0.1358954496.5, 8.0.150994944.1
Minimal sibling: 8.0.150994944.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R ${\href{/padicField/5.2.0.1}{2} }^{8}$ ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.2.0.1}{2} }^{8}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.2.0.1}{2} }^{8}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.2.0.1}{2} }^{8}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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 2.16.48.6$x^{16} + 8 x^{15} + 76 x^{14} + 192 x^{13} + 152 x^{12} + 184 x^{11} + 408 x^{10} + 368 x^{9} + 416 x^{8} + 416 x^{7} + 488 x^{6} + 224 x^{5} + 416 x^{4} + 272 x^{3} + 160 x^{2} - 96 x + 228$$8$$2$$48$$Q_8 : C_2$$[2, 3, 4]^{2}$
\(3\) Copy content Toggle raw display 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$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$