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Properties

Label 16.0.4559007230078125.1
Degree $16$
Signature $[0, 8]$
Discriminant $4.559\times 10^{15}$
Root discriminant \(9.52\)
Ramified primes $5,11,59,229$
Class number $1$
Class group trivial
Galois group $C_4^4.C_2\wr D_4$ (as 16T1823)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 3*x^14 - 6*x^13 + 11*x^12 - 19*x^11 + 26*x^10 - 30*x^9 + 33*x^8 - 30*x^7 + 26*x^6 - 19*x^5 + 11*x^4 - 6*x^3 + 3*x^2 - 2*x + 1)
 
gp: K = bnfinit(y^16 - 2*y^15 + 3*y^14 - 6*y^13 + 11*y^12 - 19*y^11 + 26*y^10 - 30*y^9 + 33*y^8 - 30*y^7 + 26*y^6 - 19*y^5 + 11*y^4 - 6*y^3 + 3*y^2 - 2*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 + 3*x^14 - 6*x^13 + 11*x^12 - 19*x^11 + 26*x^10 - 30*x^9 + 33*x^8 - 30*x^7 + 26*x^6 - 19*x^5 + 11*x^4 - 6*x^3 + 3*x^2 - 2*x + 1);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 2*x^15 + 3*x^14 - 6*x^13 + 11*x^12 - 19*x^11 + 26*x^10 - 30*x^9 + 33*x^8 - 30*x^7 + 26*x^6 - 19*x^5 + 11*x^4 - 6*x^3 + 3*x^2 - 2*x + 1)
 

\( x^{16} - 2 x^{15} + 3 x^{14} - 6 x^{13} + 11 x^{12} - 19 x^{11} + 26 x^{10} - 30 x^{9} + 33 x^{8} + \cdots + 1 \) 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:   \(4559007230078125\) \(\medspace = 5^{8}\cdot 11^{4}\cdot 59^{2}\cdot 229\) 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:  \(9.52\)
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:  $5^{1/2}11^{1/2}59^{1/2}229^{1/2}\approx 862.0353821044703$
Ramified primes:   \(5\), \(11\), \(59\), \(229\) 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:  \(\Q(\sqrt{229}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{199}a^{14}+\frac{34}{199}a^{13}+\frac{32}{199}a^{12}-\frac{82}{199}a^{11}+\frac{12}{199}a^{10}+\frac{97}{199}a^{9}-\frac{76}{199}a^{8}-\frac{77}{199}a^{7}-\frac{76}{199}a^{6}+\frac{97}{199}a^{5}+\frac{12}{199}a^{4}-\frac{82}{199}a^{3}+\frac{32}{199}a^{2}+\frac{34}{199}a+\frac{1}{199}$, $\frac{1}{199}a^{15}+\frac{70}{199}a^{13}+\frac{24}{199}a^{12}+\frac{14}{199}a^{11}+\frac{87}{199}a^{10}+\frac{9}{199}a^{9}-\frac{80}{199}a^{8}-\frac{45}{199}a^{7}+\frac{94}{199}a^{6}+\frac{97}{199}a^{5}-\frac{92}{199}a^{4}+\frac{34}{199}a^{3}-\frac{59}{199}a^{2}+\frac{39}{199}a-\frac{34}{199}$ 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:   \( -1 \)  (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:   $a$, $\frac{107}{199}a^{15}+\frac{10}{199}a^{14}+\frac{69}{199}a^{13}-\frac{296}{199}a^{12}+\frac{280}{199}a^{11}-\frac{521}{199}a^{10}+\frac{341}{199}a^{9}-\frac{365}{199}a^{8}+\frac{584}{199}a^{7}-\frac{55}{199}a^{6}+\frac{603}{199}a^{5}-\frac{172}{199}a^{4}+\frac{231}{199}a^{3}-\frac{222}{199}a^{2}+\frac{135}{199}a-\frac{46}{199}$, $\frac{195}{199}a^{15}-\frac{334}{199}a^{14}+\frac{503}{199}a^{13}-\frac{1033}{199}a^{12}+\frac{1860}{199}a^{11}-\frac{3162}{199}a^{10}+\frac{4182}{199}a^{9}-\frac{4743}{199}a^{8}+\frac{5003}{199}a^{7}-\frac{4245}{199}a^{6}+\frac{3631}{199}a^{5}-\frac{2446}{199}a^{4}+\frac{1382}{199}a^{3}-\frac{502}{199}a^{2}+\frac{229}{199}a-\frac{198}{199}$, $\frac{12}{199}a^{15}-\frac{103}{199}a^{14}+\frac{124}{199}a^{13}-\frac{222}{199}a^{12}+\frac{455}{199}a^{11}-\frac{789}{199}a^{10}+\frac{1261}{199}a^{9}-\frac{1490}{199}a^{8}+\frac{1620}{199}a^{7}-\frac{1591}{199}a^{6}+\frac{1322}{199}a^{5}-\frac{1146}{199}a^{4}+\frac{496}{199}a^{3}-\frac{223}{199}a^{2}-\frac{49}{199}a-\frac{113}{199}$, $\frac{82}{199}a^{15}-\frac{118}{199}a^{14}+\frac{136}{199}a^{13}-\frac{415}{199}a^{12}+\frac{675}{199}a^{11}-\frac{1048}{199}a^{10}+\frac{1431}{199}a^{9}-\frac{1572}{199}a^{8}+\frac{1814}{199}a^{7}-\frac{1433}{199}a^{6}+\frac{1284}{199}a^{5}-\frac{1199}{199}a^{4}+\frac{524}{199}a^{3}-\frac{455}{199}a^{2}+\frac{181}{199}a-\frac{120}{199}$, $\frac{4}{199}a^{15}-\frac{88}{199}a^{14}+\frac{74}{199}a^{13}-\frac{133}{199}a^{12}+\frac{307}{199}a^{11}-\frac{509}{199}a^{10}+\frac{853}{199}a^{9}-4a^{8}+\frac{825}{199}a^{7}-\frac{896}{199}a^{6}+\frac{409}{199}a^{5}-\frac{429}{199}a^{4}-\frac{210}{199}a^{3}+\frac{132}{199}a^{2}-\frac{50}{199}a+\frac{174}{199}$, $\frac{87}{199}a^{15}-\frac{123}{199}a^{14}+\frac{117}{199}a^{13}-\frac{256}{199}a^{12}+\frac{558}{199}a^{11}-\frac{872}{199}a^{10}+\frac{792}{199}a^{9}-3a^{8}+\frac{382}{199}a^{7}+\frac{14}{199}a^{6}-\frac{109}{199}a^{5}+\frac{470}{199}a^{4}-\frac{289}{199}a^{3}+\frac{284}{199}a^{2}+\frac{7}{199}a-\frac{96}{199}$ 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:  \( 9.23010708309 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 9.23010708309 \cdot 1}{2\cdot\sqrt{4559007230078125}}\cr\approx \mathstrut & 0.166027699478 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 3*x^14 - 6*x^13 + 11*x^12 - 19*x^11 + 26*x^10 - 30*x^9 + 33*x^8 - 30*x^7 + 26*x^6 - 19*x^5 + 11*x^4 - 6*x^3 + 3*x^2 - 2*x + 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 - 2*x^15 + 3*x^14 - 6*x^13 + 11*x^12 - 19*x^11 + 26*x^10 - 30*x^9 + 33*x^8 - 30*x^7 + 26*x^6 - 19*x^5 + 11*x^4 - 6*x^3 + 3*x^2 - 2*x + 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 - 2*x^15 + 3*x^14 - 6*x^13 + 11*x^12 - 19*x^11 + 26*x^10 - 30*x^9 + 33*x^8 - 30*x^7 + 26*x^6 - 19*x^5 + 11*x^4 - 6*x^3 + 3*x^2 - 2*x + 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 - 2*x^15 + 3*x^14 - 6*x^13 + 11*x^12 - 19*x^11 + 26*x^10 - 30*x^9 + 33*x^8 - 30*x^7 + 26*x^6 - 19*x^5 + 11*x^4 - 6*x^3 + 3*x^2 - 2*x + 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

$C_4^4.C_2\wr D_4$ (as 16T1823):

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 32768
The 230 conjugacy class representatives for $C_4^4.C_2\wr D_4$
Character table for $C_4^4.C_2\wr D_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.275.1, 8.2.4461875.1

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

Degree 16 siblings: data not computed
Degree 32 siblings: data not computed
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 $16$ ${\href{/padicField/3.8.0.1}{8} }^{2}$ R ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.4.0.1}{4} }^{2}$ R $16$ ${\href{/padicField/17.8.0.1}{8} }^{2}$ ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{3}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{3}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.4.0.1}{4} }^{2}$ ${\href{/padicField/53.8.0.1}{8} }^{2}$ R

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
\(5\) Copy content Toggle raw display 5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
\(11\) Copy content Toggle raw display 11.4.0.1$x^{4} + 8 x^{2} + 10 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.0.1$x^{4} + 8 x^{2} + 10 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.8.4.1$x^{8} + 60 x^{6} + 20 x^{5} + 970 x^{4} - 280 x^{3} + 4664 x^{2} - 5460 x + 2325$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
\(59\) Copy content Toggle raw display $\Q_{59}$$x + 57$$1$$1$$0$Trivial$[\ ]$
$\Q_{59}$$x + 57$$1$$1$$0$Trivial$[\ ]$
$\Q_{59}$$x + 57$$1$$1$$0$Trivial$[\ ]$
$\Q_{59}$$x + 57$$1$$1$$0$Trivial$[\ ]$
59.2.0.1$x^{2} + 58 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.0.1$x^{2} + 58 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.4.0.1$x^{4} + 2 x^{2} + 40 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
59.4.2.1$x^{4} + 116 x^{3} + 3486 x^{2} + 7076 x + 201725$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(229\) Copy content Toggle raw display $\Q_{229}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{229}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{229}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{229}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{229}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{229}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{229}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{229}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$