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Properties

Label 16.0.109...000.9
Degree $16$
Signature $[0, 8]$
Discriminant $1.100\times 10^{20}$
Root discriminant \(17.89\)
Ramified primes $2,5$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $D_{8}$ (as 16T13)

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Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^14 - 4*x^12 + 32*x^10 + 99*x^8 + 96*x^6 + 44*x^4 - 20*x^2 + 25)
 
Copy content gp:K = bnfinit(y^16 - 4*y^14 - 4*y^12 + 32*y^10 + 99*y^8 + 96*y^6 + 44*y^4 - 20*y^2 + 25, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^14 - 4*x^12 + 32*x^10 + 99*x^8 + 96*x^6 + 44*x^4 - 20*x^2 + 25);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 4*x^14 - 4*x^12 + 32*x^10 + 99*x^8 + 96*x^6 + 44*x^4 - 20*x^2 + 25)
 

\( x^{16} - 4x^{14} - 4x^{12} + 32x^{10} + 99x^{8} + 96x^{6} + 44x^{4} - 20x^{2} + 25 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $16$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $[0, 8]$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(109951162777600000000\) \(\medspace = 2^{48}\cdot 5^{8}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(17.89\)
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:(1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2^{3}5^{1/2}\approx 17.88854381999832$
Ramified primes:   \(2\), \(5\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$:   $D_8$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is Galois over $\Q$.
This is not a CM field.
Maximal CM subfield:  \(\Q(i, \sqrt{5})\)

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5}a^{10}+\frac{2}{5}a^{6}+\frac{1}{5}a^{2}$, $\frac{1}{5}a^{11}+\frac{2}{5}a^{7}+\frac{1}{5}a^{3}$, $\frac{1}{55}a^{12}+\frac{2}{55}a^{10}+\frac{2}{5}a^{8}-\frac{6}{55}a^{6}-\frac{14}{55}a^{4}+\frac{27}{55}a^{2}+\frac{2}{11}$, $\frac{1}{55}a^{13}+\frac{2}{55}a^{11}+\frac{2}{5}a^{9}-\frac{6}{55}a^{7}-\frac{14}{55}a^{5}+\frac{27}{55}a^{3}+\frac{2}{11}a$, $\frac{1}{487135}a^{14}-\frac{251}{97427}a^{12}-\frac{1948}{28655}a^{10}+\frac{6106}{97427}a^{8}+\frac{1128}{5731}a^{6}-\frac{27429}{97427}a^{4}+\frac{132347}{487135}a^{2}-\frac{12843}{97427}$, $\frac{1}{2435675}a^{15}-\frac{18969}{2435675}a^{13}-\frac{4032}{143275}a^{11}-\frac{846313}{2435675}a^{9}-\frac{45418}{143275}a^{7}+\frac{110851}{2435675}a^{5}+\frac{141204}{2435675}a^{3}+\frac{146583}{487135}a$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Ideal class group:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $7$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( \frac{12}{5731} a^{14} + \frac{49}{5731} a^{12} - \frac{390}{5731} a^{10} - \frac{424}{5731} a^{8} + \frac{5402}{5731} a^{6} + \frac{11040}{5731} a^{4} + \frac{7450}{5731} a^{2} - \frac{542}{5731} \)  (order $4$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $\frac{253}{44285}a^{14}-\frac{20721}{487135}a^{12}+\frac{361}{5731}a^{10}+\frac{9643}{44285}a^{8}-\frac{47}{28655}a^{6}-\frac{628681}{487135}a^{4}-\frac{714061}{487135}a^{2}-\frac{57216}{97427}$, $\frac{6696}{2435675}a^{15}-\frac{51329}{2435675}a^{13}+\frac{2553}{143275}a^{11}+\frac{416742}{2435675}a^{9}+\frac{1797}{143275}a^{7}-\frac{2526834}{2435675}a^{5}-\frac{5296966}{2435675}a^{3}-\frac{898672}{487135}a$, $\frac{14426}{2435675}a^{15}-\frac{98349}{2435675}a^{13}+\frac{6623}{143275}a^{11}+\frac{640302}{2435675}a^{9}-\frac{1028}{13025}a^{7}-\frac{1899079}{2435675}a^{5}-\frac{2271061}{2435675}a^{3}-\frac{2392}{487135}a$, $\frac{48002}{2435675}a^{15}+\frac{6119}{487135}a^{14}-\frac{183193}{2435675}a^{13}-\frac{35754}{487135}a^{12}-\frac{18614}{143275}a^{11}+\frac{147}{28655}a^{10}+\frac{1855239}{2435675}a^{9}+\frac{337792}{487135}a^{8}+\frac{321339}{143275}a^{7}+\frac{12103}{28655}a^{6}+\frac{2308347}{2435675}a^{5}-\frac{1158629}{487135}a^{4}-\frac{5723317}{2435675}a^{3}-\frac{1611619}{487135}a^{2}-\frac{1539319}{487135}a-\frac{166439}{97427}$, $\frac{19526}{2435675}a^{15}+\frac{4008}{487135}a^{14}-\frac{77524}{2435675}a^{13}-\frac{5167}{97427}a^{12}-\frac{3127}{143275}a^{11}+\frac{2206}{28655}a^{10}+\frac{460102}{2435675}a^{9}+\frac{18671}{97427}a^{8}+\frac{123742}{143275}a^{7}+\frac{1349}{5731}a^{6}+\frac{2792921}{2435675}a^{5}-\frac{20062}{97427}a^{4}+\frac{895189}{2435675}a^{3}+\frac{133901}{487135}a^{2}-\frac{48462}{487135}a+\frac{37568}{97427}$, $\frac{13479}{2435675}a^{15}-\frac{396}{44285}a^{14}+\frac{18439}{2435675}a^{13}+\frac{988}{44285}a^{12}-\frac{22658}{143275}a^{11}+\frac{328}{2605}a^{10}+\frac{274503}{2435675}a^{9}-\frac{17789}{44285}a^{8}+\frac{241068}{143275}a^{7}-\frac{712}{521}a^{6}+\frac{6583194}{2435675}a^{5}-\frac{37132}{44285}a^{4}+\frac{2753656}{2435675}a^{3}+\frac{24028}{44285}a^{2}-\frac{115873}{487135}a+\frac{1910}{8857}$, $\frac{22291}{2435675}a^{15}-\frac{2548}{487135}a^{14}-\frac{49084}{2435675}a^{13}+\frac{18077}{487135}a^{12}-\frac{20442}{143275}a^{11}-\frac{221}{5731}a^{10}+\frac{1088332}{2435675}a^{9}-\frac{141121}{487135}a^{8}+\frac{184727}{143275}a^{7}+\frac{7317}{28655}a^{6}+\frac{3286586}{2435675}a^{5}+\frac{355662}{487135}a^{4}-\frac{22296}{2435675}a^{3}+\frac{638966}{487135}a^{2}-\frac{308497}{487135}a-\frac{38079}{97427}$ Copy content Toggle raw display (assuming GRH)
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 4648.44617215 \) (assuming GRH)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 4648.44617215 \cdot 1}{4\cdot\sqrt{109951162777600000000}}\cr\approx \mathstrut & 0.269207310065 \end{aligned}\] (assuming GRH)

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^14 - 4*x^12 + 32*x^10 + 99*x^8 + 96*x^6 + 44*x^4 - 20*x^2 + 25) 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))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^16 - 4*x^14 - 4*x^12 + 32*x^10 + 99*x^8 + 96*x^6 + 44*x^4 - 20*x^2 + 25, 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)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^14 - 4*x^12 + 32*x^10 + 99*x^8 + 96*x^6 + 44*x^4 - 20*x^2 + 25); 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)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 4*x^14 - 4*x^12 + 32*x^10 + 99*x^8 + 96*x^6 + 44*x^4 - 20*x^2 + 25); 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

$D_8$ (as 16T13):

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A solvable group of order 16
The 7 conjugacy class representatives for $D_{8}$
Character table for $D_{8}$

Intermediate fields

\(\Q(\sqrt{-5}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-1}) \), \(\Q(i, \sqrt{5})\), 4.2.1600.1 x2, 4.0.1280.1 x2, 8.0.40960000.2, 8.2.2621440000.2 x4, 8.0.2097152000.4 x4

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(b)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 8 siblings: 8.2.2621440000.2, 8.0.2097152000.4
Minimal sibling: 8.0.2097152000.4

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.8.0.1}{8} }^{2}$ R ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.2.0.1}{2} }^{8}$ ${\href{/padicField/13.2.0.1}{2} }^{8}$ ${\href{/padicField/17.2.0.1}{2} }^{8}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.8.0.1}{8} }^{2}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.2.0.1}{2} }^{8}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }^{2}$ ${\href{/padicField/53.2.0.1}{2} }^{8}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_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 2.2.8.48c1.120$x^{16} + 8 x^{15} + 44 x^{14} + 168 x^{13} + 494 x^{12} + 1144 x^{11} + 2156 x^{10} + 3344 x^{9} + 4325 x^{8} + 4672 x^{7} + 4224 x^{6} + 3168 x^{5} + 1956 x^{4} + 968 x^{3} + 384 x^{2} + 112 x + 29$$8$$2$$48$$D_{8}$$$[2, 3, 4]^{2}$$
\(5\) Copy content Toggle raw display 5.2.2.2a1.2$x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
5.2.2.2a1.2$x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
5.2.2.2a1.2$x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
5.2.2.2a1.2$x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)