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Properties

Label 16.0.604...088.2
Degree $16$
Signature $[0, 8]$
Discriminant $6.045\times 10^{23}$
Root discriminant \(30.64\)
Ramified prime $2$
Class number $1$
Class group trivial
Galois group $C_2^7.C_8$ (as 16T1155)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 16*x^12 + 32*x^10 + 72*x^8 - 128*x^6 + 32*x^4 + 192*x^2 + 8)
 
gp: K = bnfinit(y^16 - 16*y^12 + 32*y^10 + 72*y^8 - 128*y^6 + 32*y^4 + 192*y^2 + 8, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 16*x^12 + 32*x^10 + 72*x^8 - 128*x^6 + 32*x^4 + 192*x^2 + 8);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 16*x^12 + 32*x^10 + 72*x^8 - 128*x^6 + 32*x^4 + 192*x^2 + 8)
 

\( x^{16} - 16x^{12} + 32x^{10} + 72x^{8} - 128x^{6} + 32x^{4} + 192x^{2} + 8 \) 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:   \(604462909807314587353088\) \(\medspace = 2^{79}\) 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:  \(30.64\)
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:  $2^{2849/512}\approx 47.32245861429085$
Ramified primes:   \(2\) 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:  \(\Q(\sqrt{2}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{2}a^{8}$, $\frac{1}{2}a^{9}$, $\frac{1}{2}a^{10}$, $\frac{1}{4}a^{11}$, $\frac{1}{4}a^{12}$, $\frac{1}{4}a^{13}$, $\frac{1}{5748988}a^{14}+\frac{22621}{5748988}a^{12}-\frac{99111}{410642}a^{10}-\frac{314754}{1437247}a^{8}+\frac{71422}{1437247}a^{6}+\frac{24486}{205321}a^{4}-\frac{407756}{1437247}a^{2}+\frac{402818}{1437247}$, $\frac{1}{5748988}a^{15}+\frac{22621}{5748988}a^{13}+\frac{7099}{821284}a^{11}-\frac{314754}{1437247}a^{9}+\frac{71422}{1437247}a^{7}+\frac{24486}{205321}a^{5}-\frac{407756}{1437247}a^{3}+\frac{402818}{1437247}a$ 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:   $\frac{88}{8927}a^{14}-\frac{73}{8927}a^{12}-\frac{1246}{8927}a^{10}+\frac{8105}{17854}a^{8}+\frac{2112}{8927}a^{6}-\frac{13016}{8927}a^{4}+\frac{16048}{8927}a^{2}+\frac{4395}{8927}$, $\frac{8355}{2874494}a^{14}+\frac{949}{5748988}a^{12}-\frac{12812}{205321}a^{10}+\frac{132113}{2874494}a^{8}+\frac{546610}{1437247}a^{6}-\frac{43693}{205321}a^{4}-\frac{1051980}{1437247}a^{2}+\frac{461079}{1437247}$, $\frac{39581}{5748988}a^{14}-\frac{10770}{1437247}a^{12}-\frac{49465}{410642}a^{10}+\frac{995091}{2874494}a^{8}+\frac{1215913}{2874494}a^{6}-\frac{345396}{205321}a^{4}+\frac{893574}{1437247}a^{2}+\frac{558287}{1437247}$, $\frac{9627}{2874494}a^{14}+\frac{58893}{5748988}a^{12}-\frac{14910}{205321}a^{10}-\frac{243081}{2874494}a^{8}+\frac{1151056}{1437247}a^{6}+\frac{36428}{205321}a^{4}-\frac{3565404}{1437247}a^{2}-\frac{964287}{1437247}$, $\frac{61501}{5748988}a^{14}-\frac{40975}{5748988}a^{12}-\frac{30542}{205321}a^{10}+\frac{594089}{1437247}a^{8}+\frac{297590}{1437247}a^{6}-\frac{116049}{205321}a^{4}+\frac{1121147}{1437247}a^{2}-\frac{116721}{1437247}$, $\frac{37673}{2874494}a^{15}+\frac{70405}{5748988}a^{14}-\frac{43269}{1437247}a^{13}+\frac{161829}{5748988}a^{12}-\frac{46318}{205321}a^{11}-\frac{37885}{205321}a^{10}+\frac{2552973}{2874494}a^{9}-\frac{125001}{2874494}a^{8}+\frac{309244}{1437247}a^{7}+\frac{2413151}{1437247}a^{6}-\frac{913634}{205321}a^{5}+\frac{267035}{205321}a^{4}+\frac{2682790}{1437247}a^{3}-\frac{1926849}{1437247}a^{2}+\frac{4691870}{1437247}a-\frac{793761}{1437247}$, $\frac{2406063}{5748988}a^{15}+\frac{149936}{205321}a^{14}+\frac{1548367}{1437247}a^{13}+\frac{1045233}{821284}a^{12}-\frac{2162247}{410642}a^{11}-\frac{2177331}{205321}a^{10}-\frac{8965771}{2874494}a^{9}+\frac{509745}{205321}a^{8}+\frac{57446664}{1437247}a^{7}+\frac{29826873}{410642}a^{6}+\frac{10727570}{205321}a^{5}+\frac{7468858}{205321}a^{4}+\frac{25606723}{1437247}a^{3}-\frac{4511908}{205321}a^{2}+\frac{3720072}{1437247}a+\frac{214399}{205321}$ 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:  \( 302852.196243 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 302852.196243 \cdot 1}{2\cdot\sqrt{604462909807314587353088}}\cr\approx \mathstrut & 0.473102071824 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 16*x^12 + 32*x^10 + 72*x^8 - 128*x^6 + 32*x^4 + 192*x^2 + 8)
 
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 - 16*x^12 + 32*x^10 + 72*x^8 - 128*x^6 + 32*x^4 + 192*x^2 + 8, 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 - 16*x^12 + 32*x^10 + 72*x^8 - 128*x^6 + 32*x^4 + 192*x^2 + 8);
 
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 - 16*x^12 + 32*x^10 + 72*x^8 - 128*x^6 + 32*x^4 + 192*x^2 + 8);
 
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

$C_2^7.C_8$ (as 16T1155):

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 1024
The 40 conjugacy class representatives for $C_2^7.C_8$
Character table for $C_2^7.C_8$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.4.2147483648.2

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

Degree 16 siblings: 16.8.604462909807314587353088.12, 16.4.604462909807314587353088.16, 16.8.604462909807314587353088.22, 16.8.604462909807314587353088.23, 16.4.604462909807314587353088.22, 16.12.604462909807314587353088.4, 16.8.604462909807314587353088.26, 16.4.604462909807314587353088.44, 16.12.604462909807314587353088.5, 16.4.604462909807314587353088.56, some data not computed
Degree 32 siblings: data not computed
Minimal sibling: 16.4.604462909807314587353088.16

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R $16$ $16$ ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{4}$ $16$ $16$ ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ $16$ ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ $16$ ${\href{/padicField/31.2.0.1}{2} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ $16$ ${\href{/padicField/41.8.0.1}{8} }^{2}$ $16$ ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ $16$ $16$

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 2.16.79.2522$x^{16} + 32 x^{15} + 16 x^{14} + 56 x^{12} + 32 x^{9} + 20 x^{8} + 48 x^{6} + 32 x^{5} + 98$$16$$1$$79$16T1155$[2, 3, 7/2, 4, 17/4, 19/4, 5, 41/8, 43/8, 6]$