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Properties

Label 16.0.594...496.2
Degree $16$
Signature $[0, 8]$
Discriminant $5.945\times 10^{19}$
Root discriminant \(17.21\)
Ramified primes $2,7$
Class number $2$
Class group [2]
Galois group $C_2^2\wr C_2$ (as 16T39)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^14 + 12*x^12 - 12*x^10 + 24*x^8 - 62*x^6 + 73*x^4 - 20*x^2 + 4)
 
gp: K = bnfinit(y^16 - 6*y^14 + 12*y^12 - 12*y^10 + 24*y^8 - 62*y^6 + 73*y^4 - 20*y^2 + 4, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 6*x^14 + 12*x^12 - 12*x^10 + 24*x^8 - 62*x^6 + 73*x^4 - 20*x^2 + 4);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 6*x^14 + 12*x^12 - 12*x^10 + 24*x^8 - 62*x^6 + 73*x^4 - 20*x^2 + 4)
 

\( x^{16} - 6x^{14} + 12x^{12} - 12x^{10} + 24x^{8} - 62x^{6} + 73x^{4} - 20x^{2} + 4 \) 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:   \(59447875862838378496\) \(\medspace = 2^{32}\cdot 7^{12}\) 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:  \(17.21\)
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:  $2^{2}7^{3/4}\approx 17.214068282635402$
Ramified primes:   \(2\), \(7\) 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 not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{11}+\frac{1}{4}a^{9}+\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{11}-\frac{1}{4}a^{10}-\frac{1}{2}a^{9}+\frac{1}{4}a^{8}+\frac{1}{4}a^{7}-\frac{1}{2}a^{6}+\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}$, $\frac{1}{1336}a^{14}+\frac{35}{334}a^{12}-\frac{32}{167}a^{10}+\frac{5}{334}a^{8}-\frac{99}{334}a^{6}+\frac{119}{668}a^{4}+\frac{85}{1336}a^{2}-\frac{151}{668}$, $\frac{1}{2672}a^{15}+\frac{35}{668}a^{13}-\frac{16}{167}a^{11}-\frac{1}{4}a^{10}+\frac{5}{668}a^{9}+\frac{1}{4}a^{8}+\frac{235}{668}a^{7}-\frac{1}{2}a^{6}+\frac{119}{1336}a^{5}-\frac{1}{4}a^{4}-\frac{1251}{2672}a^{3}-\frac{1}{4}a^{2}+\frac{517}{1336}a-\frac{1}{2}$ 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:   \( -1 \)  (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:   $\frac{187}{2672}a^{15}-\frac{151}{334}a^{13}+\frac{557}{668}a^{11}-\frac{401}{668}a^{9}+\frac{513}{334}a^{7}-\frac{5469}{1336}a^{5}+\frac{11887}{2672}a^{3}+\frac{487}{1336}a$, $\frac{315}{1336}a^{15}+\frac{67}{1336}a^{14}-\frac{829}{668}a^{13}-\frac{153}{668}a^{12}+\frac{715}{334}a^{11}+\frac{275}{668}a^{10}-\frac{1359}{668}a^{9}-\frac{83}{167}a^{8}+\frac{3261}{668}a^{7}+\frac{595}{668}a^{6}-\frac{7939}{668}a^{5}-\frac{773}{334}a^{4}+\frac{15753}{1336}a^{3}+\frac{3691}{1336}a^{2}-\frac{1139}{668}a-\frac{765}{668}$, $\frac{485}{2672}a^{15}-\frac{140}{167}a^{13}+\frac{857}{668}a^{11}-\frac{915}{668}a^{9}+\frac{563}{167}a^{7}-\frac{10087}{1336}a^{5}+\frac{18513}{2672}a^{3}-\frac{1759}{1336}a$, $\frac{315}{1336}a^{15}-\frac{43}{167}a^{14}-\frac{829}{668}a^{13}+\frac{803}{668}a^{12}+\frac{715}{334}a^{11}-\frac{1225}{668}a^{10}-\frac{1359}{668}a^{9}+\frac{309}{167}a^{8}+\frac{3261}{668}a^{7}-\frac{3197}{668}a^{6}-\frac{7939}{668}a^{5}+\frac{6993}{668}a^{4}+\frac{15753}{1336}a^{3}-\frac{1484}{167}a^{2}-\frac{1139}{668}a+\frac{294}{167}$, $\frac{45}{1336}a^{15}-\frac{11}{668}a^{14}-\frac{95}{334}a^{13}-\frac{37}{668}a^{12}+\frac{419}{668}a^{11}+\frac{36}{167}a^{10}-\frac{385}{668}a^{9}-\frac{53}{668}a^{8}+\frac{194}{167}a^{7}+\frac{181}{668}a^{6}-\frac{913}{334}a^{5}-\frac{70}{167}a^{4}+\frac{5495}{1336}a^{3}+\frac{451}{334}a^{2}-\frac{1117}{668}a-\frac{88}{167}$, $\frac{73}{334}a^{15}+\frac{315}{1336}a^{14}-\frac{234}{167}a^{13}-\frac{829}{668}a^{12}+\frac{1869}{668}a^{11}+\frac{715}{334}a^{10}-\frac{1589}{668}a^{9}-\frac{1359}{668}a^{8}+\frac{1653}{334}a^{7}+\frac{3261}{668}a^{6}-\frac{9507}{668}a^{5}-\frac{7939}{668}a^{4}+\frac{10907}{668}a^{3}+\frac{15753}{1336}a^{2}-\frac{169}{334}a-\frac{1139}{668}$, $\frac{187}{1336}a^{14}-\frac{135}{334}a^{12}+\frac{28}{167}a^{10}-\frac{67}{334}a^{8}+\frac{525}{334}a^{6}-\frac{1795}{668}a^{4}-\frac{137}{1336}a^{2}-\frac{181}{668}$ 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:  \( 4489.36686801 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 4489.36686801 \cdot 2}{2\cdot\sqrt{59447875862838378496}}\cr\approx \mathstrut & 1.41434667321 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^14 + 12*x^12 - 12*x^10 + 24*x^8 - 62*x^6 + 73*x^4 - 20*x^2 + 4)
 
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 - 6*x^14 + 12*x^12 - 12*x^10 + 24*x^8 - 62*x^6 + 73*x^4 - 20*x^2 + 4, 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 - 6*x^14 + 12*x^12 - 12*x^10 + 24*x^8 - 62*x^6 + 73*x^4 - 20*x^2 + 4);
 
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 - 6*x^14 + 12*x^12 - 12*x^10 + 24*x^8 - 62*x^6 + 73*x^4 - 20*x^2 + 4);
 
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

$C_2^2\wr C_2$ (as 16T39):

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 $C_2^2\wr C_2$
Character table for $C_2^2\wr C_2$

Intermediate fields

\(\Q(\sqrt{-14}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{2}) \), 4.0.2744.1, 4.0.21952.1, 4.0.10976.1, 4.0.1372.1, 4.2.1792.1 x2, 4.0.1568.1 x2, \(\Q(\sqrt{2}, \sqrt{-7})\), 8.0.157351936.3, 8.0.1927561216.2, 8.0.481890304.1, 8.0.1927561216.8, 8.0.1927561216.7, 8.0.7710244864.5, 8.0.120472576.1

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

Galois closure: deg 32
Degree 8 siblings: 8.0.30118144.1, 8.0.481890304.1, 8.0.1927561216.1, 8.0.7710244864.5, 8.0.1927561216.2, 8.0.120472576.1, 8.0.120472576.2, 8.0.481890304.2
Degree 16 siblings: 16.4.59447875862838378496.1, 16.0.59447875862838378496.1, 16.0.232218265089212416.1, 16.0.3715492241427398656.1, 16.0.59447875862838378496.5, 16.0.3715492241427398656.3
Minimal sibling: 8.0.30118144.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.4.0.1}{4} }^{4}$ ${\href{/padicField/5.4.0.1}{4} }^{4}$ R ${\href{/padicField/11.2.0.1}{2} }^{8}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.2.0.1}{2} }^{8}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.2.0.1}{2} }^{8}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.2.0.1}{2} }^{8}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.2.0.1}{2} }^{8}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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 2.4.8.3$x^{4} + 6 x^{2} + 4 x + 14$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.3$x^{4} + 6 x^{2} + 4 x + 14$$4$$1$$8$$C_2^2$$[2, 3]$
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}$
\(7\) Copy content Toggle raw display 7.4.3.1$x^{4} + 7$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
7.4.3.1$x^{4} + 7$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
7.4.3.1$x^{4} + 7$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
7.4.3.1$x^{4} + 7$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$