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Properties

Label 16.0.239...736.1
Degree $16$
Signature $[0, 8]$
Discriminant $2.393\times 10^{21}$
Root discriminant \(21.69\)
Ramified primes $2,3$
Class number $2$
Class group [2]
Galois group $C_4^2:C_2$ (as 16T30)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 4*x^12 - 96*x^10 + 150*x^8 + 96*x^6 + 4*x^4 + 1)
 
gp: K = bnfinit(y^16 + 4*y^12 - 96*y^10 + 150*y^8 + 96*y^6 + 4*y^4 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 4*x^12 - 96*x^10 + 150*x^8 + 96*x^6 + 4*x^4 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 + 4*x^12 - 96*x^10 + 150*x^8 + 96*x^6 + 4*x^4 + 1)
 

\( x^{16} + 4x^{12} - 96x^{10} + 150x^{8} + 96x^{6} + 4x^{4} + 1 \) 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:   \(2393397489569403764736\) \(\medspace = 2^{52}\cdot 3^{12}\) 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:  \(21.69\)
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:  $2^{13/4}3^{3/4}\approx 21.686448086636275$
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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $4$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}+\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{4}a^{8}-\frac{1}{4}$, $\frac{1}{4}a^{9}-\frac{1}{4}a$, $\frac{1}{40}a^{10}-\frac{1}{8}a^{8}-\frac{3}{20}a^{6}+\frac{1}{20}a^{4}+\frac{1}{8}a^{2}+\frac{3}{40}$, $\frac{1}{40}a^{11}-\frac{1}{8}a^{9}+\frac{1}{10}a^{7}-\frac{1}{4}a^{6}-\frac{1}{5}a^{5}-\frac{1}{4}a^{4}-\frac{1}{8}a^{3}+\frac{1}{4}a^{2}+\frac{13}{40}a+\frac{1}{4}$, $\frac{1}{40}a^{12}-\frac{1}{40}a^{8}-\frac{1}{5}a^{6}-\frac{1}{8}a^{4}+\frac{1}{5}a^{2}+\frac{1}{8}$, $\frac{1}{80}a^{13}-\frac{1}{80}a^{12}+\frac{9}{80}a^{9}-\frac{9}{80}a^{8}-\frac{1}{10}a^{7}+\frac{1}{10}a^{6}-\frac{1}{16}a^{5}+\frac{1}{16}a^{4}+\frac{1}{10}a^{3}-\frac{1}{10}a^{2}-\frac{1}{16}a+\frac{1}{16}$, $\frac{1}{80}a^{14}-\frac{1}{80}a^{12}-\frac{1}{80}a^{10}-\frac{7}{80}a^{8}-\frac{17}{80}a^{6}-\frac{7}{80}a^{4}+\frac{17}{80}a^{2}+\frac{3}{16}$, $\frac{1}{80}a^{15}-\frac{1}{80}a^{12}-\frac{1}{80}a^{11}+\frac{1}{40}a^{9}-\frac{9}{80}a^{8}-\frac{1}{16}a^{7}-\frac{3}{20}a^{6}+\frac{1}{10}a^{5}-\frac{3}{16}a^{4}+\frac{1}{16}a^{3}+\frac{3}{20}a^{2}-\frac{1}{8}a+\frac{5}{16}$ 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

$C_{2}$, 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:   \( \frac{1}{4} a^{14} - \frac{3}{20} a^{12} + \frac{21}{20} a^{10} - \frac{123}{5} a^{8} + \frac{1043}{20} a^{6} - \frac{63}{20} a^{4} - \frac{109}{20} a^{2} + \frac{9}{10} \)  (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:   $a$, $\frac{33}{80}a^{15}-\frac{13}{80}a^{13}+\frac{133}{80}a^{11}-\frac{3221}{80}a^{9}+\frac{6203}{80}a^{7}+\frac{1121}{80}a^{5}-\frac{849}{80}a^{3}-\frac{127}{80}a$, $\frac{1}{8}a^{14}+\frac{1}{20}a^{12}+\frac{21}{40}a^{10}-\frac{59}{5}a^{8}+\frac{563}{40}a^{6}+\frac{341}{20}a^{4}+\frac{371}{40}a^{2}+\frac{17}{10}$, $\frac{27}{80}a^{15}+\frac{1}{8}a^{14}+\frac{3}{80}a^{13}+\frac{1}{20}a^{12}+\frac{101}{80}a^{11}+\frac{21}{40}a^{10}-\frac{2579}{80}a^{9}-\frac{59}{5}a^{8}+\frac{3733}{80}a^{7}+\frac{563}{40}a^{6}+\frac{3717}{80}a^{5}+\frac{341}{20}a^{4}-\frac{741}{80}a^{3}+\frac{371}{40}a^{2}-\frac{501}{80}a+\frac{27}{10}$, $\frac{1}{8}a^{15}-\frac{1}{8}a^{14}-\frac{7}{40}a^{13}+\frac{1}{5}a^{12}+\frac{9}{20}a^{11}-\frac{21}{40}a^{10}-\frac{127}{10}a^{9}+\frac{64}{5}a^{8}+\frac{1413}{40}a^{7}-\frac{1523}{40}a^{6}-\frac{379}{40}a^{5}+\frac{101}{5}a^{4}-\frac{239}{10}a^{3}+\frac{589}{40}a^{2}-\frac{93}{20}a+\frac{4}{5}$, $\frac{19}{40}a^{15}-\frac{11}{80}a^{14}-\frac{1}{4}a^{13}-\frac{1}{80}a^{12}+\frac{39}{20}a^{11}-\frac{43}{80}a^{10}-\frac{1867}{40}a^{9}+\frac{1059}{80}a^{8}+\frac{3813}{40}a^{7}-\frac{1533}{80}a^{6}+\frac{53}{20}a^{5}-\frac{1231}{80}a^{4}-\frac{35}{4}a^{3}-\frac{493}{80}a^{2}+\frac{51}{40}a-\frac{67}{80}$, $\frac{29}{80}a^{15}+\frac{1}{2}a^{14}-\frac{23}{80}a^{13}-\frac{1}{8}a^{12}+\frac{25}{16}a^{11}+\frac{81}{40}a^{10}-\frac{2879}{80}a^{9}-\frac{97}{2}a^{8}+\frac{1319}{16}a^{7}+\frac{871}{10}a^{6}-\frac{309}{16}a^{5}+\frac{1077}{40}a^{4}-\frac{429}{80}a^{3}-\frac{45}{8}a^{2}+\frac{207}{80}a+\frac{7}{10}$ 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:  \( 46193.7886141 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 46193.7886141 \cdot 2}{4\cdot\sqrt{2393397489569403764736}}\cr\approx \mathstrut & 1.14679322423 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 + 4*x^12 - 96*x^10 + 150*x^8 + 96*x^6 + 4*x^4 + 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^12 - 96*x^10 + 150*x^8 + 96*x^6 + 4*x^4 + 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^12 - 96*x^10 + 150*x^8 + 96*x^6 + 4*x^4 + 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^12 - 96*x^10 + 150*x^8 + 96*x^6 + 4*x^4 + 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

$C_4^2:C_2$ (as 16T30):

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 14 conjugacy class representatives for $C_4^2:C_2$
Character table for $C_4^2:C_2$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{6}) \), 4.2.18432.2, \(\Q(i, \sqrt{6})\), 4.2.18432.1, 8.0.1358954496.8, 8.0.764411904.4, 8.0.12230590464.2

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 sibling: 16.4.2393397489569403764736.1
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 ${\href{/padicField/5.2.0.1}{2} }^{6}{,}\,{\href{/padicField/5.1.0.1}{1} }^{4}$ ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.2.0.1}{2} }^{8}$ ${\href{/padicField/29.2.0.1}{2} }^{6}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.2.0.1}{2} }^{6}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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 2.16.52.12$x^{16} + 8 x^{14} + 4 x^{10} + 8 x^{9} + 10 x^{8} + 8 x^{6} + 8 x^{5} + 26$$16$$1$$52$16T30$[2, 3, 3, 4]^{2}$
\(3\) Copy content Toggle raw display 3.8.6.1$x^{8} + 9$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
3.8.6.1$x^{8} + 9$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$