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Properties

Label 16.0.334...576.1
Degree $16$
Signature $[0, 8]$
Discriminant $3.342\times 10^{19}$
Root discriminant \(16.61\)
Ramified primes $2,3,11$
Class number $2$
Class group [2]
Galois group $C_2^2 \times D_4$ (as 16T25)

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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 + 49*x^12 - 104*x^10 + 160*x^8 - 104*x^6 + 49*x^4 - 8*x^2 + 1)
 
gp: K = bnfinit(y^16 - 8*y^14 + 49*y^12 - 104*y^10 + 160*y^8 - 104*y^6 + 49*y^4 - 8*y^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^14 + 49*x^12 - 104*x^10 + 160*x^8 - 104*x^6 + 49*x^4 - 8*x^2 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^14 + 49*x^12 - 104*x^10 + 160*x^8 - 104*x^6 + 49*x^4 - 8*x^2 + 1)
 

\( x^{16} - 8x^{14} + 49x^{12} - 104x^{10} + 160x^{8} - 104x^{6} + 49x^{4} - 8x^{2} + 1 \) 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:  $[0, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(33418400425706520576\) \(\medspace = 2^{32}\cdot 3^{12}\cdot 11^{4}\) 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:  \(16.61\)
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:  $2^{2}3^{3/4}11^{1/2}\approx 30.241078459545175$
Ramified primes:   \(2\), \(3\), \(11\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $8$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is a CM field.
Reflex fields:  unavailable$^{128}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{6}-\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{3}a^{9}-\frac{1}{3}a^{7}-\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{6}-\frac{1}{3}a^{4}+\frac{1}{3}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{7}-\frac{1}{3}a^{5}+\frac{1}{3}a$, $\frac{1}{96}a^{12}-\frac{1}{12}a^{6}-\frac{47}{96}$, $\frac{1}{96}a^{13}-\frac{1}{12}a^{7}-\frac{47}{96}a$, $\frac{1}{3168}a^{14}+\frac{1}{1584}a^{12}-\frac{1}{33}a^{10}-\frac{1}{396}a^{8}-\frac{61}{198}a^{6}+\frac{10}{33}a^{4}-\frac{911}{3168}a^{2}+\frac{721}{1584}$, $\frac{1}{3168}a^{15}+\frac{1}{1584}a^{13}-\frac{1}{33}a^{11}-\frac{1}{396}a^{9}-\frac{61}{198}a^{7}+\frac{10}{33}a^{5}-\frac{911}{3168}a^{3}+\frac{721}{1584}a$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$

Class group and class number

$C_{2}$, which has order $2$

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:   \( -\frac{1997}{3168} a^{15} + \frac{7705}{1584} a^{13} - \frac{973}{33} a^{11} + \frac{22589}{396} a^{9} - \frac{16585}{198} a^{7} + \frac{1348}{33} a^{5} - \frac{56189}{3168} a^{3} + \frac{217}{1584} a \)  (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:   $\frac{283}{1584}a^{14}-\frac{4445}{3168}a^{12}+\frac{281}{33}a^{10}-\frac{3385}{198}a^{8}+\frac{9785}{396}a^{6}-\frac{434}{33}a^{4}+\frac{7771}{1584}a^{2}-\frac{1997}{3168}$, $\frac{967}{3168}a^{14}-\frac{469}{198}a^{12}+\frac{158}{11}a^{10}-\frac{11131}{396}a^{8}+\frac{4051}{99}a^{6}-\frac{216}{11}a^{4}+\frac{20887}{3168}a^{2}+\frac{233}{198}$, $\frac{241}{1584}a^{15}-\frac{4085}{3168}a^{13}+\frac{266}{33}a^{11}-\frac{3871}{198}a^{9}+\frac{13037}{396}a^{7}-\frac{944}{33}a^{5}+\frac{24913}{1584}a^{3}-\frac{10661}{3168}a-1$, $\frac{839}{1584}a^{15}-\frac{13507}{3168}a^{13}+\frac{863}{33}a^{11}-\frac{11135}{198}a^{9}+\frac{34303}{396}a^{7}-\frac{1865}{33}a^{5}+\frac{38759}{1584}a^{3}-\frac{6451}{3168}a+1$, $\frac{299}{792}a^{15}+\frac{221}{1584}a^{14}-\frac{4711}{1584}a^{13}-\frac{3307}{3168}a^{12}+\frac{199}{11}a^{11}+\frac{69}{11}a^{10}-\frac{3632}{99}a^{9}-\frac{2201}{198}a^{8}+\frac{10633}{198}a^{7}+\frac{6235}{396}a^{6}-\frac{307}{11}a^{5}-\frac{63}{11}a^{4}+\frac{6923}{792}a^{3}+\frac{4589}{1584}a^{2}+\frac{2105}{1584}a+\frac{101}{3168}$, $\frac{493}{792}a^{15}-\frac{19}{99}a^{14}-\frac{7961}{1584}a^{13}+\frac{4823}{3168}a^{12}+\frac{340}{11}a^{11}-\frac{305}{33}a^{10}-\frac{6664}{99}a^{9}+\frac{1868}{99}a^{8}+\frac{21047}{198}a^{7}-\frac{10535}{396}a^{6}-\frac{837}{11}a^{5}+\frac{410}{33}a^{4}+\frac{31885}{792}a^{3}-\frac{49}{99}a^{2}-\frac{14777}{1584}a-\frac{8473}{3168}$, $\frac{493}{792}a^{15}+\frac{19}{99}a^{14}-\frac{7961}{1584}a^{13}-\frac{4823}{3168}a^{12}+\frac{340}{11}a^{11}+\frac{305}{33}a^{10}-\frac{6664}{99}a^{9}-\frac{1868}{99}a^{8}+\frac{21047}{198}a^{7}+\frac{10535}{396}a^{6}-\frac{837}{11}a^{5}-\frac{410}{33}a^{4}+\frac{31885}{792}a^{3}+\frac{49}{99}a^{2}-\frac{14777}{1584}a+\frac{8473}{3168}$ 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:  \( 9987.741474015005 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 9987.741474015005 \cdot 2}{24\cdot\sqrt{33418400425706520576}}\cr\approx \mathstrut & 0.349729417036342 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^14 + 49*x^12 - 104*x^10 + 160*x^8 - 104*x^6 + 49*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 + 49*x^12 - 104*x^10 + 160*x^8 - 104*x^6 + 49*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 + 49*x^12 - 104*x^10 + 160*x^8 - 104*x^6 + 49*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 + 49*x^12 - 104*x^10 + 160*x^8 - 104*x^6 + 49*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

$C_2^2\times D_4$ (as 16T25):

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 20 conjugacy class representatives for $C_2^2 \times D_4$
Character table for $C_2^2 \times D_4$

Intermediate fields

\(\Q(\sqrt{6}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-3}) \), 4.4.4752.1, 4.0.76032.2, 4.4.76032.1, 4.0.4752.1, \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(i, \sqrt{6})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\zeta_{8})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\zeta_{12})\), \(\Q(\zeta_{24})\), 8.0.5780865024.13, 8.0.5780865024.3, 8.8.5780865024.1, 8.0.5780865024.4, 8.0.22581504.2, 8.0.5780865024.5

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 16 siblings: deg 16, deg 16, deg 16, deg 16, deg 16, deg 16, deg 16
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 R ${\href{/padicField/5.4.0.1}{4} }^{4}$ ${\href{/padicField/7.4.0.1}{4} }^{4}$ R ${\href{/padicField/13.2.0.1}{2} }^{8}$ ${\href{/padicField/17.2.0.1}{2} }^{8}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.2.0.1}{2} }^{8}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.2.0.1}{2} }^{8}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

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 2.8.16.6$x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
2.8.16.6$x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
\(3\) Copy content Toggle raw display 3.8.6.2$x^{8} + 6 x^{5} + 6 x^{4} + 18 x^{2} + 18 x + 9$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
3.8.6.2$x^{8} + 6 x^{5} + 6 x^{4} + 18 x^{2} + 18 x + 9$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
\(11\) Copy content Toggle raw display 11.2.0.1$x^{2} + 7 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} + 7 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} + 7 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} + 7 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.2.1$x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$