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Properties

Label 16.0.115422332637413376.1
Degree $16$
Signature $[0, 8]$
Discriminant $1.154\times 10^{17}$
Root discriminant \(11.65\)
Ramified primes $2,3$
Class number $1$
Class group trivial
Galois group $D_4\times C_2$ (as 16T9)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^14 + 8*x^12 + 36*x^10 + 62*x^8 + 36*x^6 + 8*x^4 - 4*x^2 + 1)
 
gp: K = bnfinit(y^16 - 4*y^14 + 8*y^12 + 36*y^10 + 62*y^8 + 36*y^6 + 8*y^4 - 4*y^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^14 + 8*x^12 + 36*x^10 + 62*x^8 + 36*x^6 + 8*x^4 - 4*x^2 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 4*x^14 + 8*x^12 + 36*x^10 + 62*x^8 + 36*x^6 + 8*x^4 - 4*x^2 + 1)
 

\( x^{16} - 4x^{14} + 8x^{12} + 36x^{10} + 62x^{8} + 36x^{6} + 8x^{4} - 4x^{2} + 1 \) 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:   \(115422332637413376\) \(\medspace = 2^{44}\cdot 3^{8}\) 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:  \(11.65\)
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:  $2^{11/4}3^{1/2}\approx 11.651802520975762$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{8}-\frac{1}{4}$, $\frac{1}{8}a^{9}-\frac{1}{8}a^{8}+\frac{3}{8}a-\frac{3}{8}$, $\frac{1}{8}a^{10}-\frac{1}{8}a^{8}+\frac{3}{8}a^{2}-\frac{3}{8}$, $\frac{1}{8}a^{11}-\frac{1}{8}a^{8}+\frac{3}{8}a^{3}-\frac{3}{8}$, $\frac{1}{24}a^{12}+\frac{1}{24}a^{10}-\frac{1}{6}a^{6}+\frac{1}{8}a^{4}+\frac{7}{24}a^{2}-\frac{1}{3}$, $\frac{1}{48}a^{13}-\frac{1}{48}a^{12}-\frac{1}{24}a^{11}+\frac{1}{24}a^{10}-\frac{1}{16}a^{9}+\frac{1}{16}a^{8}+\frac{1}{6}a^{7}-\frac{1}{6}a^{6}+\frac{1}{16}a^{5}-\frac{1}{16}a^{4}-\frac{7}{24}a^{3}+\frac{7}{24}a^{2}+\frac{7}{48}a-\frac{7}{48}$, $\frac{1}{144}a^{14}-\frac{1}{48}a^{12}-\frac{1}{144}a^{10}+\frac{11}{144}a^{8}+\frac{19}{144}a^{6}+\frac{7}{144}a^{4}-\frac{1}{48}a^{2}-\frac{31}{144}$, $\frac{1}{144}a^{15}-\frac{1}{48}a^{12}-\frac{7}{144}a^{11}+\frac{1}{24}a^{10}+\frac{1}{72}a^{9}+\frac{1}{16}a^{8}-\frac{29}{144}a^{7}-\frac{1}{6}a^{6}+\frac{1}{9}a^{5}-\frac{1}{16}a^{4}+\frac{3}{16}a^{3}+\frac{7}{24}a^{2}-\frac{5}{72}a-\frac{7}{48}$ 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:  $3$

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:   \( -\frac{41}{144} a^{14} + \frac{53}{48} a^{12} - \frac{301}{144} a^{10} - \frac{1549}{144} a^{8} - \frac{2651}{144} a^{6} - \frac{1547}{144} a^{4} - \frac{109}{48} a^{2} + \frac{65}{144} \)  (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:   $\frac{23}{144}a^{15}-\frac{11}{16}a^{13}+\frac{217}{144}a^{11}+\frac{739}{144}a^{9}+\frac{1277}{144}a^{7}+\frac{431}{144}a^{5}-\frac{21}{16}a^{3}-\frac{239}{144}a$, $\frac{5}{36}a^{15}-\frac{11}{24}a^{13}+\frac{25}{36}a^{11}+\frac{425}{72}a^{9}+\frac{425}{36}a^{7}+\frac{745}{72}a^{5}+\frac{35}{12}a^{3}-\frac{25}{72}a$, $\frac{1}{6}a^{15}-\frac{1}{6}a^{14}-\frac{35}{48}a^{13}+\frac{35}{48}a^{12}+\frac{5}{3}a^{11}-\frac{5}{3}a^{10}+\frac{247}{48}a^{9}-\frac{247}{48}a^{8}+\frac{53}{6}a^{7}-\frac{53}{6}a^{6}+\frac{239}{48}a^{5}-\frac{239}{48}a^{4}+\frac{10}{3}a^{3}-\frac{10}{3}a^{2}+\frac{29}{48}a+\frac{19}{48}$, $\frac{19}{144}a^{15}-\frac{1}{144}a^{14}-\frac{17}{48}a^{13}+\frac{5}{48}a^{12}+\frac{41}{144}a^{11}-\frac{59}{144}a^{10}+\frac{929}{144}a^{9}+\frac{79}{144}a^{8}+\frac{1993}{144}a^{7}+\frac{293}{144}a^{6}+\frac{1807}{144}a^{5}+\frac{317}{144}a^{4}+\frac{169}{48}a^{3}+\frac{29}{48}a^{2}-\frac{133}{144}a-\frac{83}{144}$, $\frac{5}{36}a^{15}+\frac{5}{36}a^{14}-\frac{7}{12}a^{13}-\frac{7}{12}a^{12}+\frac{43}{36}a^{11}+\frac{43}{36}a^{10}+\frac{353}{72}a^{9}+\frac{353}{72}a^{8}+\frac{263}{36}a^{7}+\frac{263}{36}a^{6}+\frac{89}{36}a^{5}+\frac{89}{36}a^{4}-\frac{13}{12}a^{3}-\frac{13}{12}a^{2}-\frac{133}{72}a-\frac{61}{72}$, $\frac{5}{24}a^{15}-\frac{1}{9}a^{14}-\frac{41}{48}a^{13}+\frac{7}{16}a^{12}+\frac{15}{8}a^{11}-\frac{61}{72}a^{10}+\frac{323}{48}a^{9}-\frac{599}{144}a^{8}+\frac{109}{8}a^{7}-\frac{61}{9}a^{6}+\frac{469}{48}a^{5}-\frac{643}{144}a^{4}+\frac{119}{24}a^{3}-\frac{15}{8}a^{2}-\frac{13}{16}a+\frac{43}{144}$, $\frac{7}{72}a^{15}-\frac{1}{24}a^{14}-\frac{1}{3}a^{13}+\frac{1}{8}a^{12}+\frac{35}{72}a^{11}-\frac{5}{24}a^{10}+\frac{311}{72}a^{9}-\frac{19}{12}a^{8}+\frac{505}{72}a^{7}-\frac{115}{24}a^{6}+\frac{50}{9}a^{5}-\frac{115}{24}a^{4}+\frac{31}{24}a^{3}-\frac{13}{8}a^{2}+\frac{77}{72}a+\frac{17}{12}$ 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:  \( 978.775482333 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 978.775482333 \cdot 1}{24\cdot\sqrt{115422332637413376}}\cr\approx \mathstrut & 0.291585459827 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^14 + 8*x^12 + 36*x^10 + 62*x^8 + 36*x^6 + 8*x^4 - 4*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 - 4*x^14 + 8*x^12 + 36*x^10 + 62*x^8 + 36*x^6 + 8*x^4 - 4*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 - 4*x^14 + 8*x^12 + 36*x^10 + 62*x^8 + 36*x^6 + 8*x^4 - 4*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 - 4*x^14 + 8*x^12 + 36*x^10 + 62*x^8 + 36*x^6 + 8*x^4 - 4*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

$C_2\times D_4$ (as 16T9):

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 $D_4\times C_2$
Character table for $D_4\times C_2$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(i, \sqrt{6})\), \(\Q(\zeta_{8})\), \(\Q(\zeta_{12})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{-2}, \sqrt{3})\), 4.0.3072.1 x2, 4.0.3072.2 x2, 4.2.4608.1 x2, 4.2.4608.2 x2, \(\Q(\zeta_{24})\), 8.0.37748736.1 x2, 8.0.21233664.1 x2, 8.4.339738624.1 x2, 8.0.84934656.1 x2, 8.0.339738624.6, 8.0.339738624.3

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: 8.0.84934656.1, 8.4.339738624.1, 8.0.21233664.1, 8.0.37748736.1
Minimal sibling: 8.0.21233664.1

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 ${\href{/padicField/5.4.0.1}{4} }^{4}$ ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.2.0.1}{2} }^{8}$ ${\href{/padicField/13.2.0.1}{2} }^{8}$ ${\href{/padicField/17.2.0.1}{2} }^{8}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.2.0.1}{2} }^{8}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.2.0.1}{2} }^{8}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.2.0.1}{2} }^{8}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

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 2.16.44.1$x^{16} + 16 x^{15} + 120 x^{14} + 564 x^{13} + 1906 x^{12} + 5036 x^{11} + 10748 x^{10} + 18552 x^{9} + 25753 x^{8} + 27972 x^{7} + 22184 x^{6} + 10240 x^{5} + 400 x^{4} - 1572 x^{3} + 132 x^{2} + 108 x + 9$$8$$2$$44$$D_4\times C_2$$[2, 3, 7/2]^{2}$
\(3\) Copy content Toggle raw display 3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$